2. The Davey–Stewartson equation in Fourier space

The starting point is the DS equation, of which there are several equivalent forms available. Since the Zakharov equation is formulated in terms of the action variable, we will take the equation that is based on this variable. Hence, we introduce the action variable $a(\boldsymbol {k},t)$ defined in such a way that the surface elevation $\eta (\boldsymbol x,t)$ reads

(2.1)\begin{equation} \eta(\boldsymbol x,t)=\int {\rm d}\boldsymbol{k}\left(\frac{\omega_k}{2g}\right)^{1/2} [a(\boldsymbol{k},t)+a^*(-\boldsymbol{k},t)]\exp\left({\rm i}\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{x}\right), \end{equation}

where $\omega _k=\omega (k)=\sqrt {g k \tanh (k h)}$, with $g$ acceleration of gravity, $k=|\boldsymbol {k}|=\sqrt {k_x^2+k_y^2}$ the magnitude of the wavenumber and $h$ the constant depth. Assuming a narrow-band process with carrier wavenumber $\boldsymbol {k}=(k_0,0)$ propagating in the $x$-direction, and defining the Fourier transform according to

(2.2)\begin{equation} A(\boldsymbol{x},t) = \int {\rm d}\boldsymbol{k} a(\boldsymbol{k},t)\exp\{{\rm i}\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{x}\}, \end{equation}

at leading order, one finds from (2.1)

(2.3)\begin{equation} \eta(\boldsymbol x,t)=\left(\frac{\omega_0}{2g}\right)^{1/2} [A(\boldsymbol{x},t) \exp({{\rm i}k_0x})+A^*(\boldsymbol{x},t)\exp({-{\rm i}k_0x})], \end{equation}

where $\omega _0=\omega (k_0)$. The DS equation for the complex envelope $A=A(\boldsymbol {x},t)$ and for the auxiliary variable $Q=Q(\boldsymbol {x},t)$, related to the mean motion of the fluid, then becomes

(2.4)\begin{equation} \left.\begin{gathered} {\rm i}\frac{\partial A}{\partial t}+\lambda \frac{\partial^2A}{\partial x^2}+\mu \frac{\partial^2A}{\partial y^2}=\nu|A|^2A+\nu_1QA\\ \lambda_1\frac{\partial^2 Q}{\partial x^2}+\mu_1 \frac{\partial^2 Q}{\partial y^2}=\nu_2\frac{\partial^2|A|^2}{\partial y^2} \end{gathered}\right\}, \end{equation}

where

(2.5)\begin{equation} \left.\begin{gathered} \lambda=\frac{1}{2}\left.\frac{\partial^2\omega}{\partial k_x^2}\right|_{(k_0,0)},\quad \mu=\frac{v_g}{2k_0},\\ \nu=k_0^3\left[\frac{9T_0^4-10T_0^2+9}{8T_0^3}-\frac{1}{k_0h} \left\{\frac{(2v_g-c_0/2)^2}{c_S^2-v_g^2}+1\right\}\right],\\ \nu_1=\frac{k_0^4}{2\omega_0v_g}\left\{2c_0+v_g(1-T_0^2)\right\},\quad \lambda_1=c_S^2-v_g^2,\quad \mu_1=c_S^2,\\ \nu_2=\frac{1}{2}\frac{c_0}{T_0}c_S^2v_g\frac{2c_0+v_g(1-T_0^2)}{c_S^2-v_g^2}. \end{gathered}\right\} \end{equation}

Furthermore, $T_0=\tanh (k_0 h)$, $c_0=\omega _0/k_0$ is the phase speed, $v_g=\partial \omega /\partial k_x|_{(k_0,0)}$ is the group speed and $c_S=\sqrt {gh}$ is the shallow-water speed. We remark that the one-dimensional form of the DS equation with $Q=0$ agrees with earlier work by Hasimoto & Ono (Reference Hasimoto and Ono1972) and Whitham (Reference Whitham1974). The two-dimensional version of the DS equation was already found and studied by Benney & Roskes (Reference Benney and Roskes1969).

