3.1. Steady equilibria

From the definition adopted in § 1, a steady equilibrium is a solution of (1.1) with constant amplitudes $|b_i|$. From the Manley–Rowe relations (2.1), it corresponds to a solution of (2.18) such that $\dot q (t) = 0$ for all time. Since $q(0) = 0$, a steady equilibrium therefore corresponds in our problem to the null solution $q(t) = 0$, $\forall t \geq 0$, for which (2.18) implies $f(0)=0$. But since (2.18) is equivalent to ‘Newton's second law’:

(3.1)\begin{equation} \ddot{q} = {\tfrac{1}{2}}\, f'(q) =- U'(q), \end{equation}

steady equilibria also satisfy $U'(0) = 0$ (here and below, $f'(q) = \mathrm {d} f / \mathrm {d} q$), i.e. they are critical points of the potential energy (Arnold Reference Arnold1989, p. 99). For (2.20), conditions $f(0) = 0$ and $f'(0) = 0$ are respectively equivalent to $e=0$ and $d=0$. Since we now consider $T \not = 0$, condition $e=0$ yields either $q_1 q_2 q_3 q_4 = 0$, or $p_0 = 0$ or ${\rm \pi}$, while condition $d=0$ becomes

(3.2)\begin{equation} 2 T ((q_1 + q_2)q_3 q_4 - q_1 q_2 (q_3 + q_4)) + B \sqrt{q_1q_2q_3q_4} \cos p_0 = 0. \end{equation}

Suppose first $q_1 q_2 q_3 q_4 = 0$ and choose, without lost of generality, $q_4=0$. Condition (3.2) implies $q_1 q_2 q_3 = 0$. Therefore, a second wave must have a zero initial wave action, say $q_3 = 0$. From system (1.1) with $|b_i|^2 = q_i$, $i=1,2$, we get the steady bichromatic wave first considered by Phillips (Reference Phillips1960) and Longuet-Higgins & Phillips (Reference Longuet-Higgins and Phillips1962) (see also Zakharov Reference Zakharov1967; Hogan, Gruman & Stiassnie Reference Hogan, Gruman and Stiassnie1988; Leblanc Reference Leblanc2009):

(3.3)\begin{equation} \left. \begin{gathered} b_1(t)=\sqrt{q_1} \mathrm{e}^{\mathrm{i} \varphi_1} \mathrm{e}^{-\mathrm{i} \varOmega_1 t}, \quad \varOmega_1 = \omega_1 + T_{11}q_1 + 2 T_{12}q_2, \\ b_2(t)=\sqrt{q_2} \mathrm{e}^{\mathrm{i} \varphi_2} \mathrm{e}^{-\mathrm{i} \varOmega_2 t}, \quad \varOmega_2 = \omega_2 + T_{22}q_2 + 2 T_{21}q_1, \end{gathered} \right\} \end{equation}

where we recall that $\varphi _i = p_i(0)$. These solutions were given by Benney (Reference Benney1962) who also noticed that if in addition $q_2=0$ one recovers the Stokes wave:

(3.4)\begin{equation} b_1(t)=\sqrt{q_1} \mathrm{e}^{\mathrm{i} \varphi_1} \mathrm{e}^{-\mathrm{i} \varOmega_1 t}, \quad \varOmega_1 = \omega_1 + T_{11}q_1. \end{equation}

Turning now to the case $q_1 q_2 q_3 q_4 \not = 0$ and $p_0 = 0$ or ${\rm \pi}$, we get from (3.2):

(3.5)\begin{equation} \Delta \omega + B_1 q_1 + B_2 q_2 - B_3 q_3 - B_4 q_4 = 2 T \frac{q_1 q_2 (q_3 + q_4) - (q_1 + q_2)q_3 q_4 }{\sqrt{q_1q_2q_3q_4} \cos p_0}. \end{equation}

Since from (2.7) $p(t) = p_0$ at equilibrium, we conclude that:

Theorem 3.1 The wave quartet defined by

(3.6)\begin{equation} \left. \begin{gathered} b_i(t)=\sqrt{q_i} \mathrm{e}^{\mathrm{i} \varphi_i}\mathrm{e}^{-\mathrm{i} \varOmega_i t}, \quad q_i > 0, \quad i=1,\ldots,4,\\ \varOmega_i = \omega_i - T_{ii} q_i + 2\sum_{j=1}^4 T_{ij} q_j + \frac{2T}{q_i}\sqrt{q_1q_2q_3q_4} \cos p_0,\\ p_0 = \varphi_1 + \varphi_ 2 - \varphi_3 - \varphi_ 4 = 0 \mbox{ or } {\rm \pi}, \end{gathered} \right\} \end{equation}

and satisfying the compatibility condition (3.5) is a steady solution of (1.1) with (1.2).

3.2. Linear stability

Analogy with Newtonian dynamics allows us to formulate a simple criterion for linear stability of steady equilibria. Indeed, (3.1) yields, after linearization around $q=0$,

(3.7)\begin{equation} \ddot q - \gamma^2 q = 0, \quad \gamma^2 = {\tfrac{1}{2}} f''(0) =- U''(0), \end{equation}

where from (2.20) with (2.21) and (2.22), we have $\gamma ^2 = c$. Therefore, the null solution is linearly stable if $c < 0$ and unstable otherwise; growth is exponential if $c > 0$ or algebraic if $c = 0$.

Consider first the linear stability of the bichromatic solution (3.3) for which $q_3 = q_4 = 0$. In that case, $c=16T^2 q_1 q_2 - B^2$, where $B$ is defined in (2.9). Therefore, the bichromatic wavetrain (3.3) is exponentially unstable if

(3.8)\begin{equation} 16 T^2 q_1 q_2 > ( \Delta \omega + B_1 q_1 + B_2 q_2)^2. \end{equation}

Recalling that $q_i=|b_i|^2$ and that $B_i$ are defined in (2.9), we recover the criterion derived in Leblanc (Reference Leblanc2009, equation (15)) and recovered by Andrade & Stuhlmeier (Reference Andrade and Stuhlmeier2023a). We also recall that (3.8) characterizes respectively ‘type B’ and ‘class-Ia’ instabilities following the respective classifications of Okamura (Reference Okamura1984) for one-dimensional standing waves and of Ioualalen & Kharif (Reference Ioualalen and Kharif1994) for steady bichromatic wavetrains.

Turning now to the stability of steady wave quartets with non-zero initial wave actions $(q_1, q_2, q_3, q_4)$ satisfying (3.5) and $\cos p_0 = \pm 1$, the linear stability criterion becomes, after replacing $B$ by the right-hand side of (3.5) and $C$ by (2.19) onto the expression of $c$ in (2.21):

Theorem 3.2 The steady wave quartet (3.6) satisfying (3.5) is exponentially unstable if

(3.9)\begin{align} \frac{A}{2T} \frac{\cos p_0}{\sqrt{q_1 q_2 q_3 q_4}} > \frac{1}{4} \left( \frac{1}{q_1} + \frac{1}{q_2} - \frac{1}{q_3} - \frac{1}{q_4}\right)^2 + \frac{(q_1 + q_2)(q_3 + q_4) - q_1 q_2 - q_3 q_4 }{q_1 q_2 q_3 q_4}, \end{align}

algebraically unstable if equality holds, otherwise linearly stable.

3.3. The particular case of equal wave actions

Consider the particular case of the steady wave quartet with equal wave actions $q_1 = q_2 = q_3 = q_4 \equiv q_0>0$ and initial phase mismatch $p_0 = 0$ or ${\rm \pi}$. We recall that the existence of such steady interactions is conditioned by (3.5) which reduces in the present case to

(3.10)\begin{equation} \Delta \omega + q_0 \Delta B = 0, \end{equation}

where $\Delta B = B_1 + B_2 - B_3 - B_4$, i.e. from (2.9):

(3.11)\begin{equation} \Delta B = T_{11} + T_{22} - T_{33} - T_{44} + 4(T_{12} - T_{34}). \end{equation}

At resonance, (3.10) implies $\Delta B = 0$, while off resonance we get $q_0 = - \Delta \omega / \Delta B$. Since $q_0>0$, $\Delta \omega$ and $\Delta B$ must have opposite signs for the quartet to exist. The corresponding instability criterion may be easily deduced from (3.9). Therefore:

Criterion 3.1 Steady wave quartets (3.6) with $q_1 = q_2 = q_3 = q_4 \equiv q_0$ exist if $q_0=-\Delta \omega /\Delta B>0$. At resonance, they exist for any $q_0>0$ if $\Delta B = 0$. In both cases, they are exponentially unstable if

(3.12)\begin{equation} (A/T) \cos p_0 > 4 \quad (\,p_0=0 \mbox{ or } {\rm \pi}). \end{equation}

3.4. Lagrange theorem and Lyapunov stability

In his treatise on analytical mechanics published in 1788, Lagrange presented his famous principle on the stability of equilibrium positions, which was rigorously proved by Lejeune–Dirichlet in 1846 and generalized by Lyapunov in 1892 (see Loria & Panteley Reference Loria and Panteley2017). Lagrange theorem may be stated as (Gantmacher Reference Gantmacher1975, pp. 166–173; Arnold Reference Arnold1989, p. 99):

Lagrange theorem. If $U(0)=0$ is a strict local minimum of the potential $U(q)$, then the null solution of (2.18) is Lyapunov stable.

