2.1. Outer expansion away from $x=x_0$

Consider the KdV equation (1.1) together with an initial condition $u_0$ with the property (1.3), where $x_0$ (with $\mathrm{Im}(x_0)=y_0 \gt 0$) is one of a complex–conjugate pair. Assuming that $u_0$ is real for $x\in\mathbb R$, then a consequence of this problem formulation is that we need to consider the solution only on the real line and the upper half plane, with the understanding that the behaviour of the singularities in the lower half plane will be an appropriate reflection about the real- $x$ axis.

To begin, we consider a straightforward power series expansion in time, the first two terms of which give

(2.1)\begin{equation} u\sim u_0(x)+tu_1(x) \quad\mbox{as}\quad t\rightarrow 0^+. \end{equation}

By substituting into (1.1), we find

(2.2)\begin{equation} u_1=-6u_0u_0'-u_0^{\prime\prime\prime}, \end{equation}

where the primes (here and throughout the document) denote differentiation with respect to $x$. Therefore, we have the local behaviour

(2.3)\begin{equation} u_0\sim \frac{A_0}{(x-x_0)^2}, \quad u_1\sim \frac{12A_0(A_0+2)}{(x-x_0)^5} \quad\mbox{as}\quad x\rightarrow x_0, \end{equation}

noting that $u_1$ is three orders more singular than $u_0$ at $x=x_0$ (due to the third derivative in (2.2)). Thus, we see immediately that the outer expansion (2.1) applies along the real axis and in parts of the upper half complex plane away from $x=x_0$, but breaks down in regions where $u_0=\mathcal{O}(tu_1)$, or, in other words, where

\begin{equation*} x-x_0=\mathcal{O}(t^{1/3}). \end{equation*}

This reasoning suggests there will be an inner problem near $x=x_0$, which we consider in Subsection 3.1. (Note that the power-series expansion will also break down in a sector in the upper half plane bounded by anti-Stokes lines, as discussed below in Subsections 2.2 and 3.4.)

2.2. Exponential asymptotics argument for limit $t\rightarrow 0^+$

The terms $u_0$ and $u_1$ in (2.1) are the first two in a divergent asymptotic expansion of the form

(2.4)\begin{equation} u\sim \sum_{n=0}^\infty t^n u_n(x) \quad\mbox{as}\quad t\rightarrow 0^+, \end{equation}

whose divergence is caused by the singularities of the leading-order term $u_0(x)$ in the complex- $x$ plane. As such, the $u_n$ will be of the familiar factorial-over-power form for large $n$, due to repeatedly differentiating (three times) to obtain the next-order term. Thus, we expect there to be an exponential term in our asymptotic representation for $u$ in the limit $t\rightarrow 0^+$ that appears ‘beyond all orders’ of the original power series (2.4) [Reference Chapman, King and Adams20, Reference Olde Daalhuis, Chapman, King, Ockendon and Tew79]. This term will ‘switch on’ across Stokes lines in the $x$ plane and, in particular, will affect the solution on the real line by providing an exponentially small correction term to (2.4), which we call $U_{\textrm{dis}}$.

Importantly, the emergence of dispersive waves that travel in the negative- $x$ direction can never be described by a power series (2.4), as the amplitude of these waves turns out to be exponentially small in time compared to each term $t^n u_n$. Thus, in order to approximate the dispersive wavetrain in the limit $t\rightarrow 0^+$, we must consider this exponential contribution and observe where it switches on across the real- $x$ axis. To demonstrate how this works, we follow the framework in [Reference Chapman, King and Adams20], summarize the main results here and provide further details in Appendix A (see [Reference Chapman and Vanden-Broeck21, Reference Lustri and Chapman66–Reference Lustri, McCue and Chapman68, Reference Trinh, Chapman and Vanden-Broeck91], for example, for other studies of wave motion using this framework of exponential asymptotics).

A crucial step is to analyse the late terms in (2.4), which satisfy

(2.5)\begin{equation} (n+1)u_{n+1}=-6\sum_{j=0}^n u_j u'_{n-j}-u_n^{\prime\prime\prime}. \end{equation}

Due to the factorial-over-power divergence of (2.4), following Dingle [Reference Dingle37] we apply the ansatz

(2.6)\begin{equation} u_n\sim \frac{\mathcal{A}(x)\Gamma(2n+\gamma)}{\chi(x)^{2n+\gamma}} \quad \mbox{as}\quad n\rightarrow\infty, \end{equation}

which, after substituting into (2.5), leads to

(2.7)\begin{equation} \frac{1}{2}\chi=\left(\chi'\right)^3, \quad \frac{\gamma}{2}\mathcal{A} =3\mathcal{A}\chi'\chi'' +3\mathcal{A}'\left(\chi'\right)^2. \end{equation}

Here $\chi$ is the so-called singulant, which must vanish at singularities of $u_0$, and it suffices to consider the location $x=x_0$ (the full expansion of $u_n$ for large $n$ comprises a sum of terms of the form (2.6) associated with each singularity $x_0$). Thus, given $x_0$ is assumed to lie in the upper half plane, we explain in Appendix A.1 that the appropriate solutions are

(2.8)\begin{equation} \chi=-\frac{2}{3^{3/2}}(x-x_0)^{3/2}, \quad \mathcal{A}=\Lambda(x-x_0)^{(\gamma-1)/2}, \end{equation}

where $\Lambda$ is a constant, and therefore

(2.9)\begin{equation} u_n\sim \frac{\Lambda(-3^{3/2}/2)^{2n+\gamma} \Gamma(2n+\gamma)} {(x-x_0)^{3n+\gamma+1/2}} \quad \mbox{as}\quad n\rightarrow\infty. \end{equation}

For the initial conditions we are concerned with in this study, we have (1.3). To be consistent with this local behaviour near $x=x_0$, we must choose $\gamma=3/2$.

With $\chi$ and $\mathcal{A}$ determined, we show in Appendix A.2 that a consequence of this type of late-order behaviour for $u_n$ is that the exponentially small quantity

(2.10)\begin{equation} \pi\mathrm{i}\,\mathcal{A}\, t^{-\gamma/2}\mathrm{e}^{-\chi/t^{1/2}} =\pi\mathrm{i}\,\Lambda(x-x_0)^{1/4}t^{-3/4} \mathrm{e}^{2(x-x_0)^{3/2}/3(3t)^{1/2}} \end{equation}

switches on as we cross the Stokes line $\mathrm{arg}(x-x_0)=-2\pi/3$ from right to left (given that $x_0$ is in the upper half $x$ plane). This Stokes line is found by setting the singulant term $\chi$ to be real and positive. There will be another term like (2.10) switched on across a Stokes line born in the lower half plane, and together the sum of these provides the asymptotic behaviour of the dispersive waves on the real line in the small-time limit.

To determine the constant $\Lambda$ in (2.10), we need to match into an inner region near $x=x_0$, which we study below in Subsection 3.1. The details for this matching are provided in Appendix A.3, with the key result that

(2.11)\begin{equation} \Lambda= \frac{\mathrm{i}}{3^{3/4}\pi^{3/2}} \cos\left(\frac{\pi}{2}\sqrt{1-4A_0}\right). \end{equation}

A striking property of $\Lambda$ is that it vanishes for $A_0=-N(N+1)$, where $N\in \mathbb{N}$. Therefore, this leading-order result predicts that a train of dispersive waves is associated with each complex conjugate pair of singularities of the initial condition $u_0$, with the amplitude vanishing for $A_0=-N(N+1)$. This prediction is consistent with the $\mathrm{sech}^2$-type initial condition (1.4), for which it is known that there are no dispersive waves when $A_0$ takes on these triangular values (these are the $N$-soliton solutions, having flat tails). (More generally, if $A_0=-N(N+1)$ for an initial condition that is not of the special form (1.4), then parts of the required asymptotics to describe the dispersive waves will be different; we include a discussion on such cases in Appendix A.4.)

To take an example, consider an initial condition that has a singularity lying on the imaginary axis, which we write as $x_0=\mathrm{i}y_0$. Our analysis predicts that the term (2.10) switches on across the Stokes line $\mathrm{arg}(x-\mathrm{i}y_0)=-2\pi/3$ in the upper half $x$ plane. We would also observe an analogous term that switches on across $\mathrm{arg}(x+\mathrm{i}y_0)=2\pi/3$ in the lower half plane. These Stokes lines both hit the real axis at $x=-y_0/\sqrt{3}$. Thus, putting it together, we find the exponential contribution to our asymptotic solution on the real- $x$ axis is of the form

(2.12)\begin{align} U_{\textrm{dis}}\sim & \,\,\pi\mathrm{i}\Lambda t^{-3/4}\left( (x-\mathrm{i}y_0)^{1/4} \mathrm{e}^{2(x-\mathrm{i}y_0)^{3/2}/3(3t)^{1/2}} + (x+\mathrm{i}y_0)^{1/4} \mathrm{e}^{2(x+\mathrm{i}y_0)^{3/2}/3(3t)^{1/2}} \right) \nonumber \\ = & -(2/3^{3/4}\pi^{1/2}) \cos\left( \frac{\pi}{2}\sqrt{1-4A_0} \right) t^{-3/4} (x^2+y_0^2)^{1/8} \mathrm{e}^{ 2(x^2+y_0^2)^{3/4}\cos (3\phi/2)/(3(3t)^{1/2}) } \nonumber \\ & \times \cos\left( \frac{2}{3(3t)^{1/2}} (x^2+y_0^2)^{3/4} \sin {\textstyle\frac{3}{2}}\phi +{\textstyle\frac{1}{4}}\phi \right), \end{align}

$x \lt -y_0/\sqrt{3}$, $t\rightarrow 0^+$, where $\phi=\arctan(y_0/x)$ (note $\cos (3\phi/2) \lt 0$ for this interval in $x$). For large negative $x$, noting that

\begin{equation*} (x-\mathrm{i}y_0)^{3/2}\sim \mathrm{i}(-x)^{3/2}-{\textstyle\frac{3}{2}}y_0(-x)^{1/2}+ \mathcal{O}((-x)^{-1/2}), \end{equation*}

these waves take the form (1.5). Thus, we see dispersive waves whose amplitude is exponentially small in both limits $x\rightarrow -\infty$ and $t\rightarrow 0^+$ (compared to the initial condition $u_0(x)$, which is assumed to be $\mathcal{O}(1)$), scaling as

\begin{equation*} \mbox{const}\,(-x)^{1/4}t^{-3/4}\mathrm{e}^{-y_0(-x/t)^{1/2}}. \end{equation*}

Defining the locations of the crests of the dispersive waves to be $x_m$, with $m$ increasing from right to left, for large $m$, these propagate to the left as

(2.13)\begin{equation} x_m\sim -3\pi^{2/3}m^{2/3}t^{1/3}, \quad m\rightarrow\infty, \quad t\rightarrow 0^+, \end{equation}

so that the speed of the waves decreases algebraically as $\dot{x}_m\sim -\pi^{2/3}(t/m)^{-2/3}$ and the wavelength increases as $-x_{m+1}+x_m\sim 2\pi^{2/3}(t/m)^{1/3}$ as $t$ increases from zero. These scalings explain why it is so difficult to compute numerical solutions of the KdV model (1.1) accurately for small time using elementary techniques (such as finite differences), given the challenge of capturing fast-moving, small-amplitude waves that would immediately reflect off the boundary of any truncated domain. Such issues can be resolved, for example, via an inverse-scattering formulation, but such an approach is highly nontrivial [Reference Trogdon, Olver and Deconinck92] (and, of course, limited to integrable systems); we instead use the spin (stiff pde integrator) command [Reference Montanelli and Bootland74] in Chebfun [Reference Driscoll, Hale and Trefethen39], which is good enough for our real-valued numerical solutions.

