3+1 dimensional envelop waves and its stability in magnetized dusty plasma

It is well known that there are envelope solitary waves in unmagnetized dusty plasmas which are described by a nonlinear Schrodinger equation (NLSE). A three dimension nonlinear Schrodinger equation for small but finite amplitude dust acoustic waves is first obtained for magnetized dusty plasma in this paper. It suggest that in magnetized dusty plasmas the envelope solitary waves exist. The modulational instability for three dimensional NLSE is studied as well. The regions of stability and instability are well determined in this paper.     

Scattering of an incident acoustic wave by an infinite elastic cylinder in a shallow sea

The scattering of sound waves by an isotropic elastic circular cylinder of infinite extent in a water of finite depth is investigated taking into account the shear waves that can exist in addition to compressional waves in scatterers of solid material. The axis of the cylinder is parallel to the water level. The reflected and transmitted energies are plotted for the various values of the radius of the cylinder, and the farfield scattered pressure is obtained for various depth and presented in a graphical form. Copyright © 2008 John Wiley & Sons, Ltd.     

The Superharmonic Instability of Finite-Amplitude Surface Waves on Water of Finite Depth

Saffman's (1985) theory of the superharmonic stability of two-dimensional irrotational waves on fluid of infinite depth has been generalized to solitary and periodic waves of permanent form on finite uniform depth. The frame of reference for the calculation of the Hamiltonian for periodic waves of finite depth is found to be the frame in which the mean horizontal velocity is zero.     

Wave structure interaction problems in three-layer fluid

Wave structure interaction problems in a three-layer fluid having an elastic plate covered free surface are studied in a three-dimensional fluid domain in both the cases of finite and infinite water depths. Wave characteristics are analyzed from the dispersion relation of the associated wave motion, and approximate results are derived in both the cases of deep water and shallow water waves. Further, the expansion formulae and the associated orthogonal mode-coupling relations are derived for the velocity potentials for the wave structure interaction problems in channels of finite and infinite depths. The utility of the expansion formulae is demonstrated by (1) deriving the source potentials associated with the wave structure interaction problems in a three-layer fluid medium of finite and infinite water depths and (2) analyzing the wave scattering by a partially frozen crack in a floating ice sheet in the three-layer fluid medium in a three-dimensional channel of finite water depth. Various results derived can be used to deal with acoustic wave interaction with flexible structures and other wave structure interaction problems of similar nature arising in different branches of physics and engineering.     

Stability of periodic traveling flexural‐gravity waves in two dimensions

In this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. These waves are referred to as flexural‐gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic traveling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier–Floquet–Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze high‐frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.     

On the incoming water waves against a vertical cliff

The problems of obliquely incident surface water waves against a vertical cliff have been handled in both the cases of water of infinite as well as finite depth by straight-forward uses of appropriate Havelock-type expansion theorems. The logarithmic singularity along the shore-line has been incorporated in a direct manner, by suitably representing the Dirac’s delta function.     

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.     

The zero surface tension limit two‐dimensional water waves

We consider two‐dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases. © 2005 Wiley Periodicals, Inc.     

Amplitude modulation of nonlinear waves in a thick elastic tube filled with an inviscid fluid

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thick tube and the approximate equations of an incompressible inviscid fluid, and then utilizing the reductive perturbation technique the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves is governed by a nonlinear Schrödinger(NLS) equation. The range of modulational instability of the monochromatic wave solution with the initial deformation, material and geometrical characteristics is discussed for some elastic materials.     

Scattering of Water Waves by Two Surface-piercing Vertical Barriers

It is known that there are two infinite sets of discrete frequenciesfor which deep water waves may be either totally reflected ortotally transmitted by a pair of surface-piercing barriers ofequal length. Here the problem is further investigated usingan accurate method based on eigenfunction expansions. It isfound that for sufficiently closely spaced barriers each zeroof reflection is accompanied by a zero of transmission at anearby frequency. As the barrier spacing is increased the twolowest frequency zeros of transmission coalesce and are lost.This loss of zeros of transmission in pairs continues as thebarrier spacing is further increased. The effects of different barrier lengths and finite depth arealso considered. For barriers of different lengths the zerosof reflection no longer occur while the existence of the zerosof transmission is unaffected.     