For our purpose, stability in the sense of Lyapunov is defined by:

Lyapunov stability. The null solution is Lyapunov stable if for each $\varepsilon > 0$ there exists $\delta > 0$ such that, if $|\dot q (0)| < \delta$ initially, then $\sup (|q(t)|, |\dot q (t)|) < \varepsilon$ for all $t\geq 0$.

In the present case, $U(q) = - {\tfrac {1}{2}}(a q^4 + b q^3 + c q^2 + d q + e)$. But, as stated previously, $q=0$ is a steady equilibrium if $U(0)=0$ and $U'(0)=0$, or equivalently $d=e=0$. Therefore, at equilibrium, $U(q) = - {\tfrac {1}{2}}q^2(a q^2 + b q + c)$. But since in that case $U''(0)=-c$, then $U(q)$ admits a strict local minimum at $q=0$ if $c<0$. Therefore in our case, a linearly stable equilibrium is also Lyapunov stable. This means that if $c<0$, any sufficiently small disturbance will remain bounded.

Of course it would be necessary to be precise about what is meant by ‘sufficiently small’ but we shall not pursue that direction for at least two reasons: firstly, because my study is restricted to the interactions inside a single quartet while interactions with other waves may be destabilizing (see e.g. Okamura Reference Okamura1985); secondly, because higher-order interactions are not taken into account (see e.g. Andrade & Stuhlmeier Reference Andrade and Stuhlmeier2023a). As a consequence, we have to keep in mind that stability is only indicative because limited to the discrete four-wave interactions considered in the present study.

By contrast, linear instability criteria are meaningful in the nonlinear regime as it is known that the existence of an exponentially growing solution of the linearized equation (3.7) implies instability of the null solution in the nonlinear equation (3.1) (see Verhulst Reference Verhulst1996, p. 88). Furthermore, if other interactions were taken into account, various instability mechanisms would compete without mutual cancellation.

Therefore, the various instability criteria presented in the present study have to be considered as sufficient conditions for instability.

4.1. General properties

We have seen in § 2 that solutions of (2.18) with (2.5) are bounded. Furthermore, following Inoue (Reference Inoue1975), we get from (2.18) the following inequalities:

(4.1)\begin{equation} \left. \begin{gathered} f(0) = 16T^2 q_1 q_2 q_3 q_4 \sin^2 p_0 \geq 0,\\ f(-q_1) \leq 0, \quad f(-q_2) \leq 0, \quad f(q_3) \leq 0, \quad f(q_4) \leq 0. \end{gathered} \right\} \end{equation}

Therefore we can conclude that, by continuity, $f$ admits at least two real roots, say $\xi _-$ and $\xi _+$, verifying $\xi _- \leq 0 \leq \xi _+$; $f$ is therefore a polynomial function of degree at least equal to two: either quartic, cubic or quadratic.

Excluding the case $\xi _- = \xi _+ = 0$ corresponding to root $0$ with multiplicity at least equal to 2 corresponding to a steady solution since $d=0$ and $e=0$ as explained in § 3.1, we restrict from now on our discussion to the cases where either $\xi _- \leq 0 < \xi _+$ or $\xi _- < 0 \leq \xi _+$. Without lost of generality, suppose that $\xi _-$ and $\xi _+$ are the closest roots from $0$. Equation (2.18) shows that unsteady solutions exist if $f(q) \geq 0$; since $q(0) = 0$ and $f(0) \geq 0$, Jeffreys & Jeffreys (Reference Jeffreys and Jeffreys1956, pp. 667–668), Bretherton (Reference Bretherton1964) and Pars (Reference Pars1965, pp. 4–6) showed that $\xi _- \leq q(t) \leq \xi _+$, $\forall t \geq 0$, and that the particle ‘velocity’ vanishes on the boundaries of this interval: $\dot q ( \xi _\pm ) = 0$. If $\xi _-$ and $\xi _+$ are both simple roots, they are turning points, i.e. the sign of $\dot q(t)$ changes on turning points and the direction of the particle is reversed; therefore $q(t)$ is periodic. If the multiplicity of either $\xi _-$ or $\xi _+$, say $\xi _-$, is strictly greater than one, then $\lim _{t\to \pm \infty } q(t) = \xi _-$ and the solution is a breather. Finally if $\xi _-$ and $\xi _+$ are both double roots, then either $\lim _{t\to \pm \infty } q(t) = \xi _\pm$ or $\lim _{t\to \pm \infty } q(t) = \xi _\mp$ and the solution is a pump. Finally, from (2.6), we have also

(4.2)\begin{equation} -\min(q_1,q_2) \leq \xi_- \leq q(t) \leq \xi_+ \leq \min (q_3,q_4). \end{equation}

Equation (2.18) with (2.5) may be integrated (see e.g. Craik Reference Craik1985, p. 138):

(4.3)\begin{equation} t \equiv t(q) ={\pm} \int_0^q \frac{{\rm d} \xi}{\sqrt{f(\xi)}} ={\pm} \int_0^q \frac{{\rm d} \xi}{\sqrt{-2 U(\xi)}} , \end{equation}

from which $q \equiv q(t)$ may formally be obtained by inversion. The sign indeterminacy above shows that for each fixed values of the parameters, (2.18) admits two solutions, say $\{ Q_- (t),\, Q_+ (t) \}$, such that $Q_-(t) = Q_+(-t)$. Now, let $Q_+$ be the solution such that at the initial time $\dot Q_+ (0) \geq 0$. Therefore, $\dot Q_-(0) = - \dot Q_+ (0) \leq 0$. The solution to choose is determined because of (2.7) from which we get

(4.4)\begin{equation} \dot q (0) =-4T \sqrt{q_1 q_2 q_3 q_4} \sin p_0. \end{equation}

Thus, if $-4T \sqrt {q_1 q_2 q_3 q_4} \sin p_0 \geq 0$, then $q(t)=Q_+ (t)$. Else $q(t)=Q_-(t)=Q_+(-t)$. Finally, if $q(t)$ is periodic, the period $\tau$ is

(4.5)\begin{equation} \tau = 2 \int_{\xi_-}^{\xi_+} \frac{{\rm d} \xi}{\sqrt{f(\xi)}} = 2 \int_{\xi_-}^{\xi_+} \frac{{\rm d} \xi}{\sqrt{-2 U(\xi)}} . \end{equation}

In the quartic case, $f$ in (2.20) with $a \not =0$ may be factorized as

(4.6)\begin{equation} f(q) = a (q-\xi_1)(q-\xi_2)(q-\xi_3)(q-\xi_4), \end{equation}

where $\xi _1$, $\xi _2$, $\xi _3$ and $\xi _4$ are the four roots of $f$. (The labelling of the roots $\xi _i$ is independent of the labelling of the initial wave actions $q_i$.) Since $f$ is real-valued, roots are either real or complex conjugate by pairs. The roots of a quartic polynomial may formally be obtained by quadrature with Ferrari's method (published by Cardan in 1545) but formulae, which are too lengthy to be reported here, are implemented in computer algebra systems. Simple expressions are given in Appendix A in the case $q_1=q_2=q_3=q_4$.

The nature of the solutions of the Bretherton equation (2.18) depends on the sign of $a$ and on the nature of the roots, as illustrated in figure 1 (see also Turner Reference Turner1980). If $a=0$, the potential is either cubic or quadratic and the solutions are postponed to Appendix B.

Figure 1. The possible configurations for bounded unsteady solutions of (2.18) satisfying (2.5). Motion occurs in the potential well defined by $q \in [\xi _-, \xi _+]$, where $\xi _\pm$ such that $\xi _- \leq 0 < \xi _+$ or $\xi _- < 0 \leq \xi _+$ are the nearest roots around zero between which $U(q) = - \tfrac {1}{2} f(q) \leq 0$. Quartic potential (4.6) with $a>0$: (a) periodic solution (4.8); (b) breather solution (4.14); (c) pump solution (4.16). Quartic potential (4.6) with $a<0$: (d) periodic solution (4.21); (e) breather solution (4.23); (f) rational breather solution (4.25); (g) periodic solution (4.27). Cubic potential (B1) with $b>0$: (h) periodic solution (B2); (i) breather solution (B4). Any other possibility may be deduced by symmetry with respect to the vertical axis. The case of quadratic potential with periodic solution (B6) has been omitted.