2.3. Numerical tests for the exponential asymptotics

To test (2.12) against numerics, we consider the small-time expansion

\begin{equation*} u\sim u_0+tu_1+U_{\textrm{dis}} \quad\mbox{as}\quad t\rightarrow 0^+, \end{equation*}

i.e., we retain only the first two terms of the algebraic expansion (2.1) but include the leading-order exponential correction (2.12) (in order to capture the dominant oscillatory contribution). For the initial condition (1.4), we have

\begin{equation*} u_1=4A_0(3(A_0+2)-2\,\mathrm{cosh}^2x)\,\mathrm{sinh}\,x \,\mathrm{sech}^5x \end{equation*}

and $y_0=\pi/2$, while for (1.6) we have

\begin{equation*} u_1=\frac{96A_0x(-5x^2+4A_0+3)}{(1+x^2)^5} \end{equation*}

and $y_0=1$. We also make use of a scaled version of (2.12), namely $U_{\textrm{scaled}}=U_{\textrm{dis}}/K$, where

(2.14)\begin{equation} K=t^{-3/4} (x^2+y_0^2)^{1/8} \mathrm{e}^{ 2(x^2+y_0^2)^{3/4}\cos (3\phi/2)/(3(3t)^{1/2})} \end{equation}

is chosen so that $U_{\textrm{scaled}}$ does not decay as $x\rightarrow -\infty$, in the comparison with the numerics.

For example, in Figure 2(a), we show a numerical solution of (1.1), $u_{\mathrm{num}}$, computed with the $\mathrm{sech}^2$-type initial condition (1.4), with $A_0=-1/4$, for a representative early time, $t=0.02$. The very small dispersive waves we see in the inset on the left panel are propagating to the left. In the middle panel, we plot $u_{\mathrm{num}}-(u_0+tu_1)$ at the specific time $t=0.02$, and compare with $U_{\textrm{dis}}$ from (2.12). We see that for these parameter values, the comparison is very good, which is a strong test for both our numerics and asymptotics. As another test, we plot in the right panel the scaled versions $u_{\mathrm{num}}-(u_0+tu_1)/K$ and $U_{\textrm{scaled}}=U_{\textrm{dis}}/K$, where $K$ is defined in (2.14). The agreement is excellent.

Figure 2. (a) [left panel] Numerical solution of the KdV model (1.1) on the real line, computed at $t=0.02$ using the $\mathrm{sech}^2$-type initial condition (1.4) with $A_0=-1/4$; [middle panel] plots of $u_{\mathrm{num}}-(u_0+tu_1)$ (blue) and $U_{\textrm{dis}}$ (red circles) versus $x$ using same parameters as in left panel; and [right panel] plots of $\left(u_{\mathrm{num}}-(u_0+tu_1)\right)/K$ (blue) and $U_{\textrm{scaled}}$, again with same parameters as left panel. (b) Same as (a), except that $A_0=-3/4$.

We have generated a number of other examples like that presented in Figure 2(a) using the initial condition (1.4), computed for other values of $A_0 \lt 0$, and the agreement between numerics and asymptotics is also excellent. One such example is for $A_0=-3/4$, as shown in Figure 2(b). This sweep of parameters includes the special cases $A_0=-N(N+1)$, where $N\in\mathbb N$, for which we have exact $N$-soliton solutions with no dispersive waves at all. For those special choices, the amplitude of the waves is predicted by (2.12) to vanish, since

\begin{equation*} \cos\left(\frac{\pi}{2}\sqrt{1-4A_0}\right)= \cos\left(\frac{\pi}{2}\sqrt{(2N+1)^2}\right)= 0 \end{equation*}

for all $N\in\mathbb N$. Putting it together, we have presented in Figure 3(a) a comparison of our numerical estimate of the amplitude of the wavelike term $\left(u_{\mathrm{num}}-(u_0+tu_1)\right)/K$ (blue dots) with the asymptotic prediction $(2/3^{3/4}\pi^{1/2}) \cos\left(\frac{\pi}{2}\sqrt{1-4A_0}\right)$ (red solid) for simulations that take the $\mathrm{sech}^2$-type initial condition (1.4). Given the very good agreement in this plot over a range of values of $A_0$, we are confident the approximation (2.12) is correctly describing the dispersive waves, at least for the initial condition (1.4), in the small-time limit.

Figure 3. A numerically computed amplitude of the scaled dispersive waves $\left(u_{\mathrm{num}}-(u_0+tu_1)\right)/K$ (blue dots) plotted for various values of the parameter $A_0$ at a fixed time $t=0.03$, compared with the asymptotic prediction $(2/3^{3/2}\pi^{1/2})\cos\left( \frac{\pi}{2}\sqrt{1-4A_0}\right)$ (red solid). (a) The $\mathrm{sech}^2$-type initial condition (1.4); (b) the initial condition (1.6); and (c) the refined initial condition (2.16).

These types of numerical tests turn out to be slightly more subtle for the initial condition (1.6). For example, in Figure 4, we show plots that are analogous to Figure 2, except that in Figure 4, we use the initial condition (1.6). Broadly speaking, we see very good agreement between numerics and asymptotics for $A_0=-1/4$, but the agreement for $A_0=-3/4$ is not quite as good. Further, for the case $A_0=-2$, there is no agreement at all, since our asymptotic description (2.12) predicts that the dispersive waves should not appear for the precise value $A_0=-2$, whereas the numerical solutions clearly have waves (albeit of very small amplitude). Of course, since (1.6) is not of the special class of initial conditions that give rise to $N$-soliton solutions, we know that with (1.6) there cannot be solutions without dispersive waves. The explanation for the discrepancy is that the exponential asymptotics that led to (2.12) is only a first approximation. As we vary the parameter $A_0$ so that it takes values closer and closer to the triangular numbers $-N(N+1)$, higher-order correction terms will inevitably come into play and eventually dominate.

Figure 4. (a) [left panel] Numerical solution of the KdV model (1.1) on the real line, computed at $t=0.02$ using the generic initial condition (1.6) with $A_0=-1/4$; [middle panel] plots of $u_{\mathrm{num}}-(u_0+tu_1)$ (blue) and $U_{\textrm{dis}}$ (red circles) versus $x$ using same parameters as in left panel; and [right panel] plots of $\left(u_{\mathrm{num}}-(u_0+tu_1)\right)/K$ (blue) and $U_{\textrm{scaled}}$, again with same parameters as left panel. (b) Same as (a), except that $A_0=-3/4$. (c) Same as (a), except that $A_0=-2$.

Indeed, extending the local behaviour (1.3) of the initial condition about its singularity to be

(2.15)\begin{equation} u_0\sim \frac{A_0}{(x-x_0)^2}+\frac{A_1}{x-x_0}+A_2+A_3(x-x_0)+\ldots \quad\mbox{as}\quad x\rightarrow x_0, \end{equation}

the full asymptotics for the dispersive waves will depend not only on $A_0$, but also (linearly) on $A_1$, $A_2$ and $A_3$, and so on, in an increasingly complicated manner. For our special initial condition (1.4), an expansion about $x_0=\pi\mathrm{i}/2$ shows that $A_1=0$, so the first-order correction terms vanish, while for (1.6), an expansion about $x_0=\mathrm{i}$ shows that $A_1=\mathrm{i}A_0$, so the first-order correction terms in this case do not vanish. To see the effect of the difference between these two cases, we show in Figure 3(b) a plot of the scaled amplitude versus $A_0$ for the initial condition (1.6). The agreement between the asymptotic prediction (in red) and the numerics (blue dots) appears to be good only for small values of $|A_0|$. Compared with Figure 3(a) (for which $A_1=0$), it is clear that higher-order corrections (that depend linearly on $A_1$) are significant for (1.6) (for which $A_1=\mathrm{i}A_0$) as $|A_0|$ increases. To support this conclusion, we test a further initial condition, namely

(2.16)\begin{equation} u_0=-\frac{4A_0}{(1+x^2)^2}+\frac{2A_0}{1+x^2}, \end{equation}

which has the property that

\begin{equation*} u_0\sim \frac{A_0}{(x-\mathrm{i})^2}-\frac{A_0}{4} -\frac{\mathrm{i}A_0}{4}(x-\mathrm{i}) \quad\mbox{as}\quad x\rightarrow\mathrm{i}. \end{equation*}

This is a carefully constructed initial condition that has the same leading-order behaviour near $x_0=\mathrm{i}$ as (1.6), but has an additional term added so that $A_1=0$. Thus, locally near $x_0=\mathrm{i}$, the initial condition (2.16) is acting more like the special case (1.4) near $x_0=\pi\mathrm{i}/2$ (which also has $A_1=0$). We plot in Figure 3(c) the scaled amplitude versus $A_0$ for (2.16), which shows much better agreement between numerics and asymptotics than in Figure 3(b), again reinforcing the conclusion that there are important correction terms that scale with $A_1$. While the treatment of correction terms in the context of our exponential asymptotics is rather complicated, we shall summarize how the analysis works for the special case $A_0=-N(N+1)$ in Appendix A.4.

The plots in Figures 2–4 are generated for a fixed small time. As a simple check of the temporal scaling, we track the first nine dispersive wave crests for a specific example in Figure 5. An image of the wave profile is shown in part (a) and the numerically determined crest location as a function of time is plotted in (b) for the first nine crests. The latter results, reformatted on a log-log plot in (c), support the $t^{1/3}$ scaling. Further, in part (d), we have plotted the asymptotic prediction (2.13) for the first nine crests, $m=1,\ldots,9$. This prediction shows good agreement with the numerics, especially given that it holds formally for large $m$ (and small $t$), whereas these numerical results are for small $m$.

Figure 5. (a) A snapshot of a numerical solution of (1.1) with the initial condition (1.6), plotted for $t=0.3$ with the first nine wave crests indicated by red dots. (b)–(c) Numerically determined crest locations as a function of time (solid red) together with the asymptotic prediction (2.13) (blue dots) for $m=1,\ldots,9$. The slope of the hypotenuse of the triangle in the log-log plot indicates the scaling $|x_m|\sim \,\mathrm{constant}\, t^{1/3}$.

For the initial conditions we are considering here, there is a maximum (a ‘peak’) at $x=0$. This main peak can initially move to either the left or the right, depending on the initial condition and the value $A_0$. For example, we show in Figure 6 the small-time evolution of the main peak for the initial condition (1.6). Note the behaviour of the solution in the neighbourhood of this peak is not driven by the exponentially small terms (2.12), as these are only relevant to the left of $x=-y_0/\sqrt{3}=-1/\sqrt{3}$ (since the singularity is at $x=\mathrm{i}$ in this case) in the limit $t\rightarrow 0^+$. Instead, for very small times, we expect the peak to move according to the first couple of terms in the power series, namely (2.1), which suggests the peak location behaves like $x_{\mathrm{peak}}\sim -6(4A_0+3)t$. That is, we expect the peak to initially move to the left for $A_0 \lt -3/4$ and to the right for $-3/4 \lt A_0 \lt 0$. We can see this behaviour in the left panel of Figure 6, where the borderline case $A_0=-3/4$ is evident. In the right panel, these numerical results on a log-log plot support the asymptotic prediction by approaching the appropriate straight line with slope unity for large negative values of $\ln t$. Thus, an interesting observation is that the drift of the main peak of the solution scales like $t$ in the small time limit, found by tracking the local maximum of the first couple of terms in the power series expansion (2.1), while the crests of the dispersive waves scale like $t^{1/3}$ (via (2.13)), driven by the exponentially small terms that appear beyond all orders of the power series.

