Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

We present a general result of transverse nonlinear instability of 1d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1-d model and the transverse perturbation “have the same sign”. Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Boussinesq system and the Zakharov–Kuznetsov equation and we hope that it may be useful in other contexts.     

Asymptotic models for internal waves

Derived here in a systematic way, and for a large class of scaling regimes are asymptotic models for the propagation of internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on , d=1,2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators with respect to suitable small parameters that depend variously on the amplitude, wave-lengths and depth ratio of the two layers. We rigorously derive classical models and also some model systems that appear to be new. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established.     

Canonical conformal variables based method for stability of Stokes waves

We study the stability of Stokes waves in an ideal fluid of infinite depth. The perturbations that are either coperiodic with a Stokes wave (superharmonics) or integer multiples of its period (subharmonics) are considered. The eigenvalue problem is formulated using the conformal canonical Hamiltonian variables and admits numerical solution in a matrix-free manner. We find that the operator matrix of the eigenvalue problem can be factored into a product of two operators: a self-adjoint operator and an operator inverted analytically. Moreover, the self-adjoint operator matrix is efficiently inverted by a Krylov-space-based method and enjoys spectral accuracy. Application of the operator matrix associated with the eigenvalue problem requires only $O(N\log N)$ flops, where N is the number of Fourier modes needed to resolve a Stokes wave. Additionally, due to the matrix-free approach, $O(N^2)$ storage for the matrix of coefficients is no longer required. The new method is based on the shift-invert technique, and its application is illustrated in the classic examples of the Benjamin–Feir and the superharmonic instabilities. Simulations confirm numerical results of preceding works and recent theoretical work for the Benjamin–Feir instability (for small amplitude waves), and new results for large amplitude waves are shown.     

The role of thermal instability in star formation

The observations tell us that the density in the giant molecularclouds in which stars are formed is inhomogeneous on a varietyof scales, but it seems unlikely that this is due to the actionof gravitational instability. This paper describes numericalcalculations using an adaptive mesh refinement magnetohydrodynamicscode that show that thermal instability may have an importantrole to play in the formation of this structure     

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.     

Application of compact and multioperator schemes to the numerical simulation of acoustic fields generated by instability waves in supersonic jets

Acoustic fields generated by instability waves in supersonic jets were numerically simulated. A seventh-order multioperator scheme was used to solve the Euler equations linearized about the mean flow field in an axisymmetric turbulent jet. The mean field was computed using fifth-order compact approximations of the convective terms under conditions similar to experimental data. The numerical results were found to agree well with the experiment.     

Multiple time scale formalism and its application to long water waves

In the present work, by employing the multiple time scaling method, we studied the nonlinear waves in shallow-water problem and obtained a set of Korteweg–deVries equations governing the various order terms in the perturbation expansion. By seeking a travelling wave type of solutions for the evolution equations, we have obtained a set of wave speeds associated with each time parameter. It is shown that the speed corresponding to the lowest order time parameter given the wave speed of the conventional reductive perturbation method, whereas the wave speeds corresponding to the higher order time parameters give the speed correction terms. The result obtained here is exactly the same with that of Demiray [H. Demiray, Modified reductive perturbation method as applied to long water waves: Korteweg–deVries hierarchy, Int. J. Nonlinear Sci. 6 (2008) 11–20] who employed the modified reductive perturbation method.     

Water waves diffraction by a circular plate

The interaction of water waves with circular plate within the framework of a linear theory is considered. The plate lies on the free surface in water of finite depth. The integral transform technique is used to solve this problem. The problem is reduced to a system of dual integral equations for a spectral function. The way to solve these equations consists in converting them into Fredholm integral equation of the second kind. The asymptotic solutions of this equation are obtained. Representations for diffraction field and for the forces on the plate are given.     