Using the Fourier transform with wave vector $\boldsymbol {\kappa }=\{l,m\}$, e.g.

(2.6)\begin{equation} q_1= \frac{1}{4{\rm \pi}^2}\int {\rm d}\kern0.7pt \boldsymbol{x} Q(\boldsymbol{x},t) \exp\{-{\rm i}\boldsymbol{\kappa}_1\boldsymbol{\cdot}\boldsymbol{x}\}, \end{equation}

we rewrite the DS equation as follows:

(2.7)\begin{equation} \left.\begin{gathered} {\rm i}\frac{\partial a_1}{\partial t}-(\lambda l_{1}^2 +\mu m_{1}^2)a_1=\nu \int{\rm d} {\boldsymbol \kappa}_{2,3,4}a_2^*a_3a_4\delta_{1+2-3-4} +\nu_1\int {\rm d}{\boldsymbol \kappa}_{3,i}q_i a_3\delta_{1-3-i} \\ (\lambda_1 l_i^2 +\mu_1 m_i^2) q_i= \nu_2\int {\rm d}{\boldsymbol \kappa}_{2,4}(m_4-m_2)^2 a_2^*a_4\delta_{i+2-4}. \end{gathered}\right\} \end{equation}

From the second equation in (2.7), one may express $q_i$ in terms of the action variable, i.e.

(2.8)\begin{equation} q_i=\nu_2\int {\rm d}{\boldsymbol \kappa}_{2,4}\frac{(m_4-m_2)^2}{\lambda_{1} l_i^2 +\mu_1 m_{i}^2} a_2^*a_4\delta_{i+2-4}. \end{equation}

Note that, in the limit of $l_i$ and $m_i$ going to zero, to get (2.8), we are dividing by zero; we believe that this is the origin of the singularities in the subsequent analysis. Substitution (2.8) in the first equation, gives the standard form of a four-wave interaction equation

(2.9)\begin{equation} {\rm i}\frac{\partial a_1}{\partial t}=\varOmega_1 a_1 +\int{\rm d}{\boldsymbol \kappa}_{2,3,4}G_{1,2,3,4}a_2^*a_3a_4 \delta_{1+2-3-4}, \end{equation}

with the dispersion relation

(2.10)\begin{equation} \varOmega_1 = \lambda l_{1}^2 +\mu m_{1}^2, \end{equation}

while the nonlinear transfer coefficient becomes

(2.11)\begin{equation} G_{1,2,3,4} = \nu+\nu_1\frac{\nu_2(m_{4}-m_{2})^2} {\lambda_1(l_{4}-l_{2})^2+\mu_1(m_{4}-m_{2})^2}. \end{equation}

For the equation to be Hamiltonian, the coefficient must satisfy a number of symmetries: using the condition $\boldsymbol {\kappa }_1+\boldsymbol {\kappa }_2=\boldsymbol {\kappa }_3+\boldsymbol {\kappa }_4$, one may also write $G_{1,2,3,4}$ in terms of the difference modulation vector $\boldsymbol {\kappa }_1-\boldsymbol {\kappa }_3$. Moreover, $G_{1,2,3,4}$ must be invariant with respect to the exchange of labels ‘3’ and ‘4’. Combining these properties and defining

(2.12)\begin{equation} E_{i,j} = \frac{1}{4}\frac{(m_i-m_j)^2}{\lambda_1(l_i-l_j)^2+\mu_1(m_i-m_j)^2}, \end{equation}

one obtains for $G$ the symmetric form

(2.13)\begin{equation} G_{1,2,3,4} = \nu+\nu_1[\nu_2(E_{1,3}+E_{2,3}+E_{4,2}+E_{4,1})] .\end{equation}

From (2.13), it is immediately evident that the effect of the mean flow on the evolution of the modulations is significant. In the function $E$ both denominator and numerator depend on the square of the difference between the modulation wavenumbers in such a way that, if the size of the modulation wavenumbers decreases, the impact of the function $E$, and hence the mean flow, does not diminish. This is in sharp contrast with the effect of mean flow on modulations in in infinite water depth as given by the two-dimensional Dysthe equation; indeed, in Dungey & Hui (Reference Dungey and Hui1979), it has been shown show, and further discussed by Janssen (Reference Janssen1983), that the deep-water wave-induced current is a higher-order effect.

