相关论文: On evolution of multiphase nonlinear modulated wav…

Stationary solutions on a bounded interval for an initial-boundary value problem to Korteweg--de~Vries and modified Korteweg--de~Vries equation (for the last one both in focusing and defocusing cases) are constructed. The method of the…

We derive an asymptotic formula for the amplitude distribution in a fully nonlinear shallow-water solitary wave train which is formed as the long-time outcome of the initial-value problem for the Su-Gardner (or one-dimensional Green-Naghdi)…

We study the spectral stability of a family of periodic standing wave solutions to the generalized KdV (g-KdV) in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In…

We develop a general framework to describe the cubically nonlinear interaction of a unidirectional degenerate quartet of deep-water gravity waves. Starting from the discretised Zakharov equation, and thus without restriction on spectral…

In this article we address some issues related to the initial value problems for a rotating shallow water hyperbolic system of equations and the diffusive regularization of this system. For initial data close to the solution at rest, we…

We consider the extended Korteweg-de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves. In the slow time formulation this equation generates fast propagating resonant radiation due to the non-convexity of…

We prove wave breaking --- bounded solutions with unbounded derivatives --- in the nonlinear nonlocal equation which combines the dispersion relation of water waves and a nonlinearity of the shallow water equations, provided that the slope…

A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach,…

The magnetohydrodynamic equations system for heavy fluid over an arbitrary surface in shallow water approximation is studied in the present paper. It is shown that simple wave solutions exist only for underlying surfaces that are slopes of…

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg--de Vries, equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave…

The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such…

In this paper we present the unification of two existing numerical methods for the construction of solutions of the Korteweg-de Vries (KdV) equation. The first method is used to solve the Cauchy initial-value problem on the line for rapidly…

We study the Cauchy problem for the Whitