Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves

We give a detailed analysis of the interaction of two NLS-described wave packets with different carrier waves for a nonlinear wave equation. By separating the internal dynamics of each wave packet from the dynamics caused by the interaction we prove that there is almost no interaction of such wave packets. We also prove the validity of a formula for the envelope shift caused by the interaction of the wave packets.     

A new solitary wave solution for water waves with surface tension

In this paper we show that when the Froude number is less than but close to 1 and the Bond number is greater than but close to 1/3 there exists a new solitary wave solution for surface waves on water with surface tension. An approximate expression for the new solitary wave solution, which satisfies a fourth order ordinary differential equation and represents a wave of depression is presented.     

等离子体中双流体模型的调制逼近北大核心CSCD

等离子体中的双流体模型描述了丰富的等离子体动力学行为,包括离子声波和等离子体波之间的相互作用.为了描述该双流体模型小振荡波包解包络的演化,利用多尺度分析方法将非线性Schrödinger(NLS)方程作为形式逼近方程导出,并通过对该双流体模型的真实解和逼近解之间的误差,在Sobolev空间中进行了一致能量估计,最终在时间尺度O(ε^(-2))上严格证明了NLS逼近的有效性.     

Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind

The nonlinear Schrödinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The ‘derivative’ NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen-Lee-Liu equation, are studied. More precisely, the complex envelope is separated into the absolute value and the phase. The absolute value is solved in terms of a polynomial in elliptic functions while the phase is expressed in terms of elliptic integrals of the third kind. The exact periodicity condition will imply that only a countable set of elliptic function moduli is allowed. This feature contrasts sharply with other periodic solutions of envelope equations, where a continuous range of elliptic function moduli is permitted.     

Linear and nonlinear electromagnetic waves in modulated honeycomb media

Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points, which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two-dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain-wall-like mass term, we realize the subtle wave motions, which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high-accuracy numerical computations. Moreover, we also report a new type of nonlinear modes, which are localized in both directions.     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

Validity of the NLS approximation for periodic quantum graphs

We consider a nonlinear Schrödinger (NLS) equation on a spatially extended periodic quantum graph. With a multiple scaling expansion, an effective amplitude equation can be derived in order to describe slow modulations in time and space of an oscillating wave packet. Using Bloch wave analysis and Gronwall’s inequality, we estimate the distance between the macroscopic approximation which is obtained via the amplitude equation and true solutions of the NLS equation on the periodic quantum graph. Moreover, we prove an approximation result for the amplitude equations which occur at the Dirac points of the system.     

The Evolution of Nonlinear Wave Trains in Stratified Shear Flows

The propagation of an internal wave train in a stratified shear flow is investigated for a Boussinesq fluid in a horizontal channel. Linear effects are primarily reflected in the dispersion relation for the various modes. The phenomenon of Eckart resonance occurs for more realistic stratification profiles. The evolution of nonlinear internal wave packets is studied through a systematic perturbation analysis. A nonlinear Schrodinger equation for the envelope of the internal wave train is derived. Depending on the relative sign of the dispersive and nonlinear terms, a wave train may disperse or form an envelope soliton. The analysis demonstrates the existence of two types of critical layers: one the ordinary critical point where ū=c, while the other occurs where ū=c_g. In order to calculate the coefficients of the nonlinear Schrodinger equation a numerical code has been developed which computes the second-harmonic and induced mean motions. The existence of these envelope solitons and their dependence on environmental conditions are discussed.     

Weakly Nonlinear Evolution of a Wave Packet in a Zonal Mixing Layer

A nonlinear integrodifferential equation governing the amplitude evolution of a wavepacket near the critical value of the beta parameter is derived. The basic velocity profile is a hyperbolic tangent shear layer and although the neutral eigensolution is regular, all higher-order terms in the expansion of the stream function are singular at the critical point. The analysis is inviscid and in the critical layer both wave packet effects and nonlinearity are present, but the former are taken to be slightly larger. Unlike the Stuart–Watson theory, the critical layer analysis dictates the form of the amplitude equation, the outer expansion being relatively passive. A secondary instability analysis shows that the packet is unstable to sideband perturbations, but the instability is weak so its main consequence would be to produce some modulation of the packet without destroying its coherence.     