4.2. The case $a > 0$

4.2.1. Periodic solution

If $f$ admits four distinct real roots $\xi _1,\ldots, \xi _4$ such that (figure 1a)

(4.7)\begin{equation} \xi_1 < \xi_2 \leq 0 < \xi_3 < \xi_4 \quad \mbox{or} \quad \xi_1 < \xi_2 < 0 \leq \xi_3 < \xi_4, \end{equation}

then (2.18) with (2.5) has the pair of periodic solutions $\{ Q_{I}(t), Q_{I}(- t) \}$ with

(4.8)\begin{equation} \left. \begin{gathered} Q_{I} (t) = \frac{\xi_2 (\xi_3 - \xi_1) - \xi_1 (\xi_3-\xi_2)\,\mbox{sn}^2(u(t),k)}{(\xi_3-\xi_1)-(\xi_3-\xi_2) \,\mbox{sn}^2(u(t),k)},\quad u(t) = \frac{\gamma t}{2} + \mbox{sn}^{-1}(l,k) ,\\ \gamma = \sqrt{a (\xi_4-\xi_2)(\xi_3-\xi_1)}, \quad k = \sqrt{\frac{(\xi_3-\xi_2)(\xi_4-\xi_1)}{(\xi_4-\xi_2)(\xi_3-\xi_1)}}, \quad l = \sqrt{\frac{\xi_2 (\xi_3 - \xi_1)}{\xi_1 (\xi_3-\xi_2)}}, \end{gathered} \right\} \end{equation}

where $\mbox {sn}$ and $\mbox {sn}^{-1}$ are Jacobi elliptic functions defined here by (see e.g. Byrd & Friedman (Reference Byrd and Friedman1971, p. 18))

(4.9)\begin{equation} x = \mbox{sn}^{-1}(y,k) = \int_0^y \frac{{\rm d} \xi}{\sqrt{(1-\xi^2)(1-k^2 \xi^2)}}, \quad y = \mbox{sn} (x,k). \end{equation}

Period is

(4.10)\begin{equation} \tau = \frac{4}{\gamma} \,\mbox{K}(k), \quad \mbox{K}(k) = \int_0^{{\rm \pi}/2} \frac{{\rm d} \theta}{\sqrt{1-k^2 \sin^2 \theta}}, \end{equation}

where $\mbox {K}$ is the complete elliptic integral of the first kind. (Different conventions exist for the arguments of elliptic functions; we follow here the notations of Byrd & Friedman (Reference Byrd and Friedman1971). By contrast, in Wolfram Mathematica the entry for $\mbox {sn} (x, k)$ as defined in (4.9) is JacobiSN[x,m], where $m=k^2$; similarly for $\mbox {K}(k)$ defined in (4.10) for which the entry is EllipticK[m].)

Note that at the initial time

(4.11)\begin{equation} \dot Q_I (0) = \frac{\gamma \xi_1 \xi_2}{l (\xi_2 - \xi_1)} \,\mbox{cn} (\mbox{sn}^{-1}(l,k), k) \, \mbox{dn} (\mbox{sn}^{-1}(l,k), k), \end{equation}

where at fixed modulus $k$: $\mbox {cn}(x)= \sqrt {1-\mbox {sn}^2(x)}$ and $\mbox {dn}(x)= \sqrt {1-k^2 \,\mbox {sn}^2(x)}$ (Byrd & Friedman Reference Byrd and Friedman1971, p. 19). Therefore, at fixed $k$:

(4.12)\begin{equation} \mbox{cn} (\mbox{sn}^{-1}(y)) \, \mbox{dn} (\mbox{sn}^{-1}(y)) = \sqrt{1-y^2}\sqrt{1-k^2y^2}. \end{equation}

Then

(4.13)\begin{equation} \dot Q_I (0) = \frac{\gamma \xi_1 \xi_2}{l (\xi_2 - \xi_1)} \sqrt{1-l^2}\sqrt{1-k^2l^2} \geq 0. \end{equation}

Formula (254.00) in Byrd & Friedman (Reference Byrd and Friedman1971, p. 112) has been used to get (4.8), plotted in figure 2(a). Similar solutions were found by Inoue (Reference Inoue1975), Boyd & Turner (Reference Boyd and Turner1978), Shemer & Stiassnie (Reference Shemer and Stiassnie1985), Chen & Snyder (Reference Chen and Snyder1989), Cappellini & Trillo (Reference Cappellini and Trillo1991) and Stiassnie & Shemer (Reference Stiassnie and Shemer2005).

Figure 2. Unsteady solutions $q(t)$ in a quartic potential (4.6) with $a=1$ (a) or $a=-1$ (b). (a) Periodic solution (4.8) with $\xi _1=-1.2$, $\xi _2=-1$, $\xi _3=2$, $\xi _4=2.4$ (solid line); breather solution (4.14) with $\xi _{12}=-1$, $\xi _3=2$, $\xi _4=2.4$ (dotted line); pump solution (4.16) with $\xi _{12}=-1$, $\xi _{34}=2$ (dashed line). (b) Periodic solution (4.21) with $\xi _1=-1$, $\xi _2=2$, $\xi _3=2.1$, $\xi _4=2.2$ (solid line); breather solution (4.23) with $\xi _{1}=-1$, $\xi _{23}=2$, $\xi _4=2.2$ (dotted line); rational breather solution (4.25) with $\xi _{1}=-1$, $\xi _{234}=2$ (dashed line).

4.2.2. Breather solution

If $f$ admits a double real root $\xi _{12}$ and two distinct real roots $\xi _3$ and $\xi _4$ such that $\xi _{12} < 0 \leq \xi _3 < \xi _4$ (figure 1b), then (2.18) with (2.5) has the pair of breather solutions $\{ Q_{I\!I}(t), Q_{I\!I} (- t) \}$ with

(4.14)\begin{equation} \left. \begin{gathered} Q_{I\!I} (t) =\xi_{12} \frac{v^4(t) + 4 n^2 v^3(t) - 2 s v^2(t) + 4n^2 r v(t) + r^2}{ v^4(t) - 2 (8 \xi_{12}^2 n^2 + r) v^2(t) + r^2 } ,\\ v(t) = (\xi_{12} (\xi_3 + \xi_4) - 2 (\xi_3 \xi_4 - n \sqrt{\xi_3 \xi_4})) \exp(\sqrt{a}n t), \\ n = \sqrt{(\xi_3-\xi_{12})(\xi_4-\xi_{12})}, \quad r = \xi_{12}^2 (\xi_4 - \xi_3)^2, \\ s = 4 \xi_{12} (\xi_{12}^2 + \xi_3 \xi_4) (\xi_3 + \xi_4) - \xi_{12}^2 (3 \xi_3 + \xi_4) (\xi_3 + 3 \xi_4). \end{gathered} \right\} \end{equation}

At the initial time: $\dot Q_{I\!I} (0)= - \sqrt {a \xi _3 \xi _4} \xi _{12} \geq 0$. Note that from (4.1): $\xi _{12} = - q_1 = - q_2$.

Derived with the assistance of a computer algebra system and plotted in figure 2(a), solution (4.14) has not been found in the literature in this general form. However, if $\xi _3=0$, we get

(4.15)\begin{equation} Q_{I\!I} (t) =\xi_{12} - \frac{2n^2}{2 \xi_{12} - \xi_4 (1+ \cosh (\sqrt{a}n t))}, \quad n=\sqrt{-\xi_{12}(\xi_4-\xi_{12})}, \end{equation}

corresponding to expression (22a) in Inoue (Reference Inoue1975). The ‘pump-depletion’ solutions (A6) and (A7) in Cappellini & Trillo (Reference Cappellini and Trillo1991) are similar to (4.14), but correspond in their formulation to specific values of the roots $\xi _i$. Finally, connection between (4.14) and the ‘discrete Akhmediev breathers’ (5.4) and (5.15) in Andrade & Stuhlmeier (Reference Andrade and Stuhlmeier2023b) remains to be clarified since their solutions are given in implicit form, contrary to (4.14).

4.2.3. Pump solution

If $f$ admits two double real roots $\xi _{12}$ and $\xi _{34}$ such that $\xi _{12} < 0 < \xi _{34}$ (figure 1c), then (2.18) with (2.5) has the pair of pump solutions $\{ Q_{I\!I\!I} (t), \, Q_{I\!I\!I} (- t) \}$ with

(4.16)\begin{equation} Q_{I\!I\!I} (t) = \xi_{34} \frac{\mathrm{e}^{\mu t} - 1}{ \mathrm{e}^{\mu t} - m }, \quad \mu = \sqrt{a}(\xi_{34} - \xi_{12}) , \quad m = \frac{\xi_{34}}{\xi_{12}}. \end{equation}

At the initial time: $\dot Q_{I\!I\!I} (0) = - \sqrt {a} \xi _{12} \xi _{34} > 0$. The ‘shock-like’ solutions of Inoue (Reference Inoue1975) and the ‘multibreather’ solution of Andrade & Stuhlmeier (Reference Andrade and Stuhlmeier2023a,Reference Andrade and Stuhlmeierb) are equivalent to (4.16), plotted in figure 2(a).