Figure 6. (left) The location of the main peak in real solutions of (1.1) (horizontal axis) plotted against time (vertical axis). In black, numerical results are shown for the initial condition (1.6) plotted for $A_0=-0.05$, $-0.15$, $-0.25$, $-0.5$, $-0.625$, $-0.75$, $-1$ and $-2$. (right) Main peak versus time on a log-log plot for the same initial condition, which shows how the numerical results (thick curves) approach the predicted limiting behaviour that comes from analysing the first two terms in the power series (2.1), namely $\ln t \sim \ln |x_{\mathrm{peak}}| - \ln|6(4A_0+3)|$ as $t\rightarrow 0^+$ (thin solid lines). Importantly (and as might be expected), the small-time behaviour of the main peak does not come from the exponential terms that appear beyond all orders of the power series.

In summary, for our KdV problem (1.1), we see that the emergence of dispersive waves for small time can be explained by observing how exponentially small terms are switched on across the point at which the Stokes lines intersect the real- $x$ axis. As far as we are aware, this is the first asymptotic description of how dispersive waves propagate for the KdV equation in the limit that $t\rightarrow 0^+$. While these results are interesting in their own right, we also use the above information about the Stokes line structure to inform how we apply far-field conditions in the inner problem in Subsection 3.1. Further, our analysis here locates an anti-Stokes line at $\mathrm{arg}(x-x_0)=-\pi$, which is (near) where we expect there to be complex-plane singularities propagating out from each singularity of the initial condition, $x=x_0$. One of our goals for the remainder of this paper is to track the initial dynamics of the singularities born at $x=x_0$, including the important string of singularities that align (close to) $\mathrm{arg}(x-x_0)=-\pi$.

3.1. Inner region near $x=x_0$

Following on from subsection 2.1, there is an inner problem as $t\rightarrow 0^+$, which holds for

(3.1)\begin{equation} \xi\equiv\frac{x-x_0}{(3t)^{1/3}}=\mathcal{O}(1). \end{equation}

Using this new similarity-type variable, we can rewrite (2.3) as

(3.2)\begin{equation} u_0\sim \frac{A_0}{(3t)^{2/3}\xi^2}, \quad u_1\sim \frac{12A_0(A_0+2)}{(3t)^{5/3}\xi^5}, \end{equation}

which suggests the inner problem has

(3.3)\begin{equation} u=\frac{1}{(3t)^{2/3}}f(\xi,t). \end{equation}

By substituting (3.3) into (1.1), we find that $f$ satisfies the pde

(3.4)\begin{equation} 3tf_t-2f-\xi f_\xi+6ff_\xi+f_{\xi\xi\xi}=0. \end{equation}

To leading order, we write $f\sim f_0(\xi)$, so that (3.4) and (3.2) combine to give our inner problem

(3.5)\begin{equation} -2f_0-\xi \frac{\mathrm{d}f_0}{\mathrm{d}\xi} +6f_0\frac{\mathrm{d}f_0}{\mathrm{d}\xi}+\frac{\mathrm{d}^3f_0}{\mathrm{d}\xi^3}=0, \end{equation}

(3.6)\begin{equation} f_0\sim \frac{A_0}{\xi^2}+\frac{4A_0(A_0+2)}{\xi^5} \quad\mbox{as}\quad \xi\rightarrow -\mathrm{i}\infty. \end{equation}

Note that the direction of the far-field condition (3.6) comes from matching back down towards the real axis. We expect (3.6) to apply in a broader sector of the $\xi$ plane, as determined by our subsequent analysis. Further, the extent to which (3.6) may need to be supported by other far-field conditions will be explored later.

We comment here that the similarity reduction (3.3) and the corresponding differential equations (3.4) and (3.5) are also used to derive the large-time asymptotics for the KdV problem (1.1) [Reference Ablowitz and Segur3]. In that case, the similarity variable is $\xi\equiv x/(3t)^{1/3}$ (instead of (3.1)), and the appropriate solution of (3.5) is valid on the real line for $x=\mathcal{O}(t^{1/3})$ as $t\rightarrow\infty$. The relevant boundary conditions for (3.5) come from matching to an outer region as $\xi\rightarrow +\infty$. All of these variables for the large-time asymptotics are real-valued. In contrast, our study of (3.3)–(3.5) is relevant for the region of the complex plane $x-x_0=\mathcal{O}(t^{1/3})$ as $t\rightarrow 0^+$. The boundary condition (3.6) is completely different from the matching condition used for the large-time asymptotics. The variables $f$ and $\xi$ in our study of (3.3)–(3.5) are complex-valued.

The third-order nonlinear ordinary differential equation (ode) problem (3.5)–(3.6) is difficult to analyse in this form, but much progress can be made by converting (3.5) to P $_{\mathrm{II}}$ in the usual way [Reference Joshi58] (see the following subsection). Ultimately, one of the goals of this exercise is to determine the singularities of $f_0(\xi)$ in the $\xi$ plane. Our analysis then predicts that for any given singularity of $f_0$, which we call $\xi_0$, there is a double pole of $u(x,t)$ at $x=s(t)$ that emerges from $x_0$ like

(3.7)\begin{equation} s(t)\sim x_0+(3t)^{1/3}\xi_0 \quad\mbox{as}\quad t\rightarrow 0^+. \end{equation}

Before we proceed, there is a simple exact solution of (3.5)–(3.6), namely

(3.8)\begin{equation} f_0=-\frac{2}{\xi^2}, \quad A_0=-2. \end{equation}

The analysis in the present section is devoted to $A_0\neq -2$, while the special case $A_0=-2$ (for which there is this trivial exact solution for $f_0$) is treated separately in Appendix B.4.

3.2. Reduction to Painlevé II

Applying the standard (Miura-transformation) reduction [Reference Boiti and Pempinelli12, Reference Rosales81]

(3.9)\begin{equation} f_0=\frac{\mathrm{d}F}{\mathrm{d}\xi}-F^2, \end{equation}

we find that (3.5) is transformed to

(3.10)\begin{equation} \frac{\mathrm{d}^2}{\mathrm{d}\xi^2} \left(\frac{\mathrm{d}^2F}{\mathrm{d}\xi^2}-\xi F-2F^3\right)-2F\frac{\mathrm{d}}{\mathrm{d}\xi} \left(\frac{\mathrm{d}^2F}{\mathrm{d}\xi^2}-\xi F-2F^3\right)=0. \end{equation}

Before we consider (3.10) further, note that given the far-field condition (3.6) and the change of variable (3.9), we must have

(3.11)\begin{equation} F\sim \frac{\alpha}{\xi} +\frac{2\alpha(1-\alpha^2)}{\xi^4} \quad\mbox{as}\quad \xi\rightarrow -\mathrm{i}\infty, \end{equation}

where

(3.12)\begin{equation} \alpha={\textstyle\frac{1}{2}}(-1\pm\sqrt{1-4A_0}). \end{equation}

This observation is crucial for what follows.

Returning to (3.10), clearly one option is

(3.13)\begin{equation} \frac{\mathrm{d}^2F}{\mathrm{d}\xi^2}-\xi F-2F^3=\,\mbox{constant}, \end{equation}

which is P $_{\mathrm{II}}$. To determine the constant, we apply the far-field condition (3.11), which implies it is $-\alpha$. We are free to proceed with either sign in (3.12); each choice will involve different solutions for $F$ but the same result for $f_0$ via (3.9). We choose the plus sign. The other option, namely $F''-\xi F-2F^3\neq\,\mbox{constant}$, is not applicable as it is inconsistent with (3.11).

In summary, we conclude that (3.5)–(3.6) reduces to

(3.14)\begin{equation} \frac{\mathrm{d}^2F}{\mathrm{d}\xi^2}=2F^3+\xi F-\alpha, \end{equation}

(3.15)\begin{equation} F\sim \frac{\alpha}{\xi}+\frac{2\alpha(1-\alpha^2)}{\xi^4} \quad\mbox{as}\quad \xi\rightarrow -\mathrm{i}\infty, \end{equation}

where

(3.16)\begin{equation} \alpha={\textstyle\frac{1}{2}}(-1+\sqrt{1-4A_0}). \end{equation}

As mentioned above, (3.14) is P $_{\mathrm{II}}$ with the constant $\alpha$ related to our key parameter $A_0$ (sometimes (3.14) is referred to as the inhomogeneous P $_{\mathrm{II}}$ equation, with the homogenous version arising from $\alpha=0$). Motivated by the examples discussed in the Introduction, we are mostly focused on real and negative values of $A_0$, which correspond to $\alpha \gt 0$. For example, the $N$-soliton solutions with $A_0=-N(N+1)$ correspond to $\alpha=N$ (via the rational solutions summarized in Subsection 3.5). For $0\leq A_0 \leq 1/4$, we have $-1/2\leq \alpha\leq 0$, and for $A_0 \gt 1/4$, we have complex values of $\alpha$. These parameter intervals are also of some interest, although we shall restrict our exploration to real values of $\alpha$. Finally, without further investigation (or a background in studying P $_{\mathrm{II}}$), it is not obvious whether or not (3.15) is sufficient to determine our solution uniquely, which is an issue we explore in Subsection 3.4.

3.3. Reduction to Painlevé 34

In our work, we have applied the Miura-transformation (3.9) to (3.5)–(3.6) so that our inner problem is described by the P $_{\mathrm{II}}$ problem (3.14)–(3.16). Alternatively, we can set

(3.17)\begin{equation} f_0=g_0+{\textstyle\frac{1}{2}}\xi, \end{equation}

so that

\begin{equation*} g_0+2\xi \frac{\mathrm{d}g_0}{\mathrm{d}\xi} +6g_0\frac{\mathrm{d}g_0}{\mathrm{d}\xi}+\frac{\mathrm{d}^3g_0}{\mathrm{d}\xi^3}=0. \end{equation*}

Multiplying by $g_0$ and integrating, we find

(3.18)\begin{equation} g_0\frac{\mathrm{d}^2g_0}{\mathrm{d}\xi^2}=\frac{1}{2}\left( \frac{\mathrm{d}g_0}{\mathrm{d}\xi}\right)^2 -2g_0^3-\xi g_0^2+\frac{1}{2}A_0-\frac{1}{8}, \end{equation}

where the constant has been determined by enforcing the far-field condition (3.6). Equation (3.18) is known as Painlevé 34 (P $_{34}$), since an equivalent version was labelled XXXIV in section 14.33 of Ince [Reference Ince56].

Clearly, given a solution $F$ of (3.14)–(3.16), we can recover a solution of P $_{34}$ by combining (3.9) and (3.17) to give $g_0=\mathrm{d}F/\mathrm{d}\xi-F^2-\xi/2$. Therefore, all of our results for $F$ could be framed in terms of $g_0$. Note the one-to-one correspondence between solutions of P $_{\mathrm{II}}$ and P $_{34}$ was given in Ref. [Reference Fokas and Ablowitz42], for example. Equivalent links between P $_{\mathrm{II}}$ and P $_{34}$ can be established via the associate Hamiltonian, as described in section 32.6(iii) in the DLMF [77] (equation 32.6.12 is equivalent to our P $_{34}$ in (3.18)). See section 2.4 of Clarkson [Reference Clarkson24] for further details.

