On the Whitham system for the (2+1)‐dimensional nonlinear Schrödinger equation

Abstract

Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann-type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

“…Inserting the ansatz (7), the leading-order solution (2) and using (6), to leading order the averaged conservation laws (11) yield…”

“…𝑍 = 𝑘(𝑥 + 𝑞𝑦 − 𝑉𝑡)∕𝜖 is the fast variable defined in (6), 𝑉 = 𝜔∕𝑘 and 𝑘 and 𝜔 are as in (3c). Substituting this ansatz in (1), to leading order in 𝛿 we obtain a linearized ZK equation:…”

Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensions

“…While the Whitham equations for the two-dimensional version of (1.1) (hereafter referred to as the 2DNLS equation) were obtained in [5], this work differs from [5] in several important respects. First, our derivation employs a two-phase ansatz for the periodic solutions of the 2DNLS equation, which has several practical advantages.…”

“…The properties of the resulting KP-Whitham equations were then studied in [14] and the soliton limit of these equations was used in [48][49][50] to study the time evolution of a variety of piecewise-constant initial conditions in the modulation equations and, in the process, characterize the resulting dynamics of the solutions of the KP equation. Recently, the Whitham equations for the radial NLS equation [4] and those for focusing and defocusing two-dimensional nonlinear Schrödinger (2DNLS) equations [5] were also derived using a multiple scales approach.…”

“…First, our derivation employs a two-phase ansatz for the periodic solutions of the 2DNLS equation, which has several practical advantages. For one thing, it immediately yields a second conservation of waves equation in vector form that was missed in [5]. It is well known that several methods can be used to derive the Whitham equations: averaged conservation laws, averaged Lagrangian, and multiple scales perturbation theory.…”