From (4.1): $\xi _{12} = -q_1 = -q_2 = -q_{12}$ and $\xi _{34} = q_3 = q_4 = q_{34}$; from (4.6):

(4.17)\begin{equation} f(q)=a(q - \xi_{12})^2(q - \xi_{34})^2 = (16T^2 - A^2) (q + q_{12})^2(q - q_{34})^2. \end{equation}

Identification with (2.18) yields, excluding the case $A=0$, the following two compatibility conditions (Andrade & Stuhlmeier Reference Andrade and Stuhlmeier2023a):

(4.18)\begin{equation} \cos p_0 = \frac{A}{4T} \in [-1,1], \quad \Delta \omega + (B_1+B_2-A)q_{12} - (B_3+B_4-A)q_{34} = 0. \end{equation}

Finally, if $q_{12} = q_{34} = q_0>0$, the second condition in (4.18) gives $\Delta \omega + (\Delta B) q_0 = 0$, and (4.16) with $-\xi _{12} = \xi _{34} = q_0$ becomes simply

(4.19)\begin{equation} Q_{I\!I\!I} (t) = q_0 \tanh (\sqrt{a} q_0 t). \end{equation}

4.3. The case $a < 0$

4.3.1. Periodic solution (four real roots)

If $f$ admits four distinct real roots $\xi _1,\ldots, \xi _4$ such that (figure 1d)

(4.20)\begin{equation} \xi_1 \leq 0 < \xi_2 < \xi_3 \leq \xi_4 \quad \mbox{or} \quad \xi_1 < 0 \leq \xi_2 < \xi_3 \leq \xi_4, \end{equation}

then (2.18) with (2.5) has the pair of periodic solutions $\{ Q_{I\!V} (t), \, Q_{I\!V} (- t) \}$ with

(4.21)\begin{align} \left. \begin{gathered} Q_{I\!V} (t) = \frac{\xi_1 (\xi_4 - \xi_2) + \xi_4 (\xi_2-\xi_1)\,\mbox{sn}^2(u(t),k)}{(\xi_4-\xi_2)+(\xi_2-\xi_1) \,\mbox{sn}^2(u(t),k)},\quad u(t) = \frac{\gamma t}{2} + \mbox{sn}^{-1}(l, k) ,\\ \gamma = \sqrt{-a (\xi_4-\xi_2)(\xi_3-\xi_1)}, \quad k = \sqrt{\frac{(\xi_4-\xi_3)(\xi_2-\xi_1)}{(\xi_4-\xi_2)(\xi_3-\xi_1)}}, \quad l = \sqrt{-\frac{\xi_1 (\xi_4 - \xi_2)}{\xi_4 (\xi_2-\xi_1)}}. \end{gathered} \right\} \end{align}

Period is $\tau = 4 \gamma ^{-1}\mbox {K}(k)$. At the initial time:

(4.22)\begin{equation} \dot Q_{I\!V} (0) = \frac{\gamma l (\xi_2- \xi_1 ) \xi_4^2}{(\xi_4-\xi_1)(\xi_4 - \xi_2)} \sqrt{1-l^2}\sqrt{1-k^2l^2} \geq 0. \end{equation}

Formula (252.00) in Byrd & Friedman (Reference Byrd and Friedman1971, p. 103) has been used to get (4.21), plotted in figure 2(b). A similar solution is given by Chen (Reference Chen1989).

4.3.2. Breather solution

If $f$ admits a double real root $\xi _{23}$ and two distinct real roots $\xi _1$ and $\xi _4$ such that $\xi _{1} \leq 0 < \xi _{23} < \xi _4$ (figure 1e), then (2.18) with (2.5) has the pair of breather solutions $\{ Q_{V} (t), Q_{V} (- t) \}$ with

(4.23)\begin{equation} \left. \begin{gathered} Q_{V} (t) =\xi_{23} \frac{v^4(t) + 4 n^2 v^3(t) - 2 s v^2(t) + 4n^2 rv(t) + r^2}{ v^4(t) + 2 (8 \xi_{23}^2n^2 - r) v^2(t) +r^2 } ,\\ \displaystyle v(t) =- (n^2 + 2 n \sqrt{- \xi_1 \xi_4} + \xi_{23}^2 - \xi_1 \xi_4) \exp(\sqrt{-a}n t), \\ n = \sqrt{(\xi_{23}-\xi_1)(\xi_4-\xi_{23})}, \quad r = \xi_{23}^2 (\xi_4 - \xi_1)^2, \\ s = 4 \xi_{23} (\xi_{23}^2 + \xi_1 \xi_4) (\xi_1 + \xi_4) - \xi_{23}^2 (3 \xi_1 + \xi_4) (\xi_1 + 3 \xi_4). \end{gathered} \right\} \end{equation}

At the initial time: $\dot Q_{V} (0)= \sqrt {a \xi _1 \xi _4} \xi _{23} \geq 0$. Plotted in figure 2(b), solution (4.23) has not been found in the literature. If $\xi _1=0$, we get

(4.24)\begin{equation} Q_{V} (t) =\xi_{23} + \frac{2n^2}{2 \xi_{23} - \xi_4 (1+ \cosh (\sqrt{-a}n t))}, \quad n=\sqrt{\xi_{23}(\xi_4-\xi_{23})}. \end{equation}

4.3.3. Rational breather solution

If $f$ admits a triple real root $\xi _{234}$ and a single real root $\xi _1$ such that $\xi _{1} \leq 0 < \xi _{234}$ (figure 1f), then (2.18) with (2.5) has the pair of rational breather solutions $\{ Q_{V\!I}(t), \, Q_{V\!I} (- t) \}$ with

(4.25)\begin{equation} Q_{V\!I} (t) = \frac{\xi_{234} \mu t (\mu t + 4\;m )}{ \mu t( \mu t + 4 m) + 4 (1 + m^2)}, \quad \mu = \sqrt{-a} (\xi_{234} - \xi_1),\quad m = \sqrt{-\xi_1/\xi_{234}}. \end{equation}

At the initial time: $\dot Q_{V\!I} (0)= \sqrt {a \xi _1 \xi _{234}^3} \geq 0$. Plotted in figure 2(b), a solution equivalent to (4.25) is given by Turner (Reference Turner1980).

Finally, if $\xi _1=0$, (4.25) may be written as

(4.26)\begin{equation} \displaystyle Q_{V\!I} (t) = \xi_{234} \tilde Q(\tilde t), \quad \tilde Q (\tilde t) = \tilde t^2/(\tilde t^2 + 4), \quad \tilde t = \sqrt{-a} \xi_{234} t. \end{equation}

4.3.4. Periodic solution (two real roots and two complex conjugate roots)

Finally, if $f$ admits two distinct real roots $\xi _1$ and $\xi _2$ such that $\xi _1 \leq 0 < \xi _2$ or $\xi _1 < 0 \leq \xi _2$, and two complex conjugate roots $\xi _3$ and $\xi _4 = \xi _3^*$ (figure 1g), then (2.18) with (2.5) has the pair of periodic solutions $\{ Q_{V\!I\!I} (t), \, Q_{V\!I\!I} (- t) \}$ with

(4.27)\begin{align} \left. \begin{gathered} Q_{V\!I\!I} (t) = \frac{(\beta_1 \xi_2 + \beta_2 \xi_1) - (\beta_1 \xi_2 - \beta_2 \xi_1)\,\mbox{cn}(w(t),k)}{(\beta_1 + \beta_2) - (\beta_1 - \beta_2) \,\mbox{cn}(w(t),k)},\quad w(t) = \gamma t + \mbox{cn}^{-1}(l, k) ,\\ \gamma = \sqrt{- a \beta_1 \beta_2}, \quad k = \sqrt{\frac{(\xi_2-\xi_1)^2 - (\beta_2-\beta_1)^2}{4 \beta_1 \beta_2}}, \quad l = \frac{\beta_1 \xi_2 + \beta_2 \xi_1}{\beta_1 \xi_2 - \beta_2 \xi_1},\\ \beta_1 = \sqrt{(\xi_1 - x_3)^2 + y_3^2}, \quad \beta_2 = \sqrt{(\xi_2 - x_3)^2 + y_3^2},\quad x_3 = \mathrm{Re} (\xi_3), \quad y_3 = \mathrm{Im} (\xi_3). \end{gathered} \right\} \end{align}

Period is $\tau = 4 \gamma ^{-1} \mbox {K}(k)$. At the initial time:

(4.28)\begin{equation} \dot Q_{V\!I\!I} (0) =-\frac{a(\beta_1 \xi_2 - \beta_2 \xi_1)^2}{2\gamma(\xi_2-\xi_1)} \sqrt{1-l^2} \sqrt{1+k^2(l^2-1)} \geq 0, \end{equation}

since at fixed $k$, $\mbox {sn} (\mbox {cn}^{-1}(y))\,\mbox {dn} (\mbox {cn}^{-1}(y)) = \sqrt {1-y^2}\sqrt {1+k^2(y^2-1)}$.

Equation (259.00) in Byrd & Friedman (Reference Byrd and Friedman1971, p. 133) has been used to get (4.27). A similar solution is given by Turner (Reference Turner1980).