3.4. Liouville–Green (WKB) analysis and Stokes phenomenon

For simplicity, we assume for the moment that the initial condition $u_0(x)$ has a single double pole in the upper half plane at $x=x_0$ (like (1.6) does). We wish to study (3.14) subject to (3.15). The solution we are after will satisfy this far-field condition in a broader sector that includes the negative imaginary direction. To determine this sector, and also to determine whether (3.15) is sufficient to specify our solution uniquely, we shall linearize about (3.15).

To proceed with this linearization, we write

\begin{equation*} F= \frac{\alpha}{\xi}+\frac{2\alpha(1-\alpha^2)}{\xi^4}+\ldots+J, \end{equation*}

where here the ellipsis denotes higher terms in the algebraic expansion and $J$ represents the exponential correction. Linearizing about the algebraic series, to leading order, we find that

\begin{equation*} \frac{\mathrm{d}^2J}{\mathrm{d}\xi^2}=\left(\xi+\frac{6\alpha^2}{\xi^2}\right) J, \end{equation*}

which is related to the modified Bessel’s equation via a change of variables. The two possible linearly independent solutions, namely

\begin{equation*} \xi^{1/2}I_{(1+24\alpha^2)^{1/2}/3}({\textstyle\frac{2}{3}}\xi^{3/2}) \quad\mbox{and}\quad \xi^{1/2}K_{(1+24\alpha^2)^{1/2}/3}({\textstyle\frac{2}{3}}\xi^{3/2}), \end{equation*}

where $I_{\nu}(z)$ and $K_{\nu}(z)$ are modified Bessel functions of the first and second kind, have the far-field behaviour $J\sim \mathrm{const}\,\xi^{-1/4}\,\mathrm{e}^{\pm 2\xi^{3/2}/3}$. Therefore we have

\begin{equation*} F\sim \frac{\alpha}{\xi}+\frac{2\alpha(1-\alpha^2)}{\xi^4}+\ldots +\frac{\sigma_1}{\xi^{1/4}}\,\mathrm{e}^{2\xi^{3/2}/3} +\frac{\sigma_2}{\xi^{1/4}}\,\mathrm{e}^{-2\xi^{3/2}/3} \quad\mbox{as}\quad \xi\rightarrow -\mathrm{i}\infty, \end{equation*}

where $\sigma_1$ and $\sigma_2$ are important (Stokes) constants. However, since $\mathrm{e}^{-2\xi^{3/2}/3}$ grows exponentially as $\xi\rightarrow -\mathrm{i}\infty$, we must set $\sigma_2=0$ in this asymptotic expression, leaving

(3.19)\begin{equation} F\sim \frac{\alpha}{\xi}+\frac{2\alpha(1-\alpha^2)}{\xi^4}+\ldots+\frac{\sigma_1}{\xi^{1/4}}\,\mathrm{e}^{2\xi^{3/2}/3} \quad\mbox{as}\quad \xi\rightarrow -\mathrm{i}\infty. \end{equation}

This exercise demonstrates that (3.15) is acting as one boundary condition only, and to specify the solution of (3.14)–(3.15) uniquely, we need to fix $\sigma_1$ in (3.19), which is hidden beyond all orders of the algebraic expansion.

At this stage, we emphasize that (3.14) with (3.19) combine to give a one-parameter family of solutions of the type studied at length by Boutroux [Reference Boutroux16] and by others, including in Refs [Reference Fokas, Its, Kapaev and Novokshenov43, Reference Fornberg and Weideman45, Reference Its and Kapaev57, Reference Joshi and Kruskal59, Reference Joshi and Kruskal60]. We note that we are able to write out the full algebraic series in (3.19) as

(3.20)\begin{equation} F\sim \frac{\alpha}{\xi}\sum_{n=0}^\infty \frac{b_n}{\xi^{3n}} \quad\mbox{as}\quad \xi\rightarrow -\mathrm{i}\infty, \end{equation}

where the $b_n$ satisfy the recurrence relations [Reference Its and Kapaev57]

(3.21)\begin{equation} b_{n+1}=(3n+1)(3n+2)b_n-2\alpha^2\sum_{\substack{i,j,k=0\\i+j+k=n}}^{n} b_ib_jb_k,\quad b_0=1. \end{equation}

Another way to write these coefficients is via $b_n=(-2)^n(\alpha^2-1)p_n(\alpha)$, where $p_n$ is a polynomial of order $2n-2$ of the form

\begin{equation*} p_n=a^{(n)}_1 \alpha^{2n-2} - a^{(n)}_2 \alpha^{2n-4} + a^{(n)}_3 \alpha^{2n-6} - a^{(n)}_4 \alpha^{2n-8}+\ldots+(-1)^n a^{(n)}_{n-1}\alpha^2+(-1)^{n+1} a^{(n)}_{n}. \end{equation*}

As examples, the first and last coefficients $a^{(n)}_1$ and $a^{(n)}_{n}$ are A001764 and A025035 in the online encyclopedia of integer sequences [78], respectively, with large $n$ behaviour

\begin{equation*} a^{(n)}_1=\frac{(3n)!}{(2n+1)(2n)!n!}\sim \frac{3^{n+1/2}}{4{\pi}^{1/2}n^{3/2}} \left(\frac{3}{2}\right)^{2n}, \quad a^{(n)}_{n}=\frac{(3n)!}{6^n n!}\sim \frac{\sqrt{3}}{2^n}\left(\frac{3n}{\mathrm{e}}\right)^{2n}. \end{equation*}

Its & Kapaev [Reference Its and Kapaev57] show that the coefficients $b_n$ have the large $n$ behaviour

(3.22)\begin{equation} b_n \sim \frac{\sin\pi\alpha}{\alpha\pi^{3/2}}\left(\frac{3}{2}\right)^{2n+1/2} \Gamma\left(2n+{\textstyle\frac{1}{2}}\right) \quad\mbox{as}\quad n\rightarrow\infty \end{equation}

(see also [Reference Fokas, Its, Kapaev and Novokshenov43]). A key point is that for $\alpha\in \mathbb{N}$, the series (3.20) is convergent (see Subsection 3.5 below), while otherwise it is divergent.

To proceed with our argument about the one-parameter family of solutions to (3.14) with (3.19), we explore broader far-field sectors in the $\xi$ plane, referring to Figure 7. With $\xi=\rho\mathrm{e}^{\mathrm{i}\theta}$, then let the direction $\xi\rightarrow -\mathrm{i}\infty$ be $\theta=-\pi/2$. For any $\sigma_1\neq 0$, then we would have $\sigma_1\xi^{-1/4}\mathrm{e}^{2\xi^{3/2}/3}$ decaying along the ray $\theta=-\pi/2$. Along a path $\rho=\,\mathrm{const}$ (with $\rho\gg 1$), as $\theta$ increases from $\theta=-\pi/2$, this term would increase in size until it was $\mathcal{O}(1)$ at the anti-Stokes line $\theta=-\pi/3$ (dashed line in Figure 7). For $\theta \gt -\pi/3$, the term would be exponentially growing.

Figure 7. A schematic of the $\xi$ plane, where $\xi=(x-x_0)/(3t)^{1/3}$, indicating the Stokes structure for our decreasing tritronquée solution of P $_{\mathrm{II}}$.

In order to eliminate the inclusion of an exponentially growing term in the sector $-\pi \lt \theta \lt 0$, we need to force $\sigma_1=0$ in (3.19). In that case, both exponentials $\xi^{-1/4}\mathrm{e}^{2\xi^{3/2}/3}$ and $\xi^{-1/4}\mathrm{e}^{-2\xi^{3/2}/3}$ are absent along $\theta=-\pi/2$. As $\theta$ increases from $\theta=-\pi/2$, we see that $\xi^{-1/4}\mathrm{e}^{-2\xi^{3/2}/3}$ is switched on at the Stokes line $\theta=0$ (solid blue line) and becomes of $\mathcal{O}(1)$ at $\theta=\pi/3$. Transversing the other way, as $\theta$ decreases from $\theta=-\pi/2$, the exponential $\xi^{-1/4}\mathrm{e}^{2\xi^{3/2}/3}$ is switched on at the Stokes line $\theta=-2\pi/3$ (solid blue line) and becomes of $\mathcal{O}(1)$ at $\theta=-\pi$. Putting it together, a complete description of our inner problem is

(3.23)\begin{equation} \frac{\mathrm{d}^2F}{\mathrm{d}\xi^2}=2F^3+\xi F-\alpha, \end{equation}

(3.24)\begin{equation} F\sim \frac{\alpha}{\xi}+\frac{2\alpha(1-\alpha^2)}{\xi^4}+\ldots +\frac{\sigma_1}{\xi^{1/4}}\,\mathrm{e}^{2\xi^{3/2}/3} \quad\mbox{as}\quad |\xi|\rightarrow \infty, \quad -2\pi/3 \lt \mathrm{arg}(\xi) \lt 0 \end{equation}

(3.25)\begin{equation} \sigma_1=0, \quad -2\pi/3 \lt \mathrm{arg}(\xi) \lt 0, \end{equation}

where it is understood that the ellipsis represents further terms in the divergent power series (3.20); that is, there is a one-parameter family of solutions that satisfies (3.23)–(3.24), but the specific solution we are after is selected by enforcing (3.25). Furthermore, the unique solution to (3.23)–(3.25) should be pole-free region in the far field of the sector $-\pi \lt \theta \lt \pi/3$ (bounded by active anti-Stokes lines at $\theta=-\pi$ and $\theta=\pi/3$) and the power-series part of the far-field condition, namely

(3.26)\begin{equation} F\sim \frac{\alpha}{\xi}+\frac{2\alpha(1-\alpha^2)}{\xi^4} \quad\mbox{as}\quad |\xi|\rightarrow \infty, \end{equation}

applies in the sector $-\pi \lt \theta \lt \pi/3$. This type of solution with a $4\pi/3$ pole-free sector has been referred to as a decreasing tritronquée solution [Reference Fokas, Its, Kapaev and Novokshenov43] (not to be confused with increasing tritronquée solutions, which behave to leading order like $F\sim \pm (-\xi/2)^{1/2}$ as $|\xi|\rightarrow \infty$ [Reference Miller72]). Note that our solution $F(\xi)$ is equivalent to the function $y_3(x,\alpha)$ in [Reference Its and Kapaev57] or $u_3(x|\alpha)$ in [Reference Fokas, Its, Kapaev and Novokshenov43], where the Stokes multipliers $s_1=s_2=0$ and $s_3=-2\sin\pi\alpha$, using their notation.

Finally, from a practical (numerical) perspective, perhaps the best way to enforce (3.24)–(3.25) (which count as two boundary conditions) is to apply the far-field condition (3.26) along the ray $\theta=-\pi/3$ (the inactive anti-Stokes line). The reason is that for any $\sigma_1\neq 0$, the condition (3.26) would not apply along $\theta=-\pi/3$ (since, as already explained, the term $\sigma_1\xi^{-1/4}\mathrm{e}^{2\xi^{3/2}/3}$ would be $\mathcal{O}(1)$ along this ray). Therefore, applying (3.26) along $\theta=-\pi/3$ has the effect of killing off $\sigma_1\xi^{-1/4}\mathrm{e}^{2\xi^{3/2}/3}$ (as well as the other exponential $\sigma_2\xi^{-1/4}\mathrm{e}^{-2\xi^{3/2}/3}$). As an example of a numerical solution of (3.23)–(3.25) that is found by enforcing (3.26) along $\theta=-\pi/3$, we show in Figure 8 an image computed for the case $\alpha= {\textstyle\frac{1}{2}}(-1 + \sqrt{5})$ using the algorithm outlined by Fornberg & Weideman [Reference Fornberg and Weideman45]. Here the yellow dots indicate poles of the solution, which are aligned in arrays. The red lines indicate anti-Stokes lines. Note that for this value of $\alpha$, there are no poles in the third or fourth quadrant. We discuss other numerical solutions of (3.23)–(3.25) below in Subsection 3.6.