Shock waves in the modified Korteweg-de Vries-Burgers equation

Summary The asymptotics ast + of shock waves of the modified Korteweg-de Vries-Burgers (MKdV-B) equation is investigated. An attractor interpretation of shock problems for integrable systems is presented and some problems of nonlinear stability are discussed. The MKdV-B equation is considered as a nonconservative perturbation of the integrable modified Korteweg-de Vries (MKdV) equation. The MKdV equation considered here has anon-self-adjoint Lax pair. In spite of this difficulty, acomplex Whitham deformation is constructed.     

Boundedness of imaginary powers of the stokes operator in an infinite layer

In this article we prove the existence of bounded purely imaginary powers of the Stokes operator , which is defined on the space of solenoidal vector fields < q < , where is an infinite layer. It is a consequence of a special representation of the resolvent of the Stokes operator in terms of the Stokes operator on , a composition of a trace and a Poisson operator – a singular Green operator – and a negligible part.     

Unsteady stokes equations: Some complete general solutions

The completeness of solutions of homogeneous as well as non-homogeneous unsteady Stokes equations are examined. A necessary and sufficient condition for a divergence-free vector to represent the velocity field of a possible unsteady Stokes flow in the absence of body forces is derived.     

Existence and qualitative theory for stratified solitary water waves

This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation.We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.     

NONCONFORMING FINITE ELEMENT PENALTY METHOD FOR STOKES EQUATION

a special penalty method is presented to improve the accuracy of the standard penaltymethod (or solving Stokes equation with nonconforming finite element, It is shown that thismethod with a larger penalty parameter can achieve the same accuracy as the staodaxd methodwith a smaller penalty parameter. The convergence rate of the standard method is just hall order of this penalty method when using the same penalty parameter, while the extrapolationmethod proposed by Faik et al can not yield so high accuracy of convergence. At last, we alsoget the super-convergence estimates for total flux.     

Some spatial decay estimates in time-dependent stokes slow flows

This paper uses a time-weighted cross-sectional area measure in order to establish spatial decay estimates for the time-dependent Stokes slow flow of an incompressible viscous fluid in a semi- infinite cylindrical pipe of smooth cross section. The decay rate predicted in this paper depends only on the constant kinematic viscosity and the smallest positive eigenvalue of the free membrane problem as well as two geometric characteristics of the cross section of the pipe.     

A type of new posteriori error estimators for stokes problems

1. IntroductionIn the numerical approximation of PDE, it is often very importals to detect regionswhere the accuracy of the numerical solution is degraded by local singularities of the solutionof the continuous problem such as the singularity near the re-entrant corller. An obviousremedy is to refine the discretization in the critical regions, i.e., to place more gridpointswhere the solution is less regular. The question is how to identify these regions automdticallyand how to determine a goo…     

A discontinuous solution for an evolution compressible stokes system in a bounded domain

An evolution compressible Stokes system is studied in a bounded cylindrical region . The initial datum of pressure is assumed to have a jump at a specified curve C₀ in Ω. As predicted by the Rankine-Hugoniot conditions, the pressure and velocity derivatives have jump discontinuities along the characteristic plane of the curve C₀ directed by an ambient velocity vector. An explicit formula for the jump discontinuity is presented. The jump decays exponentially in time, more rapidly for smaller viscosities. Under suitable conditions of the data, a regularity of the solution is established in a compact subregion of Q away from the jump plane.     

Existence,orbital stability and instability of solitary waves for coupled BBM equations

This paper is concerned with the orbital stability/instability of solitary waves for coupled BBM equations which have Hamiltonian form. The explicit solitary wave solutions will be worked out first. Then by detailed spectral analysis and decaying estimates of solutions for the initial value problem, we obtain the orbital stability/instability of solitary waves.     

Axisymmetric stokes exterior problem and its numerical computation

We establish variational formulation and prove the existence and uniqueness of the three dimensional axisymmetric Stokes exterior problem in weighted spaces.Error estimates and convergence for P2-P0 elements with infinite element methods are also obtained.Numerical experiments are presented to verify the theoretical analysis.