5.1. The truncated quartet model

Deep-water irrotational gravity waves propagating in an inviscid incompressible fluid are governed in spectral space at third order in amplitude by the Zakharov equation (Zakharov Reference Zakharov1966, Reference Zakharov1968; Krasitskii Reference Krasitskii1990, Reference Krasitskii1994):

(5.1)\begin{align} \mathrm{i} \frac{\partial {\mathcal{B}} (\boldsymbol{k})}{\partial t} = \omega(\boldsymbol{k}) {\mathcal{B}} (\boldsymbol{k}) + \int_{\mathbb{R}^{6}} T (\boldsymbol{k},\boldsymbol{p},\boldsymbol{q},\boldsymbol{r}) {\mathcal{B}}^*(\,\boldsymbol{p}) {\mathcal{B}}(\boldsymbol{q}) {\mathcal{B}}(\boldsymbol{r}) \delta(\boldsymbol{k}+\boldsymbol{p}-\boldsymbol{q}-\boldsymbol{r}) \,\mathrm{d} \boldsymbol{p} \,\mathrm{d} \boldsymbol{q} \,\mathrm{d} \boldsymbol{r}, \end{align}

where $\boldsymbol {k} \in \mathbb {R}^{2}$, $\omega (\boldsymbol {k})=\sqrt {g|\boldsymbol {k}|}$, $\delta (\boldsymbol {k})$ is Dirac delta function and the real-valued function $T(\boldsymbol {k},\boldsymbol {p},\boldsymbol {q},\boldsymbol {r})$ is Krasitskii's kernel given in Appendix C. At leading order, ${\mathcal {B}}(\boldsymbol {k},t)$ is related to the free-surface elevation $z=\eta (\boldsymbol {x},t)$, $\boldsymbol {x} \in \mathbb {R}^{2}$, by (we follow Janssen's (Reference Janssen2004, p. 132) convention for the Fourier transform, so that expressions given here for $T$ (and $T_{ij}$) must be divided by $(2{\rm \pi} )^2$ to recover those given in Krasitskii (Reference Krasitskii1994))

(5.2)\begin{equation} \eta (\boldsymbol{x},t)= \int_{\mathbb{R}^2} \hat \eta(\boldsymbol{k},t) \mathrm{e}^{\mathrm{i} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{x}}\mathrm{d} \boldsymbol{k}, \quad \hat \eta(\boldsymbol{k},t)=\sqrt{\frac{|\boldsymbol{k}|}{2\omega(\boldsymbol{k})}} ( {\mathcal{B}}(\boldsymbol{k},t)+ {\mathcal{B}}^*(-\boldsymbol{k},t) ). \end{equation}

According to Zakharov (Reference Zakharov1966, Reference Zakharov1968), an exact solution of (5.1) is the Stokes wave:

(5.3)\begin{align} {\mathcal{B}}(\boldsymbol{k},t) = b(t) \delta (\boldsymbol{k} - \boldsymbol{k}_0), \quad b(t)=b_0 \mathrm{e}^{-\mathrm{i} \varOmega_0 t}, \quad \varOmega_0 = \omega(\boldsymbol{k}_0) + T(\boldsymbol{k}_0,\boldsymbol{k}_0,\boldsymbol{k}_0,\boldsymbol{k}_0) |b_0|^2. \end{align}

If we now consider a linear combination of waves ${\mathcal {B}}(\boldsymbol {k},t) = \sum _{i=1}^N b_i(t) \delta (\boldsymbol {k} - \boldsymbol {k}_i)$ with $N>1$, the Zakharov equation (5.1) yields a system of ordinary differential equations which is not closed, as noticed by Okamura (Reference Okamura1985); it leads indeed to the generation of higher harmonics on time scales of order $(|T||b|^2)^{-1}$. The mathematical validity of such an ansatz is therefore an open question. (This was pointed to me out by an anonymous reviewer even for $N=2$; see also discussion in Badulin et al. (Reference Badulin, Shrira, Kharif and Ioualalen1995). Zakharov (Reference Zakharov1967) already noticed that the bichromatic wave (3.3) is an approximate solution.)

If the terms corresponding to higher harmonics are neglected, one gets a truncated model consisting of a closed system of ordinary differential equations considered in various textbooks and review articles (e.g. Yuen & Lake Reference Yuen and Lake1982; Craik Reference Craik1985; Shemer & Stiassnie Reference Shemer and Stiassnie1991; Janssen Reference Janssen2004; Kartashova Reference Kartashova2010) and implicitly or explicitly used in a number of articles, including those by Saffman & Yuen (Reference Saffman and Yuen1980), Caponi, Saffman & Yuen (Reference Caponi, Saffman and Yuen1982), Okamura (Reference Okamura1984, Reference Okamura1985), Shemer & Stiassnie (Reference Shemer and Stiassnie1985), Hogan et al. (Reference Hogan, Gruman and Stiassnie1988), Badulin et al. (Reference Badulin, Shrira, Kharif and Ioualalen1995), Stiassnie & Shemer (Reference Stiassnie and Shemer2005), Leblanc (Reference Leblanc2009) and Andrade & Stuhlmeier (Reference Andrade and Stuhlmeier2023a,Reference Andrade and Stuhlmeierb):

(5.4)\begin{equation} \mathrm{i} \frac{\mathrm{d} b_i}{\mathrm{d} t} = \omega_i b_i + \sum_{j,m,n=1}^N \hat T_{ijmn} b_j^* b_m b_n , \quad i=1,\ldots, N, \end{equation}

where $\omega _i =\sqrt {g|\boldsymbol {k}_i|}$ and $\hat T_{ijmn} = T(\boldsymbol {k}_i,\boldsymbol {k}_j,\boldsymbol {k}_m,\boldsymbol {k}_n)$ if $\boldsymbol {k}_i+\boldsymbol {k}_j=\boldsymbol {k}_m+\boldsymbol {k}_n$ and $0$ otherwise. In the case of four waves satisfying (1.3), (5.4) reduces to (1.1) where $T \equiv T(\boldsymbol {k}_1,\boldsymbol {k}_2,\boldsymbol {k}_3,\boldsymbol {k}_4)$ and $T_{ij} = T(\boldsymbol {k}_i,\boldsymbol {k}_j,\boldsymbol {k}_i,\boldsymbol {k}_j)$. The ‘free-surface elevation’ of this truncated low-order model corresponding to an ‘isolated’ quartet would be

(5.5)\begin{equation} \eta_{quartet} (\boldsymbol{x},t) = \sum_{i=1}^4 \sqrt{\frac{|\boldsymbol{k}_i|}{2\omega_i}} \left( b_i(t) \mathrm{e}^{\mathrm{i} \boldsymbol{k}_i \boldsymbol{\cdot} \boldsymbol{x}} + b_i^*(t) \mathrm{e}^{-\mathrm{i} \boldsymbol{k}_i \boldsymbol{\cdot} \boldsymbol{x}} \right)\!. \end{equation}

Although the formal validity of such a model with respect to actual solutions of the Zakharov equation (5.1) is an open question, numerical and experimental results support its usefulness (Liu & Liao Reference Liu and Liao2014; Liu et al. Reference Liu, Xu, Li, Peng, Alsaedi and Liao2015; Liao et al. Reference Liao, Xu and Stiassnie2016). If the quartet is resonant, Benney's equations (1.1) obtained with the method of multiple scales are recovered.

5.2. Bidirectional standing waves and their stability

A particular case of interaction (1.3) concerns bidirectional standing waves for which

(5.6)\begin{equation} (\boldsymbol{k}_1,\boldsymbol{k}_2,\boldsymbol{k}_3,\boldsymbol{k}_4) = (\boldsymbol{k}_1,-\boldsymbol{k}_1,\boldsymbol{k}_3,-\boldsymbol{k}_3), \quad \boldsymbol{k}_1 \not= \boldsymbol{k}_3. \end{equation}

For simplicity, we consider the case where initially $q_1=q_2=q_3=q_4 \equiv q_0 >0$. From criterion 3.1 (§ 3.3), the wave quartet is steady providing that $q_0=-\Delta \omega / \Delta B >0$, where $\Delta B$ is given in (3.11). For deep water, we have (see Appendix C)

(5.7)\begin{equation} T_{11}=T_{22}=k_1^3, \quad T_{33}=T_{44}=k_3^3, \quad T_{12}=-k_1^3, \quad T_{34}=-k_3^3, \end{equation}

so that $\Delta B = - 2(k_1^3-k_3^3)$ and $-\Delta \omega / \Delta B = (\omega _1/k_1^3) \rho (k_3/k_1)$, where the function $\rho (\kappa ) = (1-\sqrt {\kappa })/(1-\kappa ^3)$ is strictly positive for any $\kappa >0$ (the discontinuity at $\kappa =1$ may be removed since $\rho (\kappa ) \to 1/6$ when $\kappa \to 1$). Therefore $q_0 = (\omega _1-\omega _3)/(k_1^3-k_3^3)$ is defined and strictly positive if $k_1 \not = k_3$. If $k_1=k_3$, then both $\Delta \omega =0$ and $\Delta B=0$ so that, from criterion 3.1, $q_0>0$ may be chosen arbitrarily.