Figure 8. An image showing $|F|$ for a numerical solution of Painlevé II with $\alpha = {\textstyle\frac{1}{2}}(-1 + \sqrt{5})$ using the algorithm from Fornberg & Weideman [Reference Fornberg and Weideman45]. The yellow dots represent the pole field for this solution. The two active anti-Stokes lines are shown in red. While solutions of P $_{\mathrm{II}}$ can be uniquely specified by the ‘initial conditions’ $F(0)$ and $F'(0)$, our scheme provides this pair as outputs. For this example, the numerically obtained values are $F(0) \approx 0.5941 + 1.0289\mathrm{i}$ and $F'(0) \approx 0.7995 - 1.3848\mathrm{i}$.

3.5. Rational solutions of P $_{\mathrm{II}}$ for $\alpha\in \mathbb{N}$

There are well-known exact rational solutions of (3.23)–(3.25), which will be important for our study. For a start, when $\alpha=1$, the solution of (3.23)–(3.25) is simply $F=1/\xi$. This scenario comes from $A_0=-2$, which happens to apply for the single-soliton solution. Here, the inner problem (3.23)–(3.25) is insufficient to describe the early time behaviour (as a singularity $\xi_0=0$ does not correspond to a moving singularity in the original variables) and higher-order terms are required. This case is considered in Appendix B.4.

More generally, there are rational solutions of P $_{\mathrm{II}}$ for all $\alpha\in \mathbb{N}$, all of which satisfy (3.23)–(3.25). Each of these solutions is of the form

(3.27)\begin{equation} F=-\frac{\mathrm{d}}{\mathrm{d}\xi}\ln\left(\frac{Q_{\alpha-1}(\xi)}{Q_\alpha(\xi)}\right), \quad \alpha\in \mathbb{N}, \end{equation}

where the $Q_n$ satisfy the recursion relation [Reference Clarkson26, Reference Clarkson and Mansfield27]

\begin{equation*} Q_{n+1}Q_{n-1}=\xi Q_n^2-4\left(Q_n\frac{\mathrm{d}^2 Q_n}{\mathrm{d}\xi^2}-\left(\frac{\mathrm{d} Q_n}{\mathrm{d}\xi}\right)^2\right), \quad Q_0=1, \quad Q_1=\xi. \end{equation*}

These $Q_n$ are referred to as Yablonskii–Vorob’ev polynomials [Reference Vorob’ev94, Reference Yablonskii97]. They are special cases of the Adler–Moser polynomials, which are associated with rational solutions of the KdV equation [Reference Adler and Moser4, Reference Airault, McKean and Moser5]; various other properties of the Yablonskii–Vorob’ev polynomials are summarized in Refs [Reference Bertola and Bothner10, Reference Clarkson25, Reference Kanedo and Ochiai62, Reference Roffelsen80, Reference Taneda87].

Using the recursion relation (or, equivalently, a determinant representation derived in [Reference Kajiwara and Ohta61]), we find

(3.28)\begin{equation} Q_2=\xi^3+4, \quad Q_3=\xi^6+20\xi^3-80, \quad Q_4=\xi(\xi^9 + 60\xi^6 + 11,200), \quad \ldots. \end{equation}

For example, the first few rational solutions are (see table 7.5.1 of [Reference Ablowitz and Clarkson2])

(3.29)\begin{equation} \!\alpha=1:\quad F= -\frac{\mathrm{d}}{\mathrm{d}\xi}\ln\left(\frac{1}{\xi}\right)=\frac{1}{\xi}, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \end{equation}

(3.30)\begin{equation} \alpha=2:\quad F= -\frac{\mathrm{d}}{\mathrm{d}\xi}\ln\left(\frac{\xi}{\xi^3+4}\right)=-\frac{1}{\xi} +\frac{3\xi^2}{\xi^3+4},\qquad\qquad\qquad\qquad\quad \end{equation}

(3.31)\begin{equation} \alpha=3:\quad F= -\frac{\mathrm{d}}{\mathrm{d}\xi}\ln\left(\frac{\xi^3+4}{\xi^6+20\xi^3-80}\right)= -\frac{3\xi^2}{\xi^3+4}+\frac{6\xi^2(\xi^3+10)}{\xi^6+20\xi^3-80}, \end{equation}

(3.32)\begin{align} \,\,\,\alpha=4:\quad F= & -\frac{\mathrm{d}}{\mathrm{d}\xi}\ln\left(\frac{\xi^6+20\xi^3-80}{\xi(\xi^9+60\xi^6+11,200)}\right)\qquad\qquad\qquad\qquad\quad \nonumber \\ = & \, \frac{1}{\xi}-\frac{6\xi^2(\xi^3+10)}{\xi^6+20\xi^3-80} +\frac{9\xi^5(\xi^3+40)}{\xi^9+60\xi^6+11,200}.\qquad\qquad \end{align}

We can easily check that each of these satisfies the far-field condition (3.24) with (3.25). Further details of rational solutions of P $_{\mathrm{II}}$ are provided in Refs [Reference Buckingham and Miller17, Reference Miller and Sheng73]

In terms of the inner problem for KdV, (3.5)–(3.6), applying the formula (3.9), these rational solutions correspond to

(3.33)\begin{equation} \alpha=1:\quad f_0=-\frac{2}{\xi^2}, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \end{equation}

(3.34)\begin{equation} \!\!\!\alpha=2:\quad f_0=-\frac{6\xi(\xi^3-8)}{(\xi^3+4)^2}, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \end{equation}

(3.35)\begin{equation} \!\alpha=3:\quad f_0=-\frac{12\xi(\xi^9+600\xi^3+1600)}{(\xi^6+20\xi^3-80)^2}, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad \end{equation}

(3.36)\begin{equation} \alpha=4:\quad f_0=-\frac{20(\xi^{18} + 48\xi^{15} + 2520\xi^{12} - 78,400\xi^9 - 1,881,600\xi^6 + 12,544,000)} {\xi^2(\xi^9+60\xi^6+11,200)^2}. \end{equation}

One conclusion is that the double poles of $f_0$ for each $\alpha\in \mathbb{N}$ are located at the zeros of the polynomials $Q_\alpha(\xi)$. These occur in an almost triangular structure [Reference Bertola and Bothner10, Reference Buckingham and Miller17, Reference Clarkson26, Reference Clarkson and Mansfield27, Reference Miller and Sheng73]. Note that, while solutions $F$ of P $_{\mathrm{II}}$ have simple poles of residue either $+1$ or $-1$, the application of (3.9) shows that only those with residue $+1$ correspond to double poles of $f_0$ (for simple poles with residue $-1$, say at $\xi=\bar{\xi}$, the function $f_0$ takes the value $\bar{\xi}/2$). This is why, for $\alpha\in \mathbb{N}$, there are $\alpha^2$ simple poles of $F$, but only ${\textstyle\frac{1}{2}}\alpha(\alpha+1)$ double poles of $f_0$.

A more direct way of deriving the rational solutions of $f_0$ is via

\begin{equation*} f_0=2\frac{\mathrm{d}^2}{\mathrm{d}\xi^2}\ln Q_\alpha(\xi), \quad \alpha\in \mathbb{N} \end{equation*}

(see Clarkson [Reference Clarkson25], for example, where $f_0$ is denoted by $W$ in equation (46) in Clarkson’s paper). This formula has the advantage of requiring only one Yablonskii–Vorob’ev polynomial $Q_n$, whereas (3.27) involves two consecutive polynomials.

For later, we shall check the result here for $\alpha=2$, which will be used to compare with the well-known 2-soliton solution of (1.1). We see above that the poles of $f_0$ for $\alpha=2$ (the zeros of $Q_2=\xi^3+4$) are at

(3.37)\begin{equation} \xi_0=-2^{2/3}, \quad 2^{2/3}\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right), \quad 2^{2/3}\left(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right), \end{equation}

where $2^{2/3}\approx 1.587$. Thus, when $\alpha=2$ (that is, $A_0=-6$), our small-time analysis (see (3.7)) suggests there are three singularities that move like

(3.38)\begin{equation} s(t)\sim x_0 - 2^{2/3}(3t)^{1/3}, \,\, s(t)\sim x_0+ 2^{2/3}\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3}, \,\, s(t)\sim x_0+ 2^{2/3}\left(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3} \end{equation}

as $t\rightarrow 0^+$. That is, they initially propagate out from $x_0$ in equispaced directions $-\pi$, $\pi/3$ and $-\pi/3$ with asymptotic speed $(2/3t)^{2/3}\approx 0.763 \,t^{-2/3}$.

Similarly, we shall later check our results here for $\alpha=3$ against the 3-soliton solution of (1.1). In this case, the poles of $f_0$ for $\alpha=3$ (the zeros of $Q_3=\xi^6+20\xi^3-80$) are at

\begin{equation*} \xi_0=(-10+6\sqrt{5})^{1/3}, \quad (-10+6\sqrt{5})^{1/3} \left(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right), \quad (-10+6\sqrt{5})^{1/3} \left(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right), \end{equation*}

(3.39)\begin{equation} -(10+6\sqrt{5})^{1/3}, \quad (10+6\sqrt{5})^{1/3} \left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right), \quad (10+6\sqrt{5})^{1/3} \left(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right), \end{equation}

where $(-10+6\sqrt{5})^{1/3}\approx 1.506$ and $10+6\sqrt{5}\approx 2.861$. Thus, for $\alpha=3$ ( $A_0=-12$), our analysis predicts there are six singularities that emerge from each $x_0$, and they propagate like (see (3.7))

(3.40)\begin{equation} s(t)\sim x_0+ (-10+6\sqrt{5})^{1/3}(3t)^{1/3}, \quad s(t)\sim x_0+ (-10+6\sqrt{5})^{1/3} \left(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3}, \end{equation}

(3.41)\begin{equation} s(t)\sim x_0+ (-10+6\sqrt{5})^{1/3} \left(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3}, \quad s(t)\sim x_0-(10+6\sqrt{5})^{1/3}(3t)^{1/3}, \end{equation}

(3.42)\begin{equation} s(t)\sim x_0+ (10+6\sqrt{5})^{1/3} \left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3}, \quad s(t)\sim x_0+ (10+6\sqrt{5})^{1/3} \left(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3}, \end{equation}

as $t\rightarrow 0^+$. This time, three singularities propagate out from $x_0$ in equispaced directions $0$, $2\pi/3$ and $-2\pi/3$ with asymptotic speed $(-10+6\sqrt{5})^{1/3}/(3t)^{2/3} \approx 0.724 \,t^{-2/3}$, while the other three propagate out in directions $-\pi$, $\pi/3$ and $-\pi/3$ with asymptotic speed $(10+6\sqrt{5})^{1/3}/(3t)^{2/3} \approx 1.375 \,t^{-2/3}$.

