Plane-wave solutions of a dissipative generalization of the vector nonlinear Schrödinger equation

Abstract

The modulational instability of perturbed plane-wave solutions of the vector nonlinear Schrödinger (VNLS) equation is examined in the presence of multiple forms of dissipation. We establish that all constant-magnitude solutions of the dissipative VNLS equation are less unstable than their counterparts in the conservative VNLS equation. We also present three families of decreasing-in-magnitude plane-wave solutions to this dissipative VNLS equation. We establish that if certain forms of dissipation are present, then all exponentially-decaying plane-wave solutions with spatial dependence are linearly unstable while those without spatial dependence are linearly stable.

Introduction

Benjamin and Feir [1] showed that a uniform train of plane waves of moderate amplitude in deep water without dissipation is unstable with respect to a small perturbation of other waves traveling in the same direction with nearly the same frequency. Since that classic paper, a significant amount of research has been dedicated to studying the stability of plane waves on deep water. In 1968, Zakharov [28] derived the (scalar) nonlinear Schrödinger (NLS) equation

$$i{\psi }_t-{\psi }_{xx}+{\psi }_{yy}-|\psi {|}^2\psi =0\text{,}$$

where ψ = ψ(x, y, t) is a complex-valued function, as an approximate model of the evolution of plane waves of moderate amplitude in deep water without dissipation. Plane-wave solutions of the NLS equation are unstable [28], [9], [21]. Dysthe [12], Trulsen and Dysthe [25], and Trulsen et al. [26] derived higher-order asymptotic generalizations of the NLS equation and found that the plane-wave solutions of these equations are also unstable. Dhar and Das [10] showed that plane-wave solutions of a coupled system of higher-order generalizations of the NLS equation are also unstable. Onorato et al. [20] and Shukla et al. [24] showed that the plane-wave solutions of the vector nonlinear Schrödinger (VNLS) equation are unstable.

Lake and Yuen [17] and Lake et al. [18] conducted thorough examinations of physical experiments of one-dimensional surface patterns in which instability growth rates were measured and compared with rates predicted by theoretical models. More recently, physical experiments of two-dimensional surface patterns were conducted by Kimmoun et al. [16], Hammack et al. [13], and Henderson et al. [14]. Surprisingly, many of the wave patterns in these two-dimensional experiments appeared to be stable. A photograph of an apparently-stable two-dimensional surface wave pattern is included in Fig. 1.

All of the theories mentioned above predict that plane-wave solutions are unstable while some of the experiments suggest that plane waves are stable. This apparent discrepancy can be explained by the fact that all of the aforementioned models are conservative and none include effects due to dissipation.

There are many generalizations of the NLS equation that contain dissipative terms including those in [19], [2], [22], [27]. Segur et al. [23] studied a dissipative generalization of the NLS equation and showed that all plane-wave solutions are stable if a certain form of dissipation is present. Further, they showed that the dissipative theory agreed well with measurements from a series of physical experiments. Craig et al. [7] and Henderson et al. [15] generalized the Segur et al. [23] results to the vector dissipative NLS equation (i.e. a pair of nonlinearly coupled dissipative NLS equations). The photograph included in Fig. 1 is from an experiment modeled by a dissipative generalization of the vector NLS equation [15]. In this experiment, the undisturbed water depth is approximately 20 cm, the wave amplitude is approximately 0.4 cm, and the carrier wave frequency is 4 Hz. These parameters establish that the VNLS equation is an appropriate model equation.

In Section 1.1, we introduce a generalization of the vector NLS equation that contains multiple forms of dissipation as a model of two-dimensional surface wave patterns on deep water. This model is more general than the model contained in [7], [15]. In Section 1.2, we present four families of plane-wave solutions to this equation, three of which decrease in magnitude.