It should be clear, see figure 1, that, at the origin the function, $E$ is not well defined. This is a potential problem because, when determining the growth rate of the Benjamin–Feir instability, we need to evaluate the nonlinear transfer function for vanishing modulation wavenumber difference, i.e. $l_i=l_j$ and $m_i=m_j$. To see this better, we consider $E$ as a function of $K_x=l_i-l_j$ and $K_y=m_i-m_j$, and set all the coefficients to 1, to get $E = {K_y^2}/(K_x^2+K_y^2)$. Now, in the limit of vanishing modulation wavenumber difference, i.e. $K_x \rightarrow 0, K_y\rightarrow 0$, one would expect to find a unique answer. However, this is not the case. Considering the straight line through the origin, $K_y=\alpha K_x$, one finds that $E=\alpha ^2/(1+\alpha ^2)$, so that the value of $E$ at the origin depends on the limiting procedure, i.e. under what angle the origin is approached. Despite this non-uniqueness, we will find that the evolution of the modulations, e.g. the linear growth rate of the modulational instability, is found to be unique.

Figure 1.

Wave-induced current function $E= {K_y^2}/(K_x^2+K_y^2)$ where $K_x=l_i-l_j$ and $K_y=m_i-m_j$ as function of $K_x$ and $K_y$.

In contrast, for the deep-water Dysthe equation the effect of the wave-induced current has, after simplification, the form $E = {K_x^2}/(\sqrt {K_x^2+K_y^2})$ and this has at the origin the unique value of zero because the numerator vanishes more rapidly towards zero than the denominator. This is immediately seen by considering once more the straight line $K_x=\alpha K_y$ so that $E=\alpha ^2 K_x/\sqrt {1+\alpha ^2}$, which vanishes unambiguously for vanishing $K_x$.

3. The Zakharov equation and its connection with the DS equation

The Zakharov equation reads

(3.1)\begin{equation} {\rm i}\frac{\partial a_1}{\partial t}=\omega_1 a_1 + \int T_{1,2,3,4}a_2^*a_3a_4 \delta_{1+2-3-4}\,{\rm d}{\boldsymbol k}_{2,3,4}, \end{equation}

where the nonlinear interaction coefficient $T_{1,2,3,4}$ is given in the Appendix. We are interested in an envelope equation so that it is useful to rewrite the absolute value of the wavevector $k$ in the dispersion relation as

(3.2)\begin{equation} k = k_x\sqrt{1+\frac{k_y^2}{k_x^2}}. \end{equation}

Because in the DS equation $k_y^2/k_x^2\ll 1$, i.e. the wavenumber in the $y$-direction is much smaller than the one in the $x$-direction, we assume that there is a dominant wave propagating in the $x$ direction with wave vector components $(k_0,0)$, and we consider the narrow-band approximation

(3.3) \begin{equation} \left.\begin{gathered} k_x=k_0+ l\\ k_y= m. \end{gathered}\right\} \end{equation}

Expanding the dispersion relation for small $l$ and $m$, we obtain

(3.4)\begin{equation} \omega_k=\omega_0+ v_g l +(\lambda l^2+\mu m^2). \end{equation}

In the Zakharov equation the first two terms can be removed by a rotation, and we are left with the dispersion relation of the DS equation, see (2.10). We now need to calculate the narrow-band version of the coefficient $T_{1,2,3,4}$. Its explicit form is given in the Appendix, and it consists of two contributions. The first one is denoted by $T^{(R)}_{1,2,3,4}$ and represents the regular contribution. It can be verified (see e.g. Janssen & Onorato Reference Janssen and Onorato2007) that the narrow-band version of the regular contribution becomes