3.6. Numerical solutions of (3.23)–(3.25)

By adapting the code used to run numerical simulations of P $_{\mathrm{II}}$ in Fornberg & Weideman [Reference Fornberg and Weideman45] (referred to as the ‘pole field solver’; see also [Reference Fornberg and Weideman44, Reference Fornberg and Weideman46]), we are able to generate numerical solutions of (3.23)–(3.25) for a selection of values of the parameter $\alpha$. Some of these results are presented in Figure 9. To obtain these results, we treat (3.23) as an initial-value problem with $F(L\mathrm{e}^{-\pi\mathrm{i}/3})$ determined by an optimal truncation of the far-field condition (3.24) using the recurrence relation in (3.21) to obtain as many terms as needed in (3.20), where $L$ is some moderately large value of $L$ ( $L=10$, say). Using the pole field solver [Reference Fornberg and Weideman45], we integrate from $\xi = L\mathrm{e}^{-\pi\mathrm{i}/3}$ towards $\xi = 0$ in order to estimate $F(0)$ and $F'(0)$. Once these estimates are obtained, the pole field solver algorithm is again used, now integrating along the entire square computational domain. In cases where $F(\xi)$ is singular at or near $\xi = 0$, instead of integrating until $\xi = 0$, we integrate until $|\xi| \lt h$ for some small $h$, and then use that stopping location as an initial value for evaluating the solution on the complete domain. Lastly, we note that if there is a singularity on or near the ray $\xi = \rho\mathrm{e}^{-\pi\mathrm{i}/3}$, the pole field solver algorithm is able to adjust the path of integration automatically, thus no adjustment is needed to handle this situation.

Figure 9. Phase portraits of solutions of P $_{\mathrm{II}}$, (3.23)–(3.25), computed for various values of $\alpha$, namely $0.25$, $0.5$, $0.75$ and $1$ in the first row, $1.25$, $1.5$, $1.75$ and $2$ in the second row and $2.25$, $2.5$, $2.75$ and $3$ in the third row. The colour denotes the phase of $F$, with red denoting real and positive, yellow imaginary and positive, light blue real and negative, and dark blue negative and imaginary. Note the images for $\alpha=1$, $2$ and $3$ in this figure are generated using the exact solutions (3.29)–(3.31), while the rest are computed using the pole field solver [Reference Fornberg and Weideman45]. In the latter case, we can see some numerical error in some images around $\mathrm{Re}(\xi)\approx 8$.

Returning to Figure 9, the most obvious distinction between some of these examples and others is that the solutions for $\alpha\in\mathbb{N}$ do not have a lattice of poles in the far field, as these are rational solutions with a finite number of poles. Note that the exact solutions for $\alpha=1$, $2$ and $3$ are given by (3.29)–(3.31), and the images in Figure 9 are generated from these exact results (numerical solutions for $\alpha=1$, $2$ and $3$ are visually indistinguishable from the exact results for $-7\lesssim\mathrm{Re}(\xi)\lesssim7$, $-7\lesssim\mathrm{Im}(\xi)\lesssim7$, but show some numerical error outside of this region). Recall that our solutions of P $_{\mathrm{II}}$ have simple poles with residue either $+1$ or $-1$. Using the solution for $\alpha=1$ in Figure 9 (right panel of the first row), namely $F=1/\xi$, as a reference, it should be clear how the colour scheme in the phase portrait displays a simple pole with residue $+1$ (with colouring red, dark blue, light blue, yellow, then red again, as we traverse once around the pole in the positive direction, starting at $\theta=0$). For example, we see the solution for $\alpha=2$ in Figure 9 (right panel of the second row), which has simple poles with residue $+1$ at the three points given by (3.37). The other pole for $\alpha=2$, located at the origin, has residue $-1$ (with colouring light blue, yellow, red, dark blue and light blue as we traverse around the pole starting at $\theta=0$). For $\alpha=3$ in Figure 9 (right panel of the third row), the six poles with residue $+1$ (3.39) form an almost triangular shape, while the three remaining poles with residue $-1$ lie inside this almost-triangle, and so on. Clear images of the pole fields for these and other rational solutions, together with the location of the simple zeros, can be found in Fornberg & Weideman [Reference Fornberg and Weideman45], for example.

For the solutions in Figure 9 for $\alpha\notin\mathbb{N}$, the lattice of poles in the far field is restricted to a sector of angle $2\pi/3$, which is why these are called tritronquée solutions. The pole-free sector in the far field is $-\pi \lt \mathrm{arg}(\xi) \lt \pi/3$. Another perspective of this lattice of poles is shown in Figure 10, where there are a few analytic landscape plots, from which it is easy to appreciate where the poles are located. To help distinguish between poles with residue $+1$ or $-1$, we have added white and black dots to a phase portrait in Figure 11. For this solution, drawn for $\alpha=5/2$, there are four poles (three with residue $+1$ and one with residue $-1$) that sit inside $-\pi \lt \mathrm{arg}(\xi) \lt \pi/3$. As $\alpha$ increases, these ‘move’ in the general direction $-\pi/3$ and ultimately are located at known positions when $\alpha=3$ (the three with residue $+1$ are the first, third and sixth poles in equation (3.39)). Thus, we see that the property of a pole-free sector only holds in the far field, not the near field. To be clear, while in this section we are discussing solutions of (3.23)–(3.25), when we apply the change of variables (3.9), it is only the poles with residue $+1$ that are important for our solution $f_0$, and therefore it is only these poles that are relevant for the original KdV problem.

Figure 10. Analytic landscape plots for numerical solutions of P $_{\mathrm{II}}$, (3.23)–(3.25), computed with $\alpha=0.5$, $1.5$ and $2.5$. The surfaces are $|F(\xi)|$, while the colour scheme is the same as in Figure 9. The clear spikes in $|F(\xi)|$ correspond to simple poles.

Figure 11. Phase portrait of P $_{\mathrm{II}}$ solution for $\alpha=5/2$. The white dashed lines indicate the boundary of the pole-free sector in the far field. The white and black dots denote poles with residues $+1$ and $-1$, respectively.

3.7. Locating singularities of Painlevé II using transseries

As we see in Figures 8–11, for each solution of (3.23)–(3.25) with $\alpha\notin\mathbb{N}$, there is a lattice of simple-pole singularities in the upper half $\xi$ plane. We are interested here in the string of singularities that align themselves in an almost-horizontal direction and tend towards the anti-Stokes line $\mathrm{Im}(\xi)=0$, $\mathrm{Re}(\xi) \lt 0$ ( $\mathrm{arg}(\xi)=-\pi$) in the far field. These are important as they will have the strongest influence on the real-line solution $u(x,t)$ of (1.1) for $x \lt 0$ in the small-time limit. Indeed, our working hypothesis is that these singularities link directly to the important properties of dispersive waves, such as their speed and wavelength. In this subsection, we summarize how to apply a standard transseries expansion to estimate the location of these singularities in the far field. The approach we take relies on transseries and is equivalent to that of Costin & Costin [Reference Costin and Costin28], who briefly use a version of P $_{\mathrm{II}}$ as an example. Other studies that apply transseries methodology for locating singularities of P $_{\mathrm{II}}$ are for the homogeneous version with $\alpha=0$, for which some of the details are different [Reference Baldino, Schiappa, Schwick and Vega8, Reference Mariño69, Reference Schiappa and Vaz82].

The starting point is to note that the solution to (3.23)–(3.25) has the behaviour

(3.43)\begin{equation} F\sim \frac{\alpha}{\xi}+\frac{2\alpha(1-\alpha^2)}{\xi^4}+\ldots +\frac{\sigma_1}{\xi^{1/4}}\,\mathrm{e}^{2\xi^{3/2}/3} \quad\mbox{as}\quad |\xi|\rightarrow \infty, \quad -\pi \lt \mathrm{arg}(\xi) \lt -2\pi/3 \end{equation}

(3.44)\begin{equation} \sigma_1\neq 0, \quad -\pi \lt \mathrm{arg}(\xi) \lt -2\pi/3, \end{equation}

where the Stokes constant $\sigma_1$ in (3.43)–(3.44) is given explicitly by

(3.45)\begin{equation} \sigma_1=\frac{\sin\pi\alpha}{\pi^{1/2}} \end{equation}

[Reference Fokas, Its, Kapaev and Novokshenov43, Reference Its and Kapaev57]. Clearly, the exponential term in (3.43) is exponentially small in the sector $-\pi \lt \mathrm{arg}(\xi) \lt -2\pi/3$, and can be interpreted as the leading-order behaviour of the non-perturbative part of the asymptotic expansion. (An abbreviated argument for (3.45) is to see from (3.22) that the terms in the algebraic terms in the asymptotic series for $F$ behave as

\begin{equation*} \frac{\alpha b_n}{\xi^{3n+1}} \sim -\mathrm{i} \frac{\sin\pi\alpha}{\pi^{3/2}} \xi^{-1/4} \frac{\Gamma(2n+1/2)}{(-2\xi^{3/2}/3)^{2n+1/2}} \quad\mbox{as}\quad n\rightarrow\infty, \end{equation*}

which is of the form $\mathcal{A}(\xi)\Gamma(2n+1/2)/\chi(\xi)^{2n+1/2}$, where $\chi$ is a singulant. Therefore, the exponentially small term

\begin{equation*} \pi\mathrm{i}\mathcal{A}\mathrm{e}^{-\chi}=\frac{\sin\pi\alpha}{\pi^{1/2}}\xi^{-1/4} \mathrm{e}^{2\xi^{3/2}/3} \end{equation*}

is switched on as we cross the Stokes line $\mathrm{arg}(\xi)=-2\pi/3$ in the clockwise direction.)

As we explain in Appendix C.1, we can extend (3.43) to a transseries of the form

(3.46)\begin{equation} F\sim \sum_{n=0}^\infty \sigma_1^n \,\mathrm{e}^{2n\xi^{3/2}/3} \sum_{m=0}^\infty \frac{F_m^{(n)}}{\xi^{-1/2+3n/4+3m/2}} = \sum_{n=0}^\infty\left(\frac{\sigma_1\,\mathrm{e}^{2\xi^{3/2}/3}} {\xi^{3/4}}\right)^n \sum_{m=0}^\infty \frac{F_m^{(n)}}{\xi^{-1/2+3m/2}}, \end{equation}

where $F_{2m}^{(0)}=0$, $F_0^{(2n)}=0$, $F_0^{(1)}=1$ and, comparing with the notation in (3.20), $F_{2m+1}^{(0)}=\alpha b_m$. The transseries (3.46) is valid in the sector $-\pi \lt \mathrm{arg}(\xi) \lt -2\pi/3$, as each term $\mathrm{e}^{2n\xi^{3/2}/3}$ is exponentially smaller than the previous one as $n$ increases.

To proceed, we define the transseries variable $\tau$ by

(3.47)\begin{equation} \tau = \frac{\sigma_1\,\mathrm{e}^{2\xi^{3/2}/3}}{\xi^{3/4}}. \end{equation}

The representation (3.46) remains well-ordered provided $|\tau|\ll 1$, which is certainly true for $|\xi|\gg 1$ in the sector $-\pi\leq \mathrm{arg}(\xi) \lt -2\pi/3$. Thus, in this region, we can swap the order of summation, so that

(3.48)\begin{equation} F\sim \sum_{m=0}^\infty \frac{1}{\xi^{-1/2+3m/2}} \sum_{n=0}^\infty \tau^n F_m^{(n)}, \end{equation}

and, furthermore, in the neighbourhood of the anti-Stokes line $\mathrm{Im}(\xi)=0$, $\mathrm{Re}(\xi) \lt 0$, we have

(3.49)\begin{equation} F\sim \sum_{m=0}^\infty \frac{G_m(\tau)}{\xi^{-1/2+3m/2}}, \end{equation}

where the $G_m(\tau)$ are functions of $\tau$ that for $-\pi\leq \mathrm{arg}(\xi) \lt -2\pi/3$ are given by the second (convergent) summation in (3.48). Crucially, the asymptotic representation (3.49) applies regardless of whether $|\tau|$ is small, which means it applies as we cross the negative real $\xi$ axis from below (where $|\tau|$ is exponentially small) to above (where $|\tau|$ is exponentially large). In particular, we shall apply (3.49) in the neighbourhood of the string of singularities that align themselves slightly above the negative real $\xi$ axis for large $|\xi|$.