In Section 2, we present the linear stability analysis for three of these families of solutions. We establish that all constant-magnitude solutions are “less unstable” than their counterparts in the conservative system. This means that all plane-wave solutions that are stable in the conservative system are also stable in the dissipative system. It also means that solutions that are unstable in the conservative system have smaller (or possibly zero) growth rates if certain forms of dissipation are present. Finally, we establish that if certain forms of dissipation are present, then all spatially-independent exponentially-decaying plane-wave solutions are stable and all spatially-dependent exponentially-decaying plane-wave solutions are unstable.

In order to model the evolution of a uniform wave train of surface waves with a two-dimensional, bi-periodic surface pattern, propagating on deep water, Hammack et al. [13] derived the two-dimensional vector nonlinear Schrödinger (VNLS) equation

$$i(A_t+{\chi }_1{A}_x+{\nu }_1{A}_y)+{\alpha }_1{A}_{xx}+{\beta }_1{A}_{yy}+{\xi }_1{A}_{xy}+{\gamma }_1|A{|}^{2}A+{\zeta }_1|B{|}^{2}A=0\text{,}$$

$$i(B_t+{\chi }_2{B}_x+{\nu }_2{B}_y)+{\alpha }_2{B}_{xx}+{\beta }_2{B}_{yy}+{\xi }_2{B}_{xy}+{\gamma }_2|A{|}^{2}B+{\zeta }_2|B{|}^{2}B=0\text{,}$$

where A = A(x, y, t) and B = B(x, y, t) are complex-valued functions and χ_j, ν_j, α_j, β_j, ξ_j, γ_j and ζ_j (j = 1, 2) are real constants. Throughout the remainder of this paper, whenever j is used it is assumed to range from 1 to 2.

In this paper we examine the following generalization of the VNLS equation which we refer to as the two-dimensional vector dissipative nonlinear Schrödinger (VDNLS) equation

$$i(A_t+{\chi }_1{A}_x+{\nu }_1{A}_y)+({\alpha }_1-i{a}_1)A_{xx}+({\beta }_1-i{b}_1)A_{yy}+{\xi }_1{A}_{xy}+({\gamma }_1+i{c}_1)|A{|}^{2}A+({\zeta }_1+i{d}_1)|B{|}^{2}A+i{f}_{1}A=0\text{,}$$

$$i(B_t+{\chi }_2{B}_x+{\nu }_2{B}_y)+({\alpha }_2-i{a}_2)B_{xx}+({\beta }_2-i{b}_2)B_{yy}+{\xi }_2{B}_{xy}+({\gamma }_2+i{c}_2)|A{|}^{2}B+({\zeta }_2+i{d}_2)|B{|}^{2}B+i{f}_{2}B=0\text{.}$$

Here A = A(x, y, t) and B = B(x, y, t) are complex-valued functions; χ_j, ν_j, α_j, β_j, ξ_j, γ_j and ζ_j are real constants; and a_j, b_j, c_j, d_j, and f_j are nonnegative real constants that represent effects due to dissipation. We consider both the VNLS and VDNLS equations with periodic boundary conditions in both spatial dimensions. We also assume that A and B have the same spatial periods.

The ad hoc addition of the dissipative terms has been justified in the scalar (B = 0) case. We assume that similar arguments apply in the vector case. Davey [8] gives a general argument for dissipative terms of the forms given in Eq. (3) when the NLS equation is used as a model of weakly nonlinear surface waves. Miles [19], Lake et al. [18], and Dias et al. [11] generalize the NLS equation to include dissipation in a physical system by adding terms of the form if₁A. Blennerhassett [2] derives an equation similar to the scalar version of Eq. (3) from the Navier-Stokes’ equations for a free surface flow with viscous free surface boundary conditions. Segur et al. [23] add a term similar to if₁A in order to account for the effects of weak dissipation. Bridges and Dias [3] and Carter and Contreras [4] examine the scalar version of Eq. (3) and show that the modulational instability is enhanced if certain forms of dissipation are present. Our current work is the vector generalization of these results.

