Whitham Modulation Theory and Two-Phase Instabilities for Generalized Nonlinear Schrödinger Equations with Full Dispersion

IF 1.9 4区 数学 Q1 MATHEMATICS, APPLIED
Patrick Sprenger, Mark A. Hoefer, Boaz Ilan
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引用次数: 0

Abstract

SIAM Journal on Applied Mathematics, Volume 84, Issue 4, Page 1337-1361, August 2024.
Abstract. The generalized nonlinear Schrödinger equation with full dispersion (FDNLS) is considered in the semiclassical regime. The Whitham modulation equations are obtained for the FDNLS equation with general linear dispersion and a generalized, local nonlinearity. Assuming the existence of a four-parameter family of two-phase solutions, a multiple-scales approach yields a system of four independent, first-order, quasi-linear conservation laws of hydrodynamic type that correspond to the slow evolution of the two wavenumbers, mass, and momentum of modulated periodic traveling waves. The modulation equations are further analyzed in the dispersionless and weakly nonlinear regimes. The ill-posedness of the dispersionless equations corresponds to the classical criterion for modulational instability (MI). For modulations of linear waves, ill-posedness coincides with the generalized MI criterion, recently identified by Amiranashvili and Tobisch [New J. Phys., 21 (2019), 033029]. A new instability index is identified by the transition from real to complex characteristics for the weakly nonlinear modulation equations. This instability is associated with long wavelength modulations of nonlinear two-phase wavetrains and can exist even when the corresponding one-phase wavetrain is stable according to the generalized MI criterion. Another interpretation is that while infinitesimal perturbations of a periodic wave may not grow, small but finite amplitude perturbations may grow, hence this index identifies a nonlinear instability mechanism for one-phase waves. Classifications of instability indices for multiple FDNLS equations with higher-order dispersion, including applications to finite-depth water waves and the discrete NLS equation, are presented and compared with direct numerical simulations.
全分散广义非线性薛定谔方程的惠瑟姆调制理论和两相不稳定性
SIAM 应用数学杂志》,第 84 卷第 4 期,第 1337-1361 页,2024 年 8 月。 摘要。在半经典体系中考虑了具有全色散的广义非线性薛定谔方程(FDNLS)。为具有一般线性色散和广义局部非线性的 FDNLS 方程求出了惠瑟姆调制方程。假设存在一个四参数的两相解系,通过多尺度方法可以得到四个独立的一阶准线性流体力学守恒定律系统,它们对应于调制周期性行波的两个波长、质量和动量的缓慢演化。我们进一步分析了无色散和弱非线性状态下的调制方程。无色散方程的非确定性对应于经典的调制不稳定性(MI)标准。对于线性波的调制,其失稳性与 Amiranashvili 和 Tobisch 最近确定的广义 MI 准则相吻合[New J. Phys., 21 (2019), 033029]。弱非线性调制方程从实数特征到复数特征的转变确定了一个新的不稳定指数。这种不稳定性与非线性两相波轨的长波长调制有关,即使根据广义 MI 准则,相应的单相波轨是稳定的,这种不稳定性也可能存在。另一种解释是,周期波的无穷小扰动可能不会增长,但振幅有限的小扰动可能会增长,因此该指数确定了单相波的非线性不稳定机制。本文介绍了具有高阶分散性的多重 FDNLS 方程的不稳定指数分类,包括对有限深度水波和离散 NLS 方程的应用,并与直接数值模拟进行了比较。
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来源期刊
CiteScore
3.60
自引率
0.00%
发文量
79
审稿时长
12 months
期刊介绍: SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.
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