Now, from (3.6):

(5.8)\begin{equation} \left. \begin{gathered} \varOmega_1 = \varOmega_2 = \omega_1 + (2 \bar{T} - k_1^3)q_0, \quad \varOmega_3 = \varOmega_4 = \omega_3 + (2 \bar{T} - k_3^3)q_0,\\ \bar{T} = T(\boldsymbol{k}_1,\boldsymbol{k}_3,\boldsymbol{k}_1,\boldsymbol{k}_3) + T(\boldsymbol{k}_1,-\boldsymbol{k}_3,\boldsymbol{k}_1,-\boldsymbol{k}_3) + T(\boldsymbol{k}_1,-\boldsymbol{k}_1,\boldsymbol{k}_3,-\boldsymbol{k}_3) \cos p_0, \end{gathered} \right\} \end{equation}

where $\bar T$ may be evaluated explicitly thanks to the expressions given in Appendix C. Choosing $\varphi _1 = \varphi _ 2 = 0$ and $\varphi _3 = \varphi _ 4 = -p_0/2$, it may be shown from (5.5) that the free-surface elevation is, at leading order,

(5.9)\begin{align} \left. \begin{gathered} \eta_{quartet}(\boldsymbol{x},t) = 2 {\mathcal{A}}_1\cos (\boldsymbol{k}_1 \boldsymbol{\cdot} \boldsymbol{x}) \cos(\varOmega_1 t) + 2 {\mathcal{A}}_3 \cos (\boldsymbol{k}_3 \boldsymbol{\cdot} \boldsymbol{x}) \cos\left(\varOmega_3 t + \frac{p_0}{2} \right),\\ {\mathcal{A}}_i = \sqrt{ \frac{2k_i q_0}{\omega_i}}, \quad i=1,3, \quad q_0 = \frac{\omega_1-\omega_3}{k_1^3-k_3^3},\quad p_0 = 0 \mbox{ or } {\rm \pi}. \end{gathered} \right\} \end{align}

Free surface (5.9) is periodic if $\varOmega _1$ and $\varOmega _3$ are commensurate. If $k_1 = k_3 \equiv k_0$ and $p_0=0$, we recover the bidirectional standing waves studied by Okamura (Reference Okamura1985), since in that case $\varOmega _1= \varOmega _3 = \omega _0 + (2 \bar T - k_0^3) q_0$, where $q_0>0$.

Let us now apply the instability criterion 3.1 (§ 3.3). Using homogeneity property (C4), we can define, without lost of generality, a function $\chi (\kappa,\psi )$ such that

(5.10)\begin{equation} \left. \begin{gathered} \frac{A(\boldsymbol{k}_1, - \boldsymbol{k}_1, \boldsymbol{k}_3 , -\boldsymbol{k}_3 )}{T(\boldsymbol{k}_1, - \boldsymbol{k}_1, \boldsymbol{k}_3 , -\boldsymbol{k}_3 )} = \frac{A(\boldsymbol{i}, - \boldsymbol{i}, \boldsymbol{\kappa} , -\boldsymbol{\kappa} )}{T(\boldsymbol{i}, - \boldsymbol{i}, \boldsymbol{\kappa} , -\boldsymbol{\kappa} )} \equiv 4 \chi (\kappa,\psi),\\ \boldsymbol{i} = (1,0), \quad \boldsymbol{\kappa} = (\kappa \cos \psi, \kappa \sin \psi),\quad \kappa = k_3/k_1. \end{gathered} \right\} \end{equation}

Since $\psi = \mbox {angle} (\boldsymbol {i},\boldsymbol {\kappa }) = \mbox {angle} (\boldsymbol {k}_1,\boldsymbol {k}_3)$, it is sufficient to consider $\psi \in [0,{\rm \pi} /2]$. Function $\chi (\kappa,\psi )$ is plotted on figure 3; since $\chi (\kappa,\psi ) > 0$, there is no instability according to (3.12) for $p_0 = {\rm \pi}$. On the contrary, if $p_0=0$, standing waves are unstable in regions where $\chi (\kappa,\psi ) > 1$, that is, when $0<\kappa <1/\kappa _c$ or $\kappa > \kappa _c$, where $\kappa _c \equiv \kappa _c(\psi )>1$ is such that $\chi (\kappa _c(\psi ),\psi )=1$. It is found numerically that $1.730<\kappa _c(\psi ) <1.861$, $\forall \psi \in [0,{\rm \pi} /2]$. Therefore, our results support the following statement:

Figure 3. Graph of $\chi (\kappa,\psi )$ defined in (5.10) for $\psi = 0$ (solid line); $\psi = {\rm \pi}/8$ (dotted line); $\psi = {\rm \pi}/2$ (dashed line). Steady bidirectional standing waves (5.9) with $p_0=0$ are exponentially unstable if $\chi (\kappa,\psi ) >1$, where $\kappa =k_3/k_1$ and $\psi = \mbox {angle} (\boldsymbol {k}_1,\boldsymbol {k}_3)$.

Theoretical predictions have been compared with numerical integration of (1.1) and results are in excellent agreement. A representative example is plotted in figure 4. Of course, the unsteady perturbed solutions that appear on this plot may be expressed with one of the explicit formulas given in § 4. To this aim, we first determine in each case the function $f$ in (2.20). From the data given in the caption of figure 4, we get $a<0$ so that $f$ may be factorized as (4.6), where the four roots $\xi _i$, $i=1,\ldots,4$, may be explicitly evaluated thanks to the expressions given in Appendix A since in both cases $q_1=q_2=q_3=q_4=q_0(1+\varepsilon )$; their numerical values are reported in table 1. Therefore, because of the Manley–Rowe relations (2.4), the unsteady periodic solutions plotted in figure 4 are $|b_1(t)|^2 =q_1 + q(t)$, where the functions $q(t)$ are given by (4.21) for $p_0=0$ (dashed line) and by (4.27) for $p_0={\rm \pi}$ (solid line). Their respective periods are given in table 1. Their graphs cannot be distinguished from their numerical counterparts, as the relative errors between numerical and exact solutions is of the order of $0.1\,\%$ at the final time of the computations.

Figure 4. Numerical integration of system (1.1) with $g=1$, $\boldsymbol {k}_1=-\boldsymbol {k}_2=(1,0)$, $\boldsymbol {k}_3=-\boldsymbol {k}_4=(0,2)$, $q_i=q_0(1+\varepsilon )$ with $q_0=(\omega _1-\omega _3)/(k_1^3-k_3^3)$ and $\varphi _1 = \varphi _2=0$, $\varphi _3= \varphi _4=-p_0/2$. Dotted line ($\varepsilon =0$, $p_0={\rm \pi}$): steady bidirectional standing waves (5.9). Dashed line ($\varepsilon =0.01$, $p_0=0$): unstable disturbance. Solid line ($\varepsilon =0.01$, $p_0={\rm \pi}$): stable disturbance.

Table 1. Truncated values of the roots $\xi _i$ of the quartic function (4.6) with $a=-25.57$ corresponding to the parameters of the unsteady periodic solutions represented in figure 4 for $\varepsilon =0.01$. Their respective exact solutions $q(t)$ are (4.21) for $p_0=0$ and (4.27) for $p_0={\rm \pi}$; their crest–trough amplitude is $\xi _2 - \xi _1$ and their period is $\tau$.

Finally, the free-surface elevation of these bifurcated solutions may be written as

(5.11)\begin{align} \left. \begin{gathered} \eta_{quartet}(\boldsymbol{x},t) = 2 {\mathcal{A}}_1 \eta_1(t) \cos (\boldsymbol{k}_1 \boldsymbol{\cdot} \boldsymbol{x}) \cos p_1(t) + 2 {\mathcal{A}}_3 \eta_3(t) \cos (\boldsymbol{k}_3 \boldsymbol{\cdot} \boldsymbol{x}) \cos p_3(t),\\ \eta_1(t) = \sqrt{1+\varepsilon + q(t)/q_0}, \quad \eta_3(t) = \sqrt{1+\varepsilon - q(t)/q_0}, \end{gathered} \right\} \end{align}

where ${\mathcal {A}}_i$ and $q_0$ are defined in (5.9) and where the phases $p_i(t)$ are integrated numerically from (2.3). Results are plotted in figure 5: they show in the unstable case (figure 5b) five large-scale beats over ten periods $\tau$ of the corresponding function $q(t)$. This is due to the fact that the amplitudes $|b_i(t)|$ involve the square root of $q(t)$; see (2.4).

Figure 5. Free-surface elevation (5.11) at $\boldsymbol {x} = (0,0)$ for $\varepsilon =0.01$. Stable disturbance $p_0={\rm \pi}$ (a); unstable disturbance $p_0=0$ (b). Data are those of figure 4 and table 1.