Under periodic boundary conditions, the VNLS equation preserves (in t) the $L_{per}^2$ norm,

$$\Vert \psi {\Vert }_{L_{\text{per}}^2}^2={\int }_0^{L_y}{\int }_0^{L_x}(|A{|}^2+|B{|}^2)dxdy\text{,}$$

where L_x and L_y are the periods of A and B in the x- and y-dimensions, respectively. In general, the VDNLS equation does not preserve the $L_{per}^2$ norm under periodic boundary conditions. Specifically, the $L_{per}^2$ norm is nonincreasing in t and

$$\frac{d}{dt}(\Vert \psi {\Vert }_{L_{\text{per}}^2}^2)=-2{\int }_0^{Lx}{\int }_0^{Ly}(a_1|A_x{|}^2+b_1|A_y{|}^2+c_1|A{|}^4+d_1|AB{|}^2+f_1|A{|}^2+a_2|B_x{|}^2+b_2|B_y{|}^2+c_2|B{|}^4+d_2|AB{|}^2+f_2|B{|}^2)dxdy\text{.}$$

It is important to notice that Eq. (5) demonstrates that solutions with spatial dependence decay to zero faster than spatially-independent solutions whenever a_j, b_j > 0 (i.e. whenever the corresponding forms of dissipation are present).

The VNLS equation admits plane-wave solutions of the form

$$A(x\text{,}y\text{,}t)=A_0{\text{e}}^{i{k}_{1}x+i{l}_{1}y+it(-{\chi }_1{k}_1-{\nu }_1{l}_1-{\xi }_1{k}_1{l}_1-{\alpha }_1{k}_1^2-{\beta }_1{l}_1^2+{\gamma }_1{A}_0^2+{\zeta }_1{B}_0^2)}\text{,}$$

$$B(x\text{,}y\text{,}t)=B_0{\text{e}}^{i{k}_{2}x+i{l}_{2}y+it(-{\chi }_2{k}_2-{\nu }_2{l}_2-{\xi }_2{k}_2{l}_2-{\alpha }_2{k}_2^2-{\beta }_2{l}_2^2+{\gamma }_2{B}_0^2+{\zeta }_2{A}_0^2)}\text{,}$$

where A₀, B₀, k_j and l_j are real constants.

We consider plane-wave solutions of the VDNLS equation of the form

$$A(x\text{,}y\text{,}t)=A_0{\text{e}}^{i{k}_{1}x+i{l}_{1}y+{\omega }_{r1}(t)+i{\omega }_{i1}(t)}\text{,}$$

$$B(x\text{,}y\text{,}t)=B_0{\text{e}}^{i{k}_{2}x+i{l}_{2}y+{\omega }_{r2}(t)+i{\omega }_{i2}(t)}\text{,}$$

where A₀, B₀, k_j and l_j are real constants and ω_rj(t) and ω_ij(t) are real-valued functions. Solutions of the form given in Eq. (6) or (7) are said to be spatially independent if k_j = l_j = 0.

Substituting Eq. (7) into Eq. (3) and separating into real and imaginary parts gives

$${\omega }_{r1}^{\prime }+c_1{A}_0^2{\text{e}}^{2{\omega }_{r1}}+d_1{B}_0^2{\text{e}}^{2{\omega }_{r2}}+G_1=0\text{,}$$

$${\omega }_{i1}^{\prime }-{\gamma }_1{A}_0^2{\text{e}}^{2{\omega }_{r1}}-{\zeta }_1{B}_0^2{\text{e}}^{2{\omega }_{r2}}+H_1=0\text{,}$$

$${\omega }_{r2}^{\prime }+d_2{B}_0^2{\text{e}}^{2{\omega }_{r2}}+c_2{A}_0^2{\text{e}}^{2{\omega }_{r1}}+G_2=0\text{,}$$

$${\omega }_{i2}^{\prime }-{\gamma }_2{A}_0^2{\text{e}}^{2{\omega }_{r1}}-{\zeta }_2{B}_0^2{\text{e}}^{2{\omega }_{r2}}+H_2=0\text{,}$$

where prime represents derivative with respect to t. Here we have introduced the following constants

$$0\le G_j=a_j{k}_j^2+b_j{l}_j^2+f_j\text{,}$$

$$H_j={\chi }_j{k}_j+{\nu }_j{l}_j+{\alpha }_j{k}_j^2+{\beta }_j{l}_j^2+{\xi }_j{k}_j{l}_j\text{.}$$