Thus, near the negative real $\xi$ axis, we substitute (3.49) into P $_{\mathrm{II}}$ (3.14) to derive odes for $G_m$, which are subject to boundary conditions that come from matching into the lower half plane, namely

(3.50)\begin{equation} G_m\sim F_m^{(0)}+F_m^{(1)}\tau+F_m^{(2)}\tau^2+\ldots \quad\mbox{as}\quad \tau\rightarrow 0. \end{equation}

We show in Appendix C.2 that the leading-order solution is

\begin{equation*} G_0=\frac{4\tau}{4-\tau^2}, \end{equation*}

which has singularities at $\tau = \tau_0 = \pm 2$ (corresponding to poles of P $_{\mathrm{II}}$ with residues $-$1 and 1, respectively). Therefore, armed with this leading-order solution only, we can see that

(3.51)\begin{equation} F\sim \frac{4\tau\xi^{1/2}}{4-\tau^2} = \frac{4\sigma_1\xi^{-3/4}\,\mathrm{e}^{2\xi^{3/2}/3}\xi^{1/2}} {4-\sigma_1^2\xi^{-3/2}\,\mathrm{e}^{4\xi^{3/2}/3}} \quad\mbox{as}\quad |\xi|\rightarrow\infty, \end{equation}

and the first approximation for the location of the singularities as $|\xi|\rightarrow\infty$ is determined by the transcendental equation

(3.52)\begin{equation} \sigma_1\xi_0^{-3/4}\,\mathrm{e}^{2\xi_0^{3/2}/3}=\tau_0^{(0)}=\pm 2. \end{equation}

Rearranging, we can write

\begin{equation*} \xi_0^{3/2}=3n\pi\mathrm{i}+\frac{3}{4}\log \left(\xi_0^{3/2}\right) +\frac{3}{2}\log \left(\frac{\tau_0^{(0)}}{\sigma_1}\right). \end{equation*}

By interpreting $\mathrm{i}=\mathrm{e}^{-3\pi\mathrm{i}/2}$, we solve this equation asymptotically in the limit $|\xi_0|\rightarrow\infty$ to give

\begin{equation*} \xi_0^{3/2}\sim 3n\pi\mathrm{i}+\frac{3}{4}\ln (3n\pi) - \frac{9\pi\mathrm{i}}{8} +\frac{3}{2}\log \left(\frac{\tau_0^{(0)}}{\sigma_1}\right) \quad\mbox{as}\quad n\rightarrow\infty, \end{equation*}

or, alternatively,

(3.53)\begin{equation} \xi_0\sim -(3\pi n)^{2/3}+\frac{1}{(3\pi n)^{1/3}} \left( \frac{3\pi}{4}+\frac{\mathrm{i}}{2}\ln (3\pi n) +\mathrm{i}\log\left(\frac{\tau_0^{(0)}}{\sigma_1}\right) \right) \quad\mbox{as}\quad n\rightarrow\infty, \end{equation}

where $\tau_0^{(0)}=\pm 2$ and $n\in\mathbb{N}$.

For our purposes, it is important to note that the Stokes constant $\sigma_1$ in (3.45) is real, but may be positive or negative, depending on $\alpha$. Further, $\sigma_1$ vanishes for $\alpha\in\mathbb{N}$. Thus, given our two choices $\tau_0^{(0)}=\pm 2$, the term $\tau_0^{(0)}/\sigma_1$ can also be positive or negative, provided $\alpha\notin\mathbb{N}$. With this in mind, we can rewrite (3.53) for the two separate cases as

(3.54)\begin{equation} \xi_0\sim -(3\pi n)^{2/3}+\frac{1}{(3\pi n)^{1/3}} \left(\frac{3\pi}{4} +\frac{\mathrm{i}}{2}\ln \left(\frac{12\pi^2 n}{\sin^2 \pi\alpha}\right) \right) \quad\mbox{as}\quad n\rightarrow\infty \quad\mbox{for}\quad \tau_0^{(0)}/\sigma_1 \gt 0, \end{equation}

(3.55)\begin{equation} \xi_0\sim -(3\pi n)^{2/3}+\frac{1}{(3\pi n)^{1/3}} \left(-\frac{\pi}{4} +\frac{\mathrm{i}}{2}\ln \left(\frac{12\pi^2 n}{\sin^2 \pi\alpha}\right) \right) \quad\mbox{as}\quad n\rightarrow\infty \quad\mbox{for}\quad \tau_0^{(0)}/\sigma_1 \lt 0. \end{equation}

For example, if we choose $\alpha=1/2$, then $\sigma_1 \gt 0$. With this parameter choice, poles with $\tau_0^{(0)}=2$ (those with residue $-1$) are located using (3.54), while poles with $\tau_0^{(0)}=-2$ (residue $+1$) are located using (3.55). We show this case in Figure 12, where we see the asymptotic formulae (3.54)–(3.55) do an excellent job of estimating the pole locations for moderately small values of $n$ even though they hold formally in the limit $n\rightarrow\infty$. (Note that by considering $G_1(\tau)$ in (3.49), we can extend (3.54)–(3.55) to include higher-order corrections. See Appendix C.3.)

Figure 12. Phase portrait of a numerical solution of (3.23)–(3.25), computed for $\alpha=1/2$. The black and white circles denote estimates of poles with residues $-1$ and $+1$ computed via (3.54) and (3.55), respectively. The closest two circles to the origin are computed using $n=1$, the next closest pair with $n=2$, and so on.

We make two further observations about our asymptotic formulae (3.54)–(3.55). Firstly, the poles alternate between residues $+1$ and $-1$ and, for very large $n$, are separated by an asymptotic distance $\pi^{2/3}/(3n)^{1/3}$. Secondly, these formulae assume that $\ln |\sigma_1|=\mathcal{O}(1)$. For values of $\alpha$ extremely close to a natural number $N$, $\sigma_1$ becomes sufficiently small in magnitude that $\ln |\sigma_1|$ becomes large and these formulae need reconsidering. Correspondingly, as $\alpha$ approaches $N$, the lattice of poles ‘moves away’ from the origin, leaving behind a block of $N^2$ poles that are generated by the relevant rational solution for $\alpha=N$.

It is worth recalling that, in terms of our original KdV problem, double poles of solutions $u(x,t)$, located at $x=s(t)$, have the small-time behaviour (3.7). Thus, we can use (3.54) and (3.55) for $\tau_0^{(0)}=-2$, to describe the leading-order location of double poles of $u(x,t)$ that propagate almost horizontally to the left like $s(t)\sim x_0-3\pi^{2/3}n^{2/3}t^{1/3}$. Each of these can be associated with a crest of the dispersive waves on the real line, linking real-valued behaviour with poles in the complex plane.

4.1. $1$-Soliton solution

An obvious example is a 1-soliton solution $u(x,t)=2\,\mathrm{sech}^2 (x-4t)$, which evolves from the initial condition $u_0=2\,\mathrm{sech}^2 x$, that is, from (1.4) with $A_0=-2$. The 1-soliton solution is simply a constant-shape travelling wave moving from left to right with speed $4$. Clearly, the solution has poles where $\cosh(x-4t)=0$, which are located at

(4.1)\begin{equation} s(t)=\left(n+{\textstyle\frac{1}{2}}\right)\pi\mathrm{i}+4t, \end{equation}

where $n$ is an integer. That is, a single pole emerges from each $x_0=(n+{\textstyle\frac{1}{2}})\,\pi\mathrm{i}$ at $t=0$ and travels horizontally in the complex plane with speed 4. Another way to write the 1-soliton solution is

\begin{equation*} u(x,t)=2\frac{\partial^2}{\partial x^2}\left(\ln U_1\right), \quad U_1=1+\mathrm{e}^{2x-8t}. \end{equation*}

With this representation, we see poles of $u$ can be determined via the zeros of $U_1$, giving the same result as (4.1), as expected.

To compare with our small-time results, we refer to Appendix B.4, since $A_0=-2$ is a special case. Further, noting that this initial condition has poles at $x_0=(n+{\textstyle\frac{1}{2}})\,\pi\mathrm{i}$ with local behaviour

\begin{equation*} u_0\sim \frac{-2}{(x-x_0)^2}+\frac{2}{3} \quad\mbox{as}\quad x\rightarrow x_0, \end{equation*}

we see from (2.15) that $A_0=-2$, $A_1=0$ and $A_2=2/3$. Following this very special case in Appendix B.4.2, our analysis predicts that the early time behaviour of singularity is described by (B.34), namely $s(t)\sim x_0 +6A_2t$ as $t\rightarrow 0^+$. Given $A_2=2/3$, this predicted speed is $6A_2=4$, which agrees with the exact solution for the 1-soliton solution. Thus, while the 1-soliton solution itself is trivial in the sense that we can easily determine the small-time behaviour of the complex singularities without any of our asymptotic analysis, this check of (B.34) provides support for the details in Appendix B.4 (which apply for other, more complicated, initial conditions that also have $A_0=-2$).

4.2. $2$-Soliton solution

The simplest less trivial example with an exact solution is a $2$-soliton solution (see [Reference Ablowitz and Clarkson2, Reference Drazin and Johnson38] and the references therein)

(4.2)\begin{equation} u(x,t)=12\frac{3+4\cosh(2x-8t)+\cosh(4x-64t)}{(3\cosh(x-28t)+\cosh(3x-36t))^2}, \end{equation}

which evolves from the initial condition

(4.3)\begin{equation} u_0=6\,\mathrm{sech}^2x. \end{equation}

This solution consists of two solitons moving to the right, with the long-time limit [Reference Ablowitz and Clarkson2, Reference Drazin and Johnson38]

\begin{equation*} u(x,t)\sim 2\,\mathrm{sech}^2(x-4t+{\textstyle\frac{1}{2}}\ln 3) +8\,\mathrm{sech}^2(2x-32t-{\textstyle\frac{1}{2}}\ln 3) \quad\mbox{as}\quad t\rightarrow\infty. \end{equation*}

That is, for sufficiently large time, the solution is dominated by a larger soliton moving with speed 16 and a smaller soliton moving with speed 4. To compare with our small-time analysis in Section 2, we note that the initial condition (4.3) has poles at $x_0=(n+{\textstyle\frac{1}{2}})\,\pi\mathrm{i}$ and is a member of (1.4) with $A_0=-6$.

