Weak Nonlinear Long Waves in Channel Flow with Internal Dissipation

This article concerns the evolution of long waves ( O (ε^−1/2) wavelength) of small [ O (ε)] amplitude in channel flow with internal dissipation. We use multiple scale expansions to derive a generalized Kuramoto–Sivashinsky (GKS) equation that governs the dominant asymptotic solution in the limit of small disturbances and marginal linear instability. We compare this solution with numerical integrations of the full quasilinear system, and show that the error is consistent with an asymptotic solution to ε^3/2 over a time interval of order ε^−3/2.     

On Nonlinear Kelvin-Helmholtz Waves Near Direct Resonance

The evolution equation for the nonlinear Kelvin-Helmholtz wave envelope with its carrier wavenumber near direct resonance is formulated directly by using the nonlinear dispersion relation. The stability of a wavetrain is examined, and the long-time evolution for an arbitrary initial condition is studied through inverse scattering transforms.     

Long Nonlinear Waves in Resonance with Topography

The evolution of periodic long surface waves over a periodic bottom topography resonant with the waves is studied. Coupled Korteweg–de Vries equations are derived and describe the evolution in terms of interaction between right- and left-traveling waves. The coupling arises from the cumulative effect of wave scattering. We discuss the various conserved quantities of the system and compute solutions for the initial value problem and for the time-periodic problem of fluid "sloshing" in a tank. Some three-dimensional extensions are discussed.     

Some Properties of Long Nonlinear Waves

This paper is concerned with certain mathematical aspects of long nonlinear wave propagation on a free surface. Some special solutions are noted and the equations are shown to have an infinite number of conservation laws.     

Long Nonlinear Surface Waves in the Presence of a Variable Current

We investigate the eigenvalue problem governing the propagation of long nonlinear surface waves when there is a current beneath the surface, y being the vertical coordinate. The amplitude of such waves evolves according to the KdV equation and it was proved by Burns [ 1 ] that their speed of propagation c is such that there is no critical layer (i.e., c lies outside the range of ). If, however, the critical layer is nonlinear, the result of Burns does not necessarily apply because the phase change of linear theory then vanishes. In this paper, we consider specific velocity profiles and determine c as a function of Froude number for modes with nonlinear critical layers. Such modes do not always exist, the case of the asymptotic suction profile being a notable example. We find, however, that singular modes can be obtained for boundary layer profiles of the Falkner–Skan similarity type, including the Blasius case. These and other examples are treated and we examine singular solutions of the Rayleigh equation to gain insight about the long wave limit of such solutions.     

Long Time Behavior of Nonlinear Strain Waves in Elastic Waveguides

The initial-boundary value problem of the propagation of nonlinear longitudinal elastic waves in an initially strained rod is considered. The rod is assumed to interact with the surrouding elastic and viscous external medium. The long time behavior of solutions is derived ancl global attractors in E_1 space is obtained.     

Evolution Equations for Long,Nonlinear Internal Waves in Stratified Shear Flows

Evolution equations for long, nonlinear internal waves are derived when the basic stratified shear flow has a slow temporal and spatial variation as well as the usual dependence on the vertical coordinate. When the horizontal waveguide has a limited vertical extent the evolution equation is a variable coefficient Korteweg-deVries equation, while in the deep fluid case the evolution equation is a variable coefficient Benjamin-Davis-Ono equation. Explicit expressions are obtained for the coefficients of these equations.     

A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

The Miyata–Choi–Camassa (MCC) system of equations describing long internal nonhydrostatic and nonlinear waves at the interface between two layers of inviscid fluids of different densities bounded by top and bottom walls is mathematically ill‐posed despite the fact that physically stable internal waves are observed matching closely those of MCC. A regularization to the MCC equations that yields a computationally simple well‐posed system for time‐dependent evolution is proposed here. The regularization is accomplished by keeping the full hyperbolic part of MCC and exchanging spatial and temporal derivatives in one of the linearized dispersive terms. Solitary waves of MCC over a wide range of parameters are used as a benchmark to check the accuracy of the model. Our model includes the possibility of a background shear, and we show that, contrary to the no shear case, solitary waves can cross the midlevel between the top and the bottom walls and may have different polarity from the case with no background shear. Time‐dependent solutions of the regularization stable model are presented, including interactions of its solitary waves, and classical and modified Korteweg‐de Vries equations for small amplitude waves with the inclusion of background shear are derived. Throughout the paper, the Boussinesq approximation is taken, although the results can be extended to the non‐Boussinesq case.     

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.     

Gravity Waves in a Channel with a Rough Bottom

We derive effective equations for the surface elevation of gravity waves in a shallow channel with a rough bottom.     