(3.5)\begin{equation} T^{(R)}_{1,1,1,1}= k_0^3\frac{9T_0^4-10T_0^2+9}{8 T_0^3}. \end{equation}

However, there is also a contribution from the non-resonant triads. This contribution is denoted by $T^{(S)}_{1,2,3,4}$; these terms involve the product of the second-order coefficients $V^{(-)}$. The difficulty of the calculation comes from the terms that, in the limit of a long wave group, give rise to an apparent singularity. For example, the term that contains at the denominator $\omega _3+\omega _{1-3}-\omega _1$ may give rise to a singular behaviour when $\boldsymbol {k}_1\simeq \boldsymbol {k}_3$.

The calculation that follows is similar to the one performed by Janssen & Onorato (Reference Janssen and Onorato2007), but now extended to the case of two-dimensional propagation. We shall only give results explicitly for the product term of the triads $(1,3,1-3)$ and $(4,2,4-2)$ and the triads $(3,1,3-1)$ and $(2,4,2-4)$. The other contributions follow by interchanging the indices 1 and 2. We need to calculate terms like the following one (see (A3)):

(3.6)\begin{equation} {-}V^{(-)}_{1,3,1-3}V^{(-)}_{4,2,4-2}\left[\frac{1}{\omega_3+\omega_{1-3}-\omega_1}+ \frac{1}{\omega_2+\omega_{4-2}-\omega_4}\right]. \end{equation}

After some algebra one finds that (see also (23) in Janssen & Onorato Reference Janssen and Onorato2007)

(3.7)\begin{equation} V^{(-)}_{1,3,1-3} \approx \frac{1}{4\sqrt{2}} \left(\frac{k_0^2-q_0^2} {\omega_0}\sqrt{g \omega_{1-3}}+2 k_0(l_1-l_3) \left(\frac{g}{\omega_{1-3}}\right)^{1/2}\right);\end{equation}

moreover, we need to compute terms such as

(3.8)\begin{equation} \frac{1}{\omega_3-\omega_1+\omega_{1-3}}. \end{equation}

To this end, it is straightforward to show that, to the leading non-trivial order, we have

(3.9) \begin{equation} \omega_{1-3}= c_s \varDelta_{1-3}, \end{equation}

with

(3.10)\begin{equation} \varDelta_{1-3}= \sqrt{(l_1-l_3)^2+(m_1-m_3)^2}, \end{equation}

while

(3.11)\begin{equation} \omega_1-\omega_3= v_g (l_1-l_3). \end{equation}

Furthermore, the wavenumber condition $\boldsymbol {k}_1+\boldsymbol {k}_2=\boldsymbol {k}_3+\boldsymbol {k}_4$ implies certain relations for the difference vectors, e.g. $l_1-l_3=l_4-l_2$ and $m_1-m_3=m_4-m_2$. As a consequence, one finds that $\omega _{1-3}=\omega _{4-2}$ while $\omega _1-\omega _3=\omega _4-\omega _2$. By interchanging 1 and 2, it is found that $\omega _{2-3}=\omega _{4-1}$ and that $\omega _2-\omega _3=\omega _4-\omega _1$. Similar relations exist for the parameter $\varDelta$, e.g. $\varDelta _{1-3}=\varDelta _{4-2}$ and $\varDelta _{2-3}=\varDelta _{4-1}$.

Putting together the first and the fourth term of $T^{(S)}_{1,2,3,4}$ denoted by $2F_{1,3}$, and the second and the third term given by $2F_{2,3}$, one then finds

(3.12)\begin{equation} T^{(S)}_{1,2,3,4} = 2F_{1,3}+2F_{2,3}, \end{equation}

with

(3.13)\begin{align} F_{i,j}=-\frac{1}{16}\frac{k_0^3 c_s^2}{c_s^2 \varDelta_{i-j}^2-v_g^2 (l_i-l_j)^2} \left[\frac{(1-T_0^2)^2 \varDelta_{i-j}^2}{T_0}+ \left(\frac{4(1-T_0^2)g v_g}{c_s^2 \omega_0}+\frac{4}{k_0 h}\right)(l_i-l_j)^2\right]. \end{align}

Furthermore, exploiting the above mentioned relations between the difference vectors, one obtains the following symmetrical form:

(3.14)\begin{equation} T^{(S)}_{1,2,3,4} = F_{1,3}+F_{2,3}+F_{4,2}+F_{4,1}, \end{equation}

so that the narrow-band version of the nonlinear interaction coefficient $T_{1,2,3,4}$ becomes

(3.15)\begin{equation} T_{1,2,3,4} \approx T^{(R)}_{1,1,1,1}+F_{1,3}+F_{2,3}+F_{4,2}+F_{4,1}, \end{equation}

where $T^{(R)}_{1,1,1,1}$ is given by (3.5). To show that the narrow-band version of the Zakharov interaction coefficient is identical to the corresponding one from the DS equation (3.3), we have to further work on the form of the function $F_{i,j}$. Straightforward algebraic manipulations show that this function is of the form

(3.16)\begin{equation} F_{i,j}= \alpha_1+\alpha_2 -4\mu_1\alpha_2E_{i,j},\quad {\rm with}\ E_{i,j}=\frac{1}{4} \frac{(m_i-m_j)^2}{\lambda_1(l_i-l_j)^2+ \mu_1(m_i-m_j)^2}, \end{equation}

where $E_{i,j}$ is identical to the form given in (2.12). Furthermore, the constants are

(3.17a,b)\begin{equation} \alpha_1=-\frac{1}{16}k_0^3 \frac{(1-T_0^2)^2}{T_0}\quad {\rm and}\quad \alpha_2=-\frac{1}{16}\frac{k_0^3}{T_0(c_s^2-v_g^2)}(2 c_0+v_g(1-T_0^2))^2. \end{equation}

Using the expressions for $\mu _1$ and $\alpha _2$, it is then straightforward to establish that the modulation dependent part of (2.13) matches the one in (3.16) since $-4\mu _1\alpha _2=\nu _1\nu _2$.

Now, it remains to establish whether the constant parts of (2.13) and (3.17a,b) agree. After some algebra, using (32)–(33) of Janssen & Onorato (Reference Janssen and Onorato2007), i.e.

(3.18)\begin{align} -\frac{1}{4}k_0^3\frac{c_s^2}{c_s^2-v_g^2}\left[\frac{(1-T_0^2)^2}{T_0}+ \frac{4(1-T_0^2)gv_g}{c_s^2\omega_0}+\frac{4 }{k_oh}\right] =-\frac{k_0^3}{k_0h}\left[\frac{(2v_g-c_0/2)^2}{c_s^2-v_g^2}+1\right], \end{align}

an alternative expression for $\alpha _2$ may be obtained which establishes a connection between $\alpha _2$ and $\alpha _1$, i.e.

(3.19)\begin{equation} \alpha_2=-\alpha_1 -\frac{1}{4}\frac{k_0^3}{k_0h}\left[\frac{(2v_g-c_0/2)^2}{c_s^2-v_g^2}+1\right]. \end{equation}

As a consequence, one may write for the narrow-band version of the nonlinear interaction coefficient

(3.20)\begin{equation} T_{1,2,3,4} \approx T_{1D}+\nu_1[\nu_2(E_{1,3}+E_{2,3}+E_{4,2}+E_{4,1})] ,\end{equation}

with

(3.21)\begin{align} T_{1D} = T^{(R)}_{1,1,1,1}+4(\alpha_1+\alpha_2)=k_0^3 \frac{9T_0^4-10T_0^2+9}{8 T_0^3}-\frac{k_0^3}{k_0h} \left[\frac{(2v_g-c_0/2)^2}{c_s^2-v_g^2}+1\right]. \end{align}

Hence, the term $T_{1D}$ agrees with the term $\nu$ in the expression for $G_{1,2,3,4}$ in (2.13), which gives the nonlinear transfer for the DS equation. Note that $T_{1D}$ is in agreement with Whitham (Reference Whitham1974), and it gives the nonlinear correction to the dispersion relation for a uniform weakly nonlinear wave train and the correction due to the one-dimensional wave-induced current. The sum of these two terms vanishes for $k_0h=1.363$, which signals for one-dimensional modulations the transition from unstable to stable. However, as already shown by Hayes (Reference Hayes1973) for two-dimensional modulations, there is still modulational instability for $k_0h<1.363$.

4. Linear stability analysis of a uniform wave train

The main topic of this section is to study the linear stability results of a uniform wave train, realizing that the nonlinear transfer coefficient is not unique in the limit of zero modulation wavenumber (see § 2). Because the growth rate will be seen to depend explicitly on the value of the $T_{1,2,3,4}$ at the carrier wavenumber $\boldsymbol {k}_0=(k_0,0)$ and at the sideband wavenumber $\boldsymbol {k}_0+\boldsymbol {\kappa }$, the main question to be answered is as follows: How does the mentioned non-uniqueness affect the stability results?

Let us now revisit the stability analysis of a plane wave. The nonlinear dispersion relation for surface gravity waves is obtained immediately from the Zakharov equation (3.1). Consider the case of a single wave

(4.1)\begin{equation} a(\boldsymbol{k},t)=\hat{a}(t)\delta(\boldsymbol{k}-\boldsymbol{k}_0), \end{equation}

then substitution of (4.1) into (3.1) gives

(4.2)\begin{equation} \frac{{\rm d} \hat{a}(t)}{{\rm d} t}=-{\rm i}T_{NL}|\hat{a}(t)|^2\hat{a}(t), \end{equation}

where $T_{NL}=T_{0,0,0,0}$. Equation (4.2) may be solved at once by writing

(4.3)\begin{equation} \hat{a}(t)=a_0 \exp({-{\rm i}\varOmega t}), \end{equation}

where $\varOmega$ denotes the correction of the dispersion relation due to nonlinearity. It is given by

(4.4)\begin{equation} \varOmega= T_{NL}|a_0|^2. \end{equation}

The dependence of the dispersion relation on the wave amplitudes is known to have a profound impact on the time evolution of a weakly nonlinear wave train, but here we just confine ourselves to discussing the short time behaviour of a nonlinear wave train by means of a linear stability analysis.

Following Crawford et al. (Reference Crawford, Lake, Saffman and Yuen1981) (see also Yuen & Lake Reference Yuen and Lake1982; Krasitskii & Kalmykov Reference Krasitskii and Kalmykov1993), we now address the question as to whether a weakly nonlinear wave train is stable or not in finite water depth (see also Brinch-Nielsen & Jonsson Reference Brinch-Nielsen and Jonsson1986; Gramstad & Trulsen Reference Gramstad and Trulsen2011). To test the stability of a uniform wave train, we perturb it by a pair of sidebands with wavenumber $\boldsymbol {k}_{\pm }=\boldsymbol {k}_0 \pm \boldsymbol {\kappa }$ and amplitude $A_{\pm }(t)$, e.g.

(4.5)\begin{equation} a= A_0\delta(\boldsymbol{k}-\boldsymbol{k}_0)+A_+\delta(\boldsymbol{k}-\boldsymbol{k}_+)+A_-\delta(\boldsymbol{k}-\boldsymbol{k}_-). \end{equation}

Assuming that the sideband amplitudes are small compared with the amplitude $A_0$ of the carrier wave and neglecting the square of small quantities, the following evolution equations for $A_{\pm }$ are found from the Zakharov equation (3.1):

(4.6)\begin{equation} {\rm i}\frac{{\rm d}}{{\rm d} t}A_{{\pm}}=T_{{\pm},\mp}a_0^2A_{{\mp}}^*\exp[-{\rm i}(\Delta\omega+ 2T_{NL}a_0^2)t]+2T_{{\pm},\pm}a_0^2A_{{\pm}}, \end{equation}

where

(4.7)\begin{equation} \left.\begin{gathered} T_{{\pm},\pm}=T(\boldsymbol{k}_0 \pm \boldsymbol{\kappa},\boldsymbol{k}_0,\boldsymbol{k}_0,\boldsymbol{k}_0 \pm \boldsymbol{\kappa}),\\ T_{{\pm},\mp}=T(\boldsymbol{k}_0 \pm \boldsymbol{\kappa},\boldsymbol{k}_0 \mp \boldsymbol{\kappa},\boldsymbol{k}_0,\boldsymbol{k}_0),\\ T_{NL} = T(\boldsymbol{k}_0,\boldsymbol{k}_0,\boldsymbol{k}_0,\boldsymbol{k}_0), \\ \Delta\omega=2\omega(\boldsymbol{k}_0)-\omega(\boldsymbol{k}_0+\boldsymbol{\kappa})-\omega(\boldsymbol{k}_0-\boldsymbol{\kappa}), \end{gathered}\right\} \end{equation}

and $a_0$ is the same quantity as given in (4.4).

By means of the substitution

(4.8)\begin{equation} \left.\begin{gathered} A_+=\hat{A}_+\exp\left[-{\rm i}\left(\tfrac{1}{2}\Delta\omega+T_{NL}a_0^2\right)t-{\rm i}\varOmega t\right], \\ A_-^*=\hat{A}_-^*\exp\left[+{\rm i}\left(\tfrac{1}{2}\Delta\omega+T_{NL}a_0^2\right)t-{\rm i}\varOmega t\right], \end{gathered}\right\} \end{equation}

where $\varOmega$ is still unknown, a set of differential equations is obtained that contains no explicit time dependence. A non-trivial solution is then found provided $\varOmega$ satisfies the dispersion relation

(4.9) \begin{align} \varOmega &= (T_{+,+}-T_{-,-})a_0^2 \nonumber\\ &\quad \pm \bigg\{\!-T_{+,-}T_{-,+}a_0^4+\left[-\tfrac{1}{2}\Delta \omega+a_0^2(T_{+,+}+T_{-,-}-T_{NL})\right]^2\!\bigg\}^{{1}/{2}}. \end{align}

We have instability provided that the term under the square root is negative (see Crawford et al. Reference Crawford, Lake, Saffman and Yuen1981).

Let us now determine the dispersion relation explicitly for the narrow-band version of the nonlinear transfer coefficient given in (3.20). Evaluating the parameters given in (4.7) with $\boldsymbol {\kappa }=(l,m)$, one finds

(4.10)\begin{equation} \left.\begin{gathered} T_{+,+} =T_{-,-}=T_{1D}+\frac{1}{2} \nu_1\nu_2\frac{ m^2}{\lambda_1 l^2+\mu_1 m^2}+2\nu_1\nu_2E_0, \\ T_{+,-}=T_{-,+}=T_{1D}+\nu_1\nu_2\frac{ m^2}{\lambda_1 l^2+\mu_1 m^2}, \\ T_{NL}=T_{1D}+4\nu_1\nu_2E_0,\\ \Delta\omega=-2(\lambda l^2+\mu m^2). \end{gathered}\right\} \end{equation}

Here, $T_{1D}$ is the part of the nonlinear transfer coefficient that is independent of the modulation wavenumber $\boldsymbol {\kappa }=(l,m)$ and $E_0$ is the value of $E$ at the origin. Now, since $T_{+,+}=T_{-,-}$ and $T_{+,-}=T_{-,+}$, the dispersion relation (4.9) simplifies to

(4.11) \begin{equation} \varOmega={\pm} \big\{-T^2_{+,-}a_0^4 +\big[-\tfrac{1}{2}\Delta\omega+a_0^2(2T_{+,+}-T_{NL})\big]^2\big\}^{{1}/{2}}. \end{equation}

The most remarkable point to make now is that the term $(2T_{+,+}-T_{NL})$, upon using (4.10), does not depend on the undefined value $E_0$ as