5.3. Steady wave quartets and their stability

We now turn to the general case (1.3), on or off resonance. Consider first Kartashova's resonant quartet (Lvov, Nazarenko & Pokorni Reference Lvov, Nazarenko and Pokorni2006; Kartashova Reference Kartashova2010, p. 78):

(5.12)\begin{equation} \boldsymbol{K}_1= (495, 90), \quad \boldsymbol{K}_2 = (64, 128), \quad \boldsymbol{K}_3 = (359, 118), \quad \boldsymbol{K}_4 = (200, 100), \end{equation}

useful for symbolic computation. Since $\Delta \omega = 0$ and $\Delta B \not = 0$ in that case, criterion 3.1 (§ 3.3) is of no use to construct a steady solution; we need another strategy. Testing $q_1=q_2\equiv q_{12}$ and $q_3=q_4 \equiv q_{34}$, the compatibility condition (3.5) with $\Delta \omega =0$ becomes

(5.13)\begin{equation} \frac{q_{34}}{q_{12}} = \frac{4T - (B_1+B_2)\cos p_0}{4T - (B_3+B_4)\cos p_0} \approx \left\{ \begin{array}{@{}ll} 0.8546, & \mbox{if} \ p_0=0,\\ 0.5857, & \mbox{if} \ p_0= {\rm \pi}. \end{array} \right. \end{equation}

Since $q_{34}/q_{12}>0$ in both cases, steady states exist. It may be checked that the instability condition (3.9) becomes also (3.12) when $q_1=q_2\equiv q_{12}$ and $q_3=q_4 \equiv q_{34}$. Since $A/T \approx -7.430< -4$ for Kartashova's resonant quartet (5.12) or for any rescaled non-trivial resonant quartet $(\lambda \boldsymbol {K}_1,\lambda \boldsymbol {K}_2,\lambda \boldsymbol {K}_3,\lambda \boldsymbol {K}_4)$ according to (C4), we conclude immediately that the corresponding steady wave quartet is stable if $p_0=0$ and unstable if $p_0={\rm \pi}$. This method may be employed for any other resonant quartet provided that the ratio $q_{34}/q_{12}$ is positive. Therefore in summary, we have proved:

Criterion 5.1 Steady resonant wave quartets (3.6) with $q_1=q_2 \equiv q_{12}$ and $q_3=q_4 \equiv q_{34}$ exist if

(5.14)\begin{equation} \frac{q_{34}}{q_{12}} = \frac{4T - (B_1+B_2)\cos p_0}{4T - (B_3+B_4)\cos p_0} > 0 \quad (\,p_0= 0 \mbox{ or } {\rm \pi}). \end{equation}

They are exponentially unstable if $(A/T) \cos p_0 > 4$.

(A companion criterion is given in Appendix D for $q_1=q_3$ and $q_2=q_4$.)

Unstable disturbed solutions can lie on the same resonant surface (figure 6a), or not. For instance, if we consider the following non-resonant (but nearly resonant) quartet:

(5.15)\begin{equation} \boldsymbol{k}_1= (500, 90), \quad \boldsymbol{k}_2 = (60, 130), \quad \boldsymbol{k}_3 = (360, 120), \quad \boldsymbol{k}_4 = (200, 100), \end{equation}

the corresponding unsteady solution is plotted in figure 6(b), showing again instability.

Figure 6. Wave action of steady Kartashova's resonant quartet $(\lambda \boldsymbol {K}_1,\lambda \boldsymbol {K}_2,\lambda \boldsymbol {K}_3,\lambda \boldsymbol {K}_4)$ given by (5.12), $\lambda =1/300$ and $p_0={\rm \pi}$ (dotted lines) versus unstable solutions according to criterion 5.1 (solid lines). Initial wave actions are in each case $q_{12}\approx 1.380$ and $q_{34} \approx 0.8084$. (a) Resonant solution with perturbed initial phase mismatch $p_0=1.05{\rm \pi}$ and same wave vectors. (b) Non-resonant solution with same phase mismatch $p_0={\rm \pi}$ and perturbed wave vectors $(\lambda \boldsymbol {k}_1,\lambda \boldsymbol {k}_2,\lambda \boldsymbol {k}_3,\lambda \boldsymbol {k}_4)$ given by (5.15) and $\lambda =1/300$. Solutions are $|b_1(t)|^2 = q_1+q(t)$ with $q(t)$ given by (a) (4.27) and (b) (4.21).

Let us now construct a steady solution for the non-resonant quartet (5.15). We first test criterion 3.1 (§ 3.3), but since in that case $\Delta \omega / \Delta B > 0$, a steady solution with $q_1 = q_2 = q_3 =q_4$ does not exist; we need one more time another strategy. We are, however, lucky with (5.15) because all the $B_i$ defined in (2.9) have the same sign: negative. Thus we can set $q_i = \alpha /B_i$ expecting $\alpha <0$ for existence. From (3.5) we get

(5.16)\begin{equation} \alpha =- \frac{\sqrt{B_1 B_2 B_3 B_4}\Delta \omega}{2T\Delta B} \cos p_0 \quad (\,p_0 = 0 \mbox{ or } {\rm \pi}). \end{equation}

Since $T>0$ for (5.15), we conclude that the only possibility to have $\alpha <0$ is $p_0=0$. Concerning stability, condition (3.9) with $q_i=\alpha /B_i$ becomes

(5.17)\begin{equation} \frac{A}{2T} \sqrt{B_1 B_2 B_3 B_4} \cos p_0 > \frac{(\Delta B)^2}{4} +(B_1 + B_2)(B_3 + B_4) - B_1 B_2 - B_3 B_4. \end{equation}

Applying this condition for quartet (5.15) and $p_0=0$ gives stability.

The method presented here may be applied to any non-resonant quartet providing that all the $B_i$ have the same sign:

Other strategies might be used to construct steady non-resonant quartets. For instance we can choose $q_1=q_2$ and $q_3=q_4$, or $q_1=q_3$ and $q_2=q_4$. In these cases, compatibility condition (3.5) would yield affine relations between $q_i$, but conditions for exponential instability resulting from (3.9) would involve $q_i$, contrary to the instability conditions given in criteria 1 to 4. We shall not pursue this issue here.

7.1. The $\varPsi$-breathers

Looking for breathers is a difficult task because two or three roots of the quartic function (4.6) have to coincide in a non-symmetric way, as compared with the X-pump solution (4.19) for which the roots were equal by pairs and opposite.

Fortunately, one branch of such breathers bifurcates from the X-pump solution studied previously. Indeed, consider the one-parameter family of resonant quartet:

(7.1)\begin{equation} \left. \begin{gathered} \boldsymbol{k}_1^{\varepsilon}= (\alpha_+, 0), \quad \boldsymbol{k}_2^{\varepsilon} = (-\alpha_-, 0), \quad \boldsymbol{k}_3^{\varepsilon} = (\beta, \gamma), \quad \boldsymbol{k}_4^{\varepsilon} = (\beta, -\gamma),\\ \alpha_\pm= k_0 \left(1 \pm \tfrac{1}{2} \varepsilon + \tfrac{1}{4} \varepsilon^2\right)^2, \quad \beta = k_0 \varepsilon\left(1 + \tfrac{1}{4} \varepsilon^2\right), \quad \gamma = k_0 \left(1 - \tfrac{1}{16} \varepsilon^4\right), \end{gathered} \right\} \end{equation}

where $\varepsilon \geq 0$, with $\varepsilon \not =2$ to avoid $\boldsymbol {k}_3^{\varepsilon }=\boldsymbol {k}_4^{\varepsilon }$, is the main parameter ($k_0 >0$ is just a scaling parameter). Quartet (7.1) is a member of the ‘tridents’ (D2) proposed by Lvov et al. (Reference Lvov, Nazarenko and Pokorni2006), with $(m,n) = (1,\varepsilon /2)$. The X-quartet (6.1) is matched for $\varepsilon =0$.

With (7.1), it is found numerically that $T<0$, $A<0$ and $\Delta B>0$ for any positive $\varepsilon \not =2$. Furthermore, $4T-A>0$ (then $a<0$) for any $\varepsilon \not =2$ such that $\varepsilon _1 < \varepsilon < \varepsilon _2$, and $4T-A<0$ (then $a>0$) otherwise, where $\varepsilon _1 \approx 0.4672$ and $\varepsilon _2 \approx 8.562$. Therefore, we may expect to find breathers of the kind (4.14) for $0<\varepsilon < \varepsilon _1$ (because $a>0$), and those of the kind (4.23) or (4.25) for $\varepsilon _1 < \varepsilon < \varepsilon _2$, $\varepsilon \not =2$ (because $a<0$). The case $\varepsilon = \varepsilon _1$ (then $a=0$) will be considered later.

To construct breathers (4.14) bifurcating from the X-pump (6.6), we consider $0 < \varepsilon < \varepsilon _1$ (so that $a>0$) and choose again initially $q_1=q_2=q_3=q_4 \equiv q_0>0$. To do so, two roots have to coincide, so we impose arbitrarily the following constraint for the negative roots: $\xi _{12} = \xi _1 = \xi _2 = - q_0<0$, while $0 \leq \xi _3 < \xi _4$. Since the initial wave actions are equal, the four roots are given explicitly by (A2), here with

(7.2)\begin{equation} \Delta \omega=0, \quad \Delta B >0, \quad T<0, \quad 4T\pm A <0, \end{equation}

since $0 < \varepsilon < \varepsilon _1$. Roots are therefore real and such that $\xi _-^{(-)} < 0 < \xi _-^{(+)}$. Since the two negative roots must coincide with $-q_0$, we get a first compatibility condition:

(7.3)\begin{equation} \sqrt{(\Delta B)^2 + 16T(4T-A)(1-\cos p_0)} = 2(A-4T) - \Delta B. \end{equation}

Positiveness of the right-hand side is ensured when $0<\varepsilon \leq \varepsilon _3$, with $\varepsilon _3 \approx 0.3705$. In that range, on squaring the above equality, we get $\cos p_0 = (A- \Delta B)/(4T) \geq 0$. Furthermore, it may be checked numerically that the constraint $\cos p_0 \leq 1$ is ensured providing that $0<\varepsilon \leq \varepsilon _4$, with $\varepsilon _4 \approx 0.3213$. Therefore, in that range, we may choose

(7.4)\begin{equation} p_0 = \arccos \left( \frac{A-\Delta B}{4T} \right)\!. \end{equation}

With (7.4), the four roots are therefore $\xi _{12}= \xi _\pm ^{(\pm )}$, $\xi _{3}= \xi _+^{(-)}$, $\xi _4= \xi _-^{(+)}$:

(7.5)\begin{equation} \xi_{12}=-q_0, \quad \xi_3 = \left( 1 - \frac{\Delta B}{A-4T} \right) q_0, \quad \xi_4 = \left( 1 - \frac{\Delta B}{A+4T} \right) q_0. \end{equation}

Note that $p_0=0$, $\xi _3=0$ and $\xi _4/q_0 = 8T/(A+4T)$ when $\varepsilon = \varepsilon _4 \approx 0.3213$, which corresponds to the end point of this branch of solutions.