Forced Waves Near Resonance at a Phase-Speed Minimum

When a dispersive wave system is subject to forcing by a moving external disturbance, a maximum or minimum of the phase speed is associated with a critical forcing speed at which the linear response is resonant. Nonlinear effects can play an important part near such resonances, and the salient characteristics of the nonlinear response depend on whether the maximum or minimum of the phase speed is realized in the long-wave limit (zero wavenumber) or at a finite wavenumber. The focus here is on the latter case that, among other physical systems, applies to gravity–capillary waves on water of finite or infinite depth. The analysis, for simplicity, is based on a forced–damped fifth-order Korteweg–de Vries equation, a model problem that features a phase-speed minimum at a finite wavenumber. When damping is not too strong compared with forcing, multiple subcritical finite-amplitude steady-solution branches coexist with the small-amplitude response predicted by linear theory. For forcing speed well below critical, the transient response from rest approaches the small-amplitude state, but at speeds close to critical, jump phenomena can occur, and reaching a time-periodic state that involves shedding of wavepacket solitary waves is also possible.     

长水波近似方程组的新精确解

依据齐次平衡法的思想 ,首先提出了求非线性发展方程精确解的新思路 ,这种方法通过改变待定函数的次序 ,优势是使求解的复杂计算得到简化 .应用本文的思路 ,可得到某些非线性偏微分方程的新解 .其次我们给出了长水波近似方程组的一些新精确解 ,其中包括椭圆周期解 ,我们推广了有关长波近似方程的已有结果 .     

The Cauchy Problem for the Nonlinear Long Longitudinal Waves Equation in a Viscoelastic Rod

We study the Cauchy problem in the space of continuous functions for some nonlinear differential equation of Sobolev type that simulates longitudinal waves in an infinite viscoelastic rod. Under consideration are the conditions for the existence of the global classical solution and the blow-up of the solution to the Cauchy problem on a finite time interval.

     

Nonlinear Waves in a Shear Flow with a Vorticity Discontinuity

We consider nonlinear finite-amplitude progressive shear-flow waves on a basic velocity profile consisting of two coflowing layers of inviscid equal-density fluid, each of uniform but different vorticity. The problem is formulated as a nonlinear integral equation describing the shape of the vorticity discontinuity in a frame of reference in which the flow is steady. Numerical solutions to this equation are presented for a range of values of the vorticity ratio Ω. For 1 > © ≥ ? 1 the theoretical maximum wave amplitude occurs when the wave crest forms a 90° corner which just touches the appropriate critical-layer stagnation point. The linearized stability of the progressive wave states to arbitrary subharmonic isovortical disturbances is studied numerically. The results indicate stability at moderate values of the wave amplitude.     

Integrable Nonlinear Equations for Water Waves in Straits of Varying Depth and Width

A considerable amount of information is currently available on the creation and propagation of large solitary waves in marine straits. In order to be able to analyze such data we develop a theoretical model, extending previous one-dimensional models to the case of straits with varying width and depth, and nonvanishing vorticity. Starting from the Euler equations for a three-dimensional homogeneous incompressible inviscid fluid, we derive, in the quasi-one-dimensional long-wave and shallow-water approximation, a generalized KadomtsevPetviashvili (GKP) equation, together with its appropriate boundary conditions. In general, the coefficients of this equation depend on the form of the bottom and on the vorticity; the sides of the straits figure only in the boundary conditions. Under certain restrictions on the vorticity and the geometry of the straits we reduce the GKP equation to one of several completely integrable partial differential equations, in order to study the evolution of solitons which originate in the straits.     

Branching Near Plane Waves in Perturbed Dispersive Systems

A Poincaré-Lindstedt type technique for partial differential equations is used to study branching phenomena in perturbed dispersive systems arising in hydrodynamic stability theory. Multi-periodic waves with two frequencies which branch from a family of neutrally stable nonlinear periodic plane waves are constructed, the second frequency as a power series expansion in ε. The branching is compared with that of the unperturbed equations described in an earlier paper for the purpose of understanding how higher order perturbation terms effect the properties of the lower order amplitude equations. We find that in general the perturbation terms alter the leading order frequency shifts, thus changing the bifurcation from pitchfork to transverse type. The method is used to study the perturbed nonlinear Schrödinger equation and the perturbed MKdV equation.     

Weakly Nonlinear Internal Waves in Shear

An evolution equation in a finite depth fluid for weakly nonlinear long internal waves is derived in a stratified and sheared medium. The equation reduces to the Korteweg-deVries equation when the depth is small compared to the wavelength, and to the Benjamin-Ono equation when the depth is large compared to the wavelength. Both the cases with and without critical levels are investigated. Numerical solutions to the evolution equation are presented to illustrate the effect of shear on the evolution of a waveform.