(4.12)\begin{equation} 2T_{+,+}-T_{NL}=T_{1D}+\nu_1\nu_2\frac{m^2}{\lambda_1 l^2+\mu_1 m^2}=T_{+,-}.\end{equation}

Since also the term $T_{+,-}$ does not depend on the undefined value $E_0$, this implies that the dispersion relation for $\varOmega$ is well defined. Further simplification, using $T_{+,-}=2T_{+,+}-T_{NL}$, gives as dispersion relation

(4.13)\begin{equation} \varOmega^2=(\lambda l^2+\mu m^2)[2T_{+,-}a_0^2+(\lambda l^2+\mu m^2)], \end{equation}

which is in perfect agreement with stability results obtained directly from the DS equation. Hayes (Reference Hayes1973) and Davey & Stewartson (Reference Davey and Stewartson1974) showed that the wave train is unstable if

(4.14)\begin{equation} T_{+,-}(\lambda l^2+\mu m^2) < 0. \end{equation}

Note that $\lambda = \omega _0''/2$ is always negative while $\mu =v_g/2k_0$ is positive, and $T_{1D}$ changes from negative to positive as $k_0h$ decreases beyond $k_0h=1.363$ (Hasimoto & Ono Reference Hasimoto and Ono1972). The two-dimensional effect involving the modulation wavenumbers $l$ and $m$ tends to enhance the Benjamin–Feir instability. Hayes (Reference Hayes1973) has shown that it is always possible to choose $l$ and $m$ in such a way that criterion (4.14) is satisfied, but instability is practically non-existent for shallow-water waves in the range $0\leq k_0h\leq 0.5$.

Finally, it is remarked that, while $T_{+,-}$ is not unique at the origin, $\varOmega$ from (4.13) is well defined because $T_{+,-}$ is multiplied by $\Delta \omega$, which vanishes at the origin.

It is clear that the parameter $T_{+,-}$ plays an important role in the modulational instability. Therefore, in order to obtain confidence in our results, we compare this parameter with alternative approaches. From (4.10), we know that, in the narrow-band approximation, we have

(4.15)\begin{equation} T_{+,-}=T_{1D}+\nu_1\nu_2\frac{m^2}{\lambda_1 l^2+\mu_1 m^2}. \end{equation}

Some effort has been made to try to validate the result given in (4.15) using direct numerical computation of the original interaction coefficient. In figure 2, we show the ratio of the narrow-band approximation in (4.15) and the numerical result obtained by computing the element matrix of the Krasitskii's interaction coefficient. This last coefficient is evaluated following Janssen & Onorato (Reference Janssen and Onorato2007), where the wavenumbers have been perturbed by a small amount in such a way that the perturbations (with size of order $10^{-5}$) satisfy the wavenumber resonance conditions. Therefore, the computation of the last coefficient is also an approximation, and one might expect deviations much larger than machine precision. The analytical result was obtained by (3.20) where from the outset we used the wavenumber condition $\boldsymbol {k}_1+\boldsymbol {k}_2=\boldsymbol {k}_3+\boldsymbol {k}_4$ to simplify (3.20), so that it only depends on the first three wavenumbers, thus

(4.16)\begin{equation} T_{1,2,3} = T_{1D} + 2\nu_1\nu_2(E_{1,3} + E_{2,3}). \end{equation}

Figure 2.

Dependence of $T_{1,2,3,4}$ on the dimensionless depth parameter $k_0h$. The case $k_1=k_0(\cos \theta,+\sin \theta )$, $k_2=k_0(\cos \theta,-\sin \theta )$, $k_3=k_4=k_0(\cos \theta,0)$ is chosen. Shown is the analytical result in (4.15) normalized with the numerical result for different values of $\theta$.

The conclusion, looking at figure 2, is that there is a reasonable agreement between the analytical result and the numerical implementation of the Krasitskii interaction coefficients. This implies that for numerical calculations of the nonlinear transfer in shallow and intermediate water, i.e. the procedure suggested by Janssen & Onorato (Reference Janssen and Onorato2007), seems to give valid results.