On stable composite capillary-gravitational waves of finite amplitude on the surface of a fluid of finite depth : PMM vol. 39, n 6, 1975, pp. 1023–1031

The problem of stable plane capillary-gravitational waves of finite amplitude on the surface of a perfect incompressible fluid stream of finite depth is considered. It is assumed that the waves are induced by pressure periodically distributed along the free surface, and that these, unlike induced waves, do not vanish when the pressure becomes constant, are transformed into free waves. Such waves are called composite; they exist similarly to free waves, for particular values of velocity of the stream.The problem, which is rigorously stated, reduces to solving a system of four nonlinear equations for two functions and two constants. One of the equations is integral and the remaining are transcendental. Pressure on the surface is defined by an infinite trigonometric series whose coefficients are proportional to integral powers of some dimensionless small parameter; these powers are by two units greater than the numbers of coefficients.The theorem of existence and uniqueness of solution is established, and the method of its proof is indicated. The derivation of solution in any approximation is presented in the form of series in powers of the indicated small parameter. Computation of the first three approximations is carried out to the end, and an approximate equation of the wave profile is presented.Composite capillary-gravitational waves in the case of fluid of infinite depth were considered by the author in [1].     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

Stability of matter–wave soliton in a time-dependent complicated trap

We examine the possibility to generate localized structures in effectively one-dimensional Gross–Pitaevskii with a time-dependent scattering length and a complicated potential. Through analytical methods invoking a generalized lens-type transformation and the Darboux transformation, we present the integrable condition for the Gross–Pitaevskii equation and obtain the exact analytical solution which describes the modulational instability and the propagation of bright solitary waves on a continuous wave background. The dynamics and stability of this solution are analyzed. Moreover, by employing the extended tanh-function method we obtain the exact analytical solutions which describes the propagation of dark and other families of solitary waves.     

The NLS approximation for two dimensional deep gravity waves

Science China Mathematics - This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic...     

The Cauchy-Poisson problem in an inviscid stratified liquid

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface response of a nonrotating inviscid stratified liquid of finite or infinite depth to the initial displacement of the free surface. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and distance from the origin of disturbance. It is shown that the asymptotic solution consists of the classical free surface gravity waves and the internal waves.     

Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schrödinger equation with time-dependent nonlinearity

In this paper, we consider a general form of nonlinear Schrödinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schrödinger equation is identified by admitting an infinite number of conservation laws. Using the Darboux transformation method, we obtain some explicit bright multi-soliton solutions in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.     

Modulational Instability and Variational Structure

We study the modulational instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension. We examine how the Jordan block structure of the associated linearized operator bifurcates for small values of the Floquet exponent to derive a criterion governing instability to long wavelengths perturbations in terms of the kinetic and potential energies, the momentum, the mass of the underlying wave, and their derivatives. The dispersion operator of the equation is allowed to be nonlocal, for which Evans function techniques may not be applicable. We illustrate the results by discussing analytically and numerically equations of Korteweg‐de Vries type.     

Model Equations for Gravity-Capillary Waves in Deep Water

The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gravity-capillary dispersion relation, but preserving exactly either a quadratic energy or a momentum. Solitary wavepacket waves are calculated for each model. Each model supports the elevation and depression waves known to exist in the Euler equations. The stability of these waves is discussed, as is the dynamics resulting from instabilities and solitary wave collisions.     

Temperature waves in a rigid heat conductor

The propagation of acceleration-temperature waves in a rigid heat conductor is investigated. The theory employed allows temperature to travel with a finite wavespeed, and the full nonlinear theory is analysed. It is shown that various types of behaviour are possible for the amplitude of the wave, including one for which the amplitude becomes infinite in a finite time. Higher order temperature waves are also briefly discussed.