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.     

On the optimal focusing of solitons and breathers in long‐wave models

Conditions of optimal (synchronized) collisions of any number of solitons and breathers are studied within the framework of the Gardner equation (GE) with positive cubic nonlinearity, which in the limits of small and large amplitudes tends to other long‐wave models, the classic and the modified Korteweg–de Vries equations. The local solution for an isolated soliton or breather within the GE is obtained. The wave amplitude in the focal point is calculated exactly. It exhibits a linear superposition of partial amplitudes of the solitons and breathers. The crucial role of the choice of proper soliton polarities and breather phases on the cumulative wave amplitude in the focal point is demonstrated. Solitons are most synchronized when they have alternating polarities. The straightforward link to the problem of synchronization of envelope solitons and breathers in the focusing nonlinear Schrödinger equation is discussed (then breathers correspond to envelope solitons propagating above a condensate).     

The Modulation of an Internal Gravity-Wave Packet,and the Resonance with the Mean Motion

An inviscid, incompressible, stably stratified fluid occupies a horizontal channel, along which an internal gravity-wave packet is propagating. The wave induced mean motions are calculated, and the equations describing the evolution of the wave amplitude derived. When the group velocity of the wave packet coincides with a long-wave speed there is a resonance, and the equations describing this resonance are derived.     

Massive neutral Dirac particle in a curved space-time with the Kerr-Schild metric

A method for constructing the solution of the general covariant Dirac equation is developed. The solution has the form of a wave packet in the case of a slightly (compared to the wave packet size) curved space-time with the Kerr-Schild metric and describes the evolution of the spin states of a massive neutral particle with a half-integer spin. The method allows reducing the Dirac equation to a system of ordinary differential equations for the spinor amplitudes (with the necessary accuracy). We propose an iterative procedure for solving the system based on expansion with respect to a small parameter equal to the ratio between the particle wave length and the characteristic spatial scale of the change of the metric. The characteristic dissipation time of a wave packet moving in a curved space-time is estimated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 343–352, November, 2000.     

Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances

The nonlinear Schrödinger (NLS) equation can be derived as an amplitude equation describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. The purpose of this paper is to prove estimates, between the formal approximation, obtained via the NLS equation, and true solutions of the original system in case of non-trivial quadratic resonances. It turns out that the approximation property (APP) holds if the approximation is stable in the system for the three-wave interaction (TWI) associated to the resonance. We construct a counterexample showing that the NLS equation can fail to approximate the original system if instability occurs for the approximation in the TWI system. In the unstable case we give some arguments why the validity of the APP can be expected for spatially localized solutions and why it cannot be expected for non-localized solutions. Although, we restrict ourselves to a nonlinear wave equation as original system we believe that the results hold in more general situations, too.     

Wave-Mean Flow Interactions in Thermally Stratified Poiseuille Flow

We consider nonlinear wave motions in thermally stratified Poiseuille flow. Attention is focused on short wavelength wave modes for which the neutral Reynolds number scales as the square of the wave number. The nonlinear evolution of a single monochromatic wave is governed by a first harmonic/mean-flow interaction theory in which the wave-induced mean flow is comparable in size to the wave component of the flow. An integrodifferential equation is derived which governs the normal variation of the wave amplitude. This equation admits finite-amplitude solutions which bifurcate supercritically from the linear neutral point(s).     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Lagrangian motion of fluid particles in gravity–capillary standing waves

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads to a change of the crest form. Moreover, we also find that the particle trajectories near the surface oscillate back and forth along the arcs which will change from concave to convex as the inverse Bond number increases. There is no mass transport of the particles in a wave period.