Instability of nonlinear dispersive solitary waves

We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula is used to find the instability criteria. These techniques have also been extended to study instability of periodic waves and of the full water wave problem.     

Well-posedness in Sobolev spaces of the full water wave problem in 3-D

We consider the motion of the interface of a 3-D inviscid, incompressible, irrotational water wave, with air region above water region and surface tension zero. We prove that the motion of the interface of the water wave is not subject to Taylor instability, as long as the interface separates the whole 3-D space into two simply connected regions. We prove further the existence and uniqueness of solutions of the full 3-D water wave problem, locally in time, for any initial interface that separates the whole 3-D space into two simply connected regions.

     

Spectral Stability of Deep Two‐Dimensional Gravity‐Capillary Water Waves

In this contribution we study the spectral stability problem for periodic traveling gravity‐capillary waves on a two‐dimensional fluid of infinite depth. We use a perturbative approach that computes the spectrum of the linearized water wave operator as an analytic function of the wave amplitude/slope. We extend the highly accurate method of Transformed Field Expansions to address surface tension in the presence of both simple and repeated eigenvalues, then numerically simulate the evolution of the spectrum as the wave amplitude is increased. We also calculate explicitly the first nonzero correction to the flat‐water spectrum, which we observe to accurately predict the stability (or instability) for all amplitudes within the disk of analyticity of the spectrum. With this observation in mind, the disk of analyticity of the flat state spectrum is numerically estimated as a function of the Bond number and the Bloch parameter, and compared to the value of the wave slope at the first finite amplitude eigenvalue collision.     

Instability of solitary waves for a generalized Benney–Luke equation

We study linear instability of solitary wave solutions of a one-dimensional generalized Benney–Luke equation, which is a formally valid approximation for describing two-way water wave propagation in the presence of surface tension. Further, we implement a finite difference numerical scheme which combines an explicit predictor and an implicit corrector step to compute solutions of the model equation which is used to validate the theory presented.     

Resonant instability and nonlinear wave interactions in electrically forced jets

We investigate the problem of linear temporal instability of the modes that satisfy the dyad resonance conditions and the associated nonlinear wave interactions in jets driven by either a constant or a variable external electric field. A mathematical model, which is developed and used for the temporally growing modes with resonance and their nonlinear wave interactions in electrically driven jet flows, leads to equations for the unknown amplitudes of such waves. These equations are solved for both water and glycerol jet cases, and the expressions for the dependent variables of the corresponding modes are determined. The results of the generated data for these dependent variables versus time indicate, in particular, that the instability resulted from the nonlinear interactions of such modes is mostly quite strong but can also lead to significant reduction in the jet radius.     

Modulational Instability in the Whitham Equation for Water Waves

We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long‐wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin–Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small‐amplitude parameter. We extend the result to related, nonlinear dispersive equations.     

Surfactant Influence on Water Wave Packets

The influence of surfactant on water wave packets is investigated. An envelope equation for a slowly varying wave packet in the potential flow equations with variable Bond number is derived. The properties of this equation depend on the relative phases of the wave packet and the distribution of surface tension. We observe that small variations in the Bond number may change the focusing nature of the envelope equation from that of the constant Bond number problem. Variations in Bond number can thus suppress, or incite, the Benjamin‐Feir instability. The existence of envelope solitary waves depends in a similar way on the Bond number variation. The envelope equation is also derived in a larger class of models.     

波浪、海洋土参数对海床稳定性影响

基于Yamamoto的多孔弹性介质模型,研究了波生底床的稳定性.通过给出的有限深底床下土响应分析解,针对三种土质底床,讨论了主要波参数和土参数对这些底床稳定性的影响.与其他土模型计算结果进行了比较,分析了海洋土内部Coulomb摩擦因素的影响.     

Dynamics of water evaporation fronts

The evolution and shapes of water evaporation fronts caused by long-wave instability of vertical flows with a phase transition in extended two-dimensional horizontal porous domains are analyzed numerically. The plane surface of the phase transition loses stability when the wave number becomes infinite or zero. In the latter case, the transition to instability is accompanied with reversible bifurcations in a subcritical neighborhood of the instability threshold and by the formation of secondary (not necessarily horizontal homogeneous) flows. An example of motion in a porous medium is considered concerning the instability of a water layer lying above a mixture of air and vapor filling a porous layer under isothermal conditions in the presence of capillary forces acting on the phase transition interface.     

Dynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water

We study the dynamics of large amplitude internal solitary waves in shallow water by using a strongly nonlinear long-wave model. We investigate higher order nonlinear effects on the evolution of solitary waves by comparing our numerical solutions of the model with weakly nonlinear solutions. We carry out the local stability analysis of solitary wave solution of the model and identify an instability mechanism of the Kelvin–Helmholtz type. With parameters in the stable range, we simulate the interaction of two solitary waves: both head-on and overtaking collisions. We also study the deformation of a solitary wave propagating over non-uniform topography and describe the process of disintegration in detail. Our numerical solutions unveil new dynamical behaviors of large amplitude internal solitary waves, to which any weakly nonlinear model is inapplicable.     