Another way of writing (4.2) is

\begin{equation*} u(x,t)=2\frac{\partial^2}{\partial x^2}\left(\ln U_2\right), \quad U_2=1+3\mathrm{e}^{2x-8t}+3\mathrm{e}^{4x-64t}+\mathrm{e}^{6x-72t}, \end{equation*}

so that clearly the singularities of $u$ can be determined by setting $x=s(t)$ and solving for zeros of $U_2$. By letting $\gamma=\mathrm{e}^{2s}$, this exercise reduces to finding roots of the cubic [Reference Drazin and Johnson38]

\begin{equation*} U_2=1+3\gamma\mathrm{e}^{-8t}+3\gamma^2\mathrm{e}^{-64t}+\gamma^3\mathrm{e}^{-72t}. \end{equation*}

By using a symbolic manipulation package (Maple, for example) or otherwise, we can use the cubic formula to derive complicated expressions for the roots $\gamma$ and then determine the small-time behaviour using the appropriate series command. We find the three behaviours (one for each solution of the cubic)

\begin{align*} \gamma\sim & -1+2^{5/3}(3t)^{1/3}+2^{7/3}(3t)^{2/3}-8t, \\ & -1-2^{5/3}\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)(3t)^{1/3} -2^{7/3}\left(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right)(3t)^{2/3}-8t, \\ & -1-2^{5/3}\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)(3t)^{1/3} -2^{7/3}\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)(3t)^{2/3}-8t. \end{align*}

For each of these, we expand $s=(\ln\gamma)/2$ for small time to give the three options

(4.4)\begin{equation} s(t)\sim \left(n+{\textstyle\frac{1}{2}}\right)\pi\mathrm{i}-2^{2/3}(3t)^{1/3}+12t, \quad \left(n+{\textstyle\frac{1}{2}}\right)\pi\mathrm{i}+ 2^{2/3}\left(\frac{1}{2}\pm\frac{\sqrt{3}}{2}\mathrm{i}\right)(3t)^{1/3}+12t \end{equation}

as $t\rightarrow 0^+$.

The $\mathcal{O}(t^{1/3})$ part of (4.4) agrees with our prediction (3.38), which comes from our inner problem for the case $A_0=-6$ (and involves the rational solution of P $_{\mathrm{II}}$ for $\alpha=2$). The $\mathcal{O}(t)$ terms in (4.4) are all $12t$; this result agrees with our prediction (B.25) from Appendix B.3 (found by considering the effect of higher-order terms in (1.3)). Further, these $\mathcal{O}(t)$ terms in (4.4) are consistent with the well-known results that, for the 2-soliton solution, there are three singularities that initially move out of each of $x_0=(n+{\textstyle\frac{1}{2}})\,\pi\mathrm{i}$ in three equally spaced directions but then all begin to move to the right. More specifically, taking the closest $x_0$ to the real- $x$ axis as an example ( $n=0$), we can label

\begin{equation*} s_1(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}-2^{2/3}(3t)^{1/3}+12t \end{equation*}

as $t\rightarrow 0^+$, which is associated with the late-time soliton $2\,\mathrm{sech}^2(x-4t+{\textstyle\frac{1}{2}}\ln 3)$ and therefore has the asymptotic behaviour $s_1(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i} -{\textstyle\frac{1}{2}}\ln 3 + 4t$ as $t\rightarrow\infty$. Further, we can label the singularities

\begin{equation*} s_2(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}+ 2^{2/3}\left(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right)(3t)^{1/3}+12t, \quad s_3(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}+ 2^{2/3}\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)(3t)^{1/3}+12t, \end{equation*}

as $t\rightarrow 0^+$, so that both $s_2(t)$ and $s_3(t)$ are associated with the late-time soliton $8\,\mathrm{sech}^2(2x-32t-{\textstyle\frac{1}{2}}\ln 3)$ and have the asymptotic behaviours $s_2(t)\sim {\textstyle\frac{1}{4}}\pi\mathrm{i} +{\textstyle\frac{1}{4}}\ln 3 + 16t$, $s_3(t)\sim {\textstyle\frac{3}{4}}\pi\mathrm{i} +{\textstyle\frac{1}{4}}\ln 3 + 16t$ as $t\rightarrow\infty$. We illustrate this behaviour in Figure 13(a), where snapshots of the $2$-soliton solution are provided at times $t=0.001$, $0.02$ and $0.2$. The accompanying phase portraits show how the singularities first move away from $x=\pi\mathrm{i}/2$ in different directions, as described above, but ultimately all propagate to the right and align themselves with the peaks of the relevant soliton.

Figure 13. Solution profiles and phase portraits for (a) the $2$-soliton solution (4.2), which evolves from $u(x,0)=6\,\mathrm{sech}^2x$ and (b) the $3$-soliton solution (4.6), which evolves from $u(x,0)=12\,\mathrm{sech}^2x$.

Another check on our 2-soliton solution is to put $x={\textstyle\frac{1}{2}}\pi\mathrm{i}+(3t)^{1/3}\xi$ into (4.2) and then take the limit $t\rightarrow 0^+$. The result is that

\begin{equation*} u(x,t)\sim -\frac{1}{(3t)^{2/3}}\frac{6\xi(\xi^3-8)}{(\xi^3+4)^2}+\frac{2\xi^9-24\xi^6+1440\xi^3-640}{(\xi^3+4)^3} \quad\mbox{as}\quad t\rightarrow 0^+. \end{equation*}

The leading-order part here agrees with the rational solution computed for $\alpha=2$, namely (3.34), as expected. The correction term can be checked by showing it is a solution of the appropriate ode problem (B.22) with (B.9), the details of which are discussed in Appendix B.3. In summary, by carefully studying the exact solution for the $2$-soliton solution we are able to check our matched asymptotics, including the correction terms, providing additional confidence that our analysis is correct.

4.3. $3$-Soliton solution

A more complicated example is a $3$-soliton solution that evolves from the initial condition

(4.5)\begin{equation} u_0=12\,\mathrm{sech}^2x, \end{equation}

and can be represented by

\begin{equation*} u(x,t)=2\frac{\partial^2}{\partial x^2}\left(\ln U_3\right), \end{equation*}

\begin{equation*} U_3=1+6\mathrm{e}^{2x-8t}+15\mathrm{e}^{4x-64t}+10\mathrm{e}^{6x-72t} +10\mathrm{e}^{6x-216t}+15\mathrm{e}^{8x-224t}+6\mathrm{e}^{10x-280t} +\mathrm{e}^{12x-288t}. \end{equation*}

By performing the two partial derivatives and simplifying, we can show that another version of this $3$-soliton solution is

(4.6)\begin{equation} u(x,t)=H_1/H_2^2, \end{equation}

where

\begin{align*} H_1&=24[126+50\cosh(2x-8t)+135\cosh(2x-56t)+25\cosh(2x-152t)+80\cosh(4x-64t)\\ &\quad +40\cosh(4x-208t)+15\cosh(6x-72t)+30\cosh(6x-216t)+10\cosh(8x-224t)\\ &\quad +\cosh(10x-280t)],\\ H_2&=15\cosh(2x-80t)+6\cosh(4x-136t)+\cosh(6x-144t)+10\cosh(72t) \end{align*}

(see [Reference Shen85]). This solution involves three solitons moving to the right, with the long-time limit

\begin{equation*} u(x,t)\sim 2\,\mathrm{sech}^2(x-4t+{\textstyle\frac{1}{2}}\ln 6) +8\,\mathrm{sech}^2(2x-32t+{\textstyle\frac{1}{2}}\ln {\textstyle\frac{5}{3}}) +18\,\mathrm{sech}^2(3x-108t-{\textstyle\frac{1}{2}}\ln 10) \quad\mbox{as}\quad t\rightarrow\infty. \end{equation*}

In other words, for late times, the smallest soliton is moving with speed $4$, the middle soliton is moving with speed $16$, while the largest soliton is moving with speed $36$.

To determine the small-time behaviour or complex singularities for the $3$-soliton solution, we set $x=s(t)$ and solve for the zeros of $U_3$, which becomes a sixth-order polynomial in terms of $\gamma=\mathrm{e}^{2s}$. Leaving out the details, by concentrating on singularities that emerge from $x_0={\textstyle\frac{1}{2}}\pi\mathrm{i}$, we have the six small-time behaviours

\begin{equation*} s_1(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}-(10+6\sqrt{5})^{1/3}(3t)^{1/3} +24t, \quad s_2(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}+(-10+6\sqrt{5})^{1/3} \left(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3} +24t, \end{equation*}

\begin{align*} s_3(t)&\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}+(-10+6\sqrt{5})^{1/3} \left(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3} +24t,\\ s_4(t)&\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}+(10+6\sqrt{5})^{1/3} \left(\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3} +24t, \end{align*}

\begin{equation*} s_5(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}+(-10+6\sqrt{5})^{1/3}(3t)^{1/3} +24t, \quad s_6(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i}+(10+6\sqrt{5})^{1/3} \left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right) (3t)^{1/3} +24t, \end{equation*}

as $t\rightarrow 0^+$. With this labelling, one of the six singularities, namely $s_1$, is associated with the slowest soliton, with $s_1(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i} -{\textstyle\frac{1}{2}}\ln 6 + 4t$ as $t\rightarrow\infty$; two of the singularities, $s_2$ and $s_3$, are associated with the intermediate-speed soliton, with $s_2(t)\sim {\textstyle\frac{1}{4}}\pi\mathrm{i} -{\textstyle\frac{1}{4}}\ln {\textstyle\frac{5}{3}} + 16t$ and $s_3(t)\sim {\textstyle\frac{3}{4}}\pi\mathrm{i} -{\textstyle\frac{1}{4}}\ln {\textstyle\frac{5}{3}} + 16t$ as $t\rightarrow\infty$; and the remaining three singularities are associated with the fastest soliton, with $s_4(t)\sim {\textstyle\frac{1}{6}}\pi\mathrm{i} +{\textstyle\frac{1}{6}}\ln 10 + 36t$, $s_5(t)\sim {\textstyle\frac{1}{2}}\pi\mathrm{i} +{\textstyle\frac{1}{6}}\ln 10 + 36t$ and $s_6(t)\sim {\textstyle\frac{5}{6}}\pi\mathrm{i} +{\textstyle\frac{1}{6}}\ln 10 + 36t$ as $t\rightarrow\infty$. This $3$-soliton solution and the singularity dynamics are illustrated in Figure 13(b). As with the $2$-soliton example, we see in the phase portraits how the singularities initial spread out from $x=\pi\mathrm{i}/2$ before propagating to the right and aligning with a soliton.

To compare these results for the $3$-soliton solution with our analysis in Section 2, we note that (4.5) is a member of (1.4) with $A_0=-12$. Thus, the relevant leading-order results are (3.40)–(3.42), which come from the roots of $f_0$ when $\alpha=3$ (that are derived via the rational solution of P $_{\mathrm{II}}$ for $\alpha=3$). The $\mathcal{O}(t^{1/3})$ parts of $s_1(t)$ to $s_6(t)$ above agree with (3.40)–(3.42), as expected, but nevertheless this comparison acts as an important check on our leading-order asymptotic analysis. Further, the $\mathcal{O}(t)$ terms in $s_1(t)$ to $s_6(t)$ above are all $24t$. These terms agree with the prediction (B.26) that comes from considering higher-order effects in Appendix B.3.

As an additional check, we set $x={\textstyle\frac{1}{2}}\pi\mathrm{i}+(3t)^{1/3}\xi$ in (4.6) and take the limit $t\rightarrow 0^+$, to give

\begin{equation*} u(x,t)\sim -\frac{1}{(3t)^{2/3}}\frac{12\xi(\xi^9+600\xi^3+1600)}{(\xi^6+20\xi^3-80)^2} \end{equation*}

\begin{equation*} +\frac{4(\xi^{18} + 12\xi^{15} + 2880\xi^{12} - 136000\xi^9 - 1075200\xi^6 - 8064000\xi^3 - 3584000)}{(\xi^6+20\xi^3-80)^3} \end{equation*}

as $t\rightarrow 0^+$. The leading-order part of this expression agrees with the rational solution of $f_0$ for $\alpha=3$, given by (3.35), as required. Further, the correction term can be shown to satisfy the appropriate problem (B.22) with (B.9), as mentioned in Appendix B.3. These tests, including those for higher-order terms, again provide confidence that our matched asymptotics is correct.