Together with $a=(4T+A)(4T-A)$, roots (7.5) completely determine the solution (4.14), referred to as the ‘$\varPsi$-breathers’ represented in figure 10 for various values of $\varepsilon$. Note that $\Delta B = 0$ for $\varepsilon =0$, so that (7.4) matches with (6.4) obtained for the X-pump solution. Expansion near $\varepsilon = 0$ gives, up to the second order:

(7.6)\begin{equation} \left. \begin{gathered} \frac{\xi_{12}}{q_0} =-1,\quad \frac{\xi_3}{q_0} = 1 - \frac{5}{46}(17 + 8 \sqrt{2}) \varepsilon^2 , \quad \frac{\xi_4}{q_0} = 1 + \frac{5}{142}(25- 8 \sqrt{2}) \varepsilon^2 ,\\ \frac{a}{k_0^6} = \frac{1}{49}(297 -64 \sqrt{2}) - \frac{8}{7}(17- 8 \sqrt{2})\varepsilon +\frac{1}{9604}(35\,017 - 86\,052 \sqrt{2}) \varepsilon^2 . \end{gathered} \right\} \end{equation}

It is interesting for the $\varPsi$-breathers to reconstruct the free-surface elevation. Since the relative $q(t)$ is given explicitly by (4.14), the relative phase $p(t)$ defined in (2.2) may be deduced from (2.7) and plotted in figure 10. Then, the four individual moduli $|b_i(t)|$ are deduced from the Manley–Rowe equation (2.4), while numerical integration of (2.3) is necessary to obtain the individual phases $p_i(t)=\arg b_i(t)$ in order reconstruct the free-surface elevation given by (5.5). This is achieved in figure 11 for the $\varPsi$-breathers defined by (7.1), (7.4) and (7.5) for different values of the parameter $\varepsilon$ that quantifies the departure from the X-pump solution (6.6). Is is clear from figure 11 that the sole considerations of $q(t)$ and $p(t)$ represented in figure 10 cannot explain the local behaviour of $\eta _{quartet}(0,t)$ and that phase mixing must be invoked to explain such destructive/constructive interferences (the ${\rm \pi} /2$ relative phase jump is also worth noting). Whether this kind of behaviour may be related or not to the occurrence of extreme events is left for future investigation.

Figure 10. The $\varPsi$-breathers corresponding to quartets (7.1) with $k_0=1$, $q_0=0.01$ and $p_0$ given by (7.4), for $\varepsilon = 0.2$ (dotted line), $\varepsilon =0.1$ (dot-dashed line), $\varepsilon =0.05$ (dashed line) and $\varepsilon =0.01$ (solid line); the grey line corresponds to the X-pump (4.19) for which $\varepsilon =0$. (a) Relative action $q(t)$. (b) Relative phase $p(t)$.

Figure 11. Free-surface elevation (5.5) at $\boldsymbol {x} = (0,0)$ for the $\varPsi$-breathers (7.1) with (7.4) for $\varepsilon = 0.01$ (a), $\varepsilon = 0.05$ (b), $\varepsilon = 0.1$ (c) and $\varepsilon = 0.2$ (d). Computations have been carried out backward and forward from $t=0$, with $g=1$, $k_0=1$ and $q_0=0.01$.

Finally, if $\varepsilon = \varepsilon _1$ or $\varepsilon = \varepsilon _2$, then $A=4T$ and $a=0$, so that $f$ in (2.18) becomes cubic and reads as (B1), where here $b=-2A (\Delta B) q_0 >0$. It may be deduced from the considerations of Appendix B in that case that if we constrain the positive roots to coincide, i.e. $\xi _{23} \equiv \xi _2 = \xi _3 = q_0$, then we get the following compatibility conditions:

(7.7)\begin{equation} \cos p_0 = 1 + \frac{\Delta B}{A} \in [-1,1], \quad \xi_1 =- \left( 1 + \frac{\Delta B}{2 A} \right)q_0 \leq 0. \end{equation}

For both $\varepsilon = \varepsilon _1$ and $\varepsilon = \varepsilon _2$ we have the same numerical value for $\Delta B / A \approx -0.8487$; both conditions above are fulfilled and the corresponding solution is breather (B4).

7.2. Looking for rational breathers

Since $a<0$ when $\varepsilon \not = 2$ lies in the range $\varepsilon _1 < \varepsilon < \varepsilon _2$, we expect to find rational breathers (4.25). Suppose that the triple root which characterizes this solution is positive, $\xi _2=\xi _3=\xi _4 \equiv \xi _{234}>0$, so that $\xi _1 \leq 0$. Now, contrary to the case $a>0$ for which the multiple roots must coincide (in absolute value) with the initial wave actions because of (4.1), roots satisfy in the present case the following inequalities:

(7.8)\begin{equation} - q_0 \leq \xi_1 \leq 0 < \xi_{234} \leq q_0. \end{equation}

However, no rational breather has been found for the resonant trident (7.1).

Thus, constraints imposed by the coincidence of three roots of the quartic function $f$ make the search for rational breathers even more difficult than for other breathers. For reasons that will be made clear later, still for the quartet (7.1), we consider the point in parameter space where $A=12T$. It is found numerically that it corresponds to $\varepsilon = \varepsilon _0$, where $\varepsilon _0 \approx 0.7475$. Now, from this point of bifurcation, we consider the following family of non-resonant quartet:

(7.9)\begin{equation} \boldsymbol{k}_1 = \boldsymbol{k}_1^{\varepsilon_0} + \boldsymbol{\kappa}, \quad \boldsymbol{k}_2 = \boldsymbol{k}_2^{\varepsilon_0} - \boldsymbol{\kappa}, \quad \boldsymbol{k}_3 = \boldsymbol{k}_3^{\varepsilon_0}, \quad \boldsymbol{k}_4 = \boldsymbol{k}_4^{\varepsilon_0}, \quad \boldsymbol{\kappa} = (k_0 \kappa_x,k_0 \kappa_y), \end{equation}

where the wave vectors $\boldsymbol {k}_i^{\varepsilon }$ are defined in (7.1) and where $(\kappa _x, \kappa _y)$ are real numbers.

Now, if we impose on the roots in (7.8) the constraints $\xi _1=0$ and $\xi _{234}=q_0>0$, and if we assume that $A=12T$ and $\cos p_0=-1$, then it may be deduced from (A2) that $q_0$ must satisfy the following compatibility condition:

(7.10)\begin{equation} q_0 =- \frac{\Delta \omega}{\Delta B + 16 T} > 0, \end{equation}

where all parameters are to be evaluated on the non-resonant manifold (7.9). In figure 12 is represented in the $(\kappa _x,\kappa _y)$ plane the curve $(\varGamma )$ defined by $A=12T$ that bifurcates from the origin $O$ corresponding to $\boldsymbol {\kappa }=0$. This curve passes through points $P_1$, $P_2$ and $P_3$ and forms a loop which crosses light-grey regions in which condition (7.10) is violated, so that rational breathers of the kind considered here exist on any point of $(\varGamma )$, except in these shadowed regions.

Figure 12. Curve $(\varGamma )$ defined by $A=12T$ in the $(\kappa _x,\kappa _y)$ plane defined in (7.9). The origin $O$ corresponds to $(\kappa _x,\kappa _y)=(0,0)$ for which (7.9) matches (7.1). The oval (dashed line) crossing $(\varGamma )$ in $O$ is the resonant curve $\Delta \omega =0$. The dashed curve is defined by $\Delta B + 16 T=0$. The light-grey region delimited by these two curves is defined by $\Delta \omega /(\Delta B + 16 T) > 0$. The dark-grey regions are defined by $a=16T^2 - A^2>0$. The coordinates of points $P_1$, $P_2$ and $P_3$ are reported in table 2.

Table 2. Coordinates and values of $a$, $q_0$ and $|\Delta \omega |$ for $P_1$, $P_2$ and $P_3$ of figure 12.

Truncated values of $a= -128 T^2$ (since $A=12T$) and $q_0$ in (7.10) are reported in table 2. Since $\xi _1=0$ and $\xi _{234}=q_0$, the corresponding rational breathers are (4.26). The values of the frequency mismatch show that for $|\Delta \omega |/(k_0^3q_0)$ to stay of order 1, one must stay on curve $(\varGamma )$ between $O$ and approximately $P_1$ to keep physical insight.