To our knowledge, an exact solution of the system given in Eq. (8) is not known. However, it does admit the following four families of solutions that are valid in restricted parameter regimes. In each of the following regimes, the constants of integration were chosen so that the plane-wave solutions of the VDNLS equation limit to the plane-wave solutions of the VNLS equation as a_j, b_j, c_j, d_j and f_j (the coefficients representing dissipation) limit to zero.

• Case 1: c_j = d_j = 0 and G_j = 0

  In this case, the solution of the system given in Eq. (8) is

  $${\omega }_{rj}(t)=0\text{,}$$

  $${\omega }_{ij}(t)=-t(H_j-{\gamma }_j{A}_0^2-{\zeta }_j{B}_0^2)\text{.}$$

  The magnitude of any solution of this form is constant in t. The family of plane-wave solutions to VNLS given in Eq. (6) is a subset of the solutions in this case. As these solutions do not decay with t, this family of solutions is conservative.

• Case 2: c₂ = c₁ > 0, d₂ = d₁ > 0 and G_j = 0

  In this case, the solution of the system given in Eq. (8) is

  $${\omega }_{rj}(t)=-\frac{1}{2}\ln (1+2t(c_1{A}_0^2+d_1{B}_0^2))\text{,}$$

  $${\omega }_{ij}(t)=-t{H}_j-\frac{{\gamma }_j{A}_0^2+{\zeta }_j{B}_0^2}{c_1{A}_0^2+d_1{B}_0^2}{\omega }_{rj}(t)\text{.}$$

  The magnitude of any solution of this form decays to zero as t^−1/2. These solutions decay (relatively) slowly and therefore can be thought of as nearly conservative.

• Case 3: c_j = d_j = 0 and G_j > 0

  In this case, the solution of the system given in Eq. (8) is

  $${\omega }_{rj}(t)=-t{G}_j\text{,}$$

  $${\omega }_{ij}(t)=-t{H}_j+\frac{{\gamma }_j{A}_0^2}{2G_1}(1-{\text{e}}^{-2G_{1}t})+\frac{{\zeta }_j{B}_0^2}{2G_2}(1-{\text{e}}^{-2G_{2}t})\text{.}$$

  The magnitude of any solution of this form decays to zero exponentially. The behavior of these solutions is dominated by dissipative effects.

• Case 4: c₂ = c₁ > 0, d₂ = d₁ > 0, G_j > 0 and B₀ = A₀

  In this case, the solution of the system given in Eq. (8) is

  $${\omega }_{rj}(t)=-t{G}_j-\frac{1}{2}\ln \left(1+\frac{c_1{A}_0^2}{G_1}(1-{\text{e}}^{-2t{G}_1})+\frac{d_1{A}_0^2}{G_2}(1-{\text{e}}^{-2t{G}_2})\right)\text{,}$$

  $${\omega }_{ij}(t)=-t{H}_j+\int \frac{A_0^2{G}_1{G}_2({\gamma }_1{\text{e}}^{-2t{G}_1}+{\zeta }_1{\text{e}}^{-2t{G}_2})}{G_1{G}_2+A_0^2(d_1{G}_1(1-{\text{e}}^{-2t{G}_2})+c_1{G}_2(1-{\text{e}}^{-2t{G}_1}))}dt\text{.}$$

  The lack of a closed-form expression for ω_ij(t) is not too troubling because the integral is bounded for all t ≥ 0 and further, it only contributes to the phase of the solution. The magnitude of any solution of this form decays to zero exponentially. The behavior of these solutions is dominated by dissipative effects.