Modulation of Gravity Waves with Shear in Water

The effect of water shear on the stability of infinitesimal perturbations (in the form of side bands) to a finite-amplitude gravity wave is investigated both numerically and analytically. The shear is modeled by a piecewise-linear velocity profile. Nonlinear cubic Schrödinger equation for the wave envelope of a slowly varying wave train is derived. It is shown that depending on the direction of propagation (along or against the shear) of the finite-amplitude waves, the effect of shear on the stability is substantially different. In most cases, however, the shear strength increase first enhances, but later suppresses, the instability.     

Transverse linear instability of solitary waves for coupled long-wave–short-wave interaction equations

In this paper, we investigate the transverse linear instability of one-dimensional solitary wave solutions of the coupled system of two-dimensional long-wave–short-wave interaction equations. We show that the one-dimensional solitary waves are linearly unstable to perturbations in the transverse direction if the coefficient of the term associated with transverse effects is negative. This transverse instability condition coincides with the non-existence condition identified in the literature for two-dimensional localized solitary wave solutions of the coupled system.     

On the linear stability of simple and semi-simple periodic waves for a system of cubic Klein–Gordon equations

We study the linear stability of traveling wave solutions for the nonlinear wave equation and coupled nonlinear wave equations. It is shown that periodic waves of the dnoidal type are spectrally unstable with respect to co-periodic perturbations. Our arguments rely on a careful spectral analysis of various self-adjoint operators, both scalar and matrix and on instability index count theory for Hamiltonian systems.     

Instability of the finite‐difference split‐step method applied to the nonlinear Schrödinger equation. I. standing soliton

We consider numerical instability that can be observed in simulations of solitons of the nonlinear Schrödinger equation (NLS) by a split‐step method (SSM) where the linear part of the evolution is solved by a finite‐difference discretization. The von Neumann analysis predicts that this method is unconditionally stable on the background of a constant‐amplitude plane wave. However, simulations show that the method can become unstable on the background of a soliton. We present an analysis explaining this instability. Both this analysis and the features and threshold of the instability are substantially different from those of the Fourier SSM, which computes the linear part of the NLS by a spectral discretization. For example, the modes responsible for the numerical instability are not similar to plane waves, as for the Fourier SSM or, more generally, in the von Neumann analysis. Instead, they are localized at the sides of the soliton. This also makes them different from “physical” (as opposed to numerical) unstable modes of nonlinear waves, which (the modes) are localized around the “core” of a solitary wave. Moreover, the instability threshold for thefinite‐difference split‐step method is considerably relaxed compared with that for the Fourier split‐step. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1002–1023, 2016     

The effects of viscosity on the linear stability of damped Stokes waves,downshifting, and rogue wave generation

We investigate a higher order nonlinear Schrödinger equation with linear damping and weak viscosity, recently proposed as a model for deep water waves exhibiting frequency downshifting. Through analysis and numerical simulations, we discuss how the viscosity affects the linear stability of the Stokes wave solution, enhances rogue wave formation, and leads to permanent downshift in the spectral peak. The novel results in this work include the analysis of the transition from the initial Benjamin–Feir instability to a predominantly oscillatory behavior, which takes place in a time interval when most rogue wave activity occurs. In addition, we propose new criteria for downshifting in the spectral peak and determine the relation between the time of permanent downshift and the location of the global minimum of the momentum and the magnitude of its second derivative.     

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.     

Transverse nonlinear instability for two-dimensional dispersive models

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.     

Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms

We report exact bright and dark solitary wave solution of the nonlinear Schrodinger equation (NLSE) in cubic–quintic non-Kerr medium adopting phase–amplitude ansatz method. We have found the solitary wave parameters along with the constraints under which bright or dark solitons may exist in such a media. Furthermore, we have also studied the modulation instability analysis both in anomalous and normal dispersion regime. The role of fourth order dispersion, cubic–quintic nonlinear parameter and self-steeping parameter on modulation instability gain has been investigated.     

自由剪切流大尺度结构的二次稳定性*

本文用二次稳定性理论研究自由剪切湍流中周期性基本流空间增长扰动的稳定性。数值结果表明三维亚谐扰动对横向波数有很强的选择性,二维亚谐波的空间增长率最大。与之相反,基本模式的三维扰动在很大的波数范围内存在不稳定性,证明β=0时存在“转移”不稳定性;当KH波的幅值A≥0.06时出现分叉现象。     

PSE在超音速边界层二次失稳问题中的应用

用抛物化稳定性方程（PSE）研究超音速边界层中的二次失稳问题．结果显示无论二维基本扰动是第一模态还是第二模态的T-S波,二次失稳机制都起作用．三维亚谐波的放大率随其展向波数和二维基本波幅值的变化关系与不可压缩边界层中所得类似．但是,即使二维波的幅值大到2％的量级,三维亚谐波的最大放大率仍远小于最不稳定的第二模态二维T-S波的放大率．因此,二次失稳应该不是导致超音速边界层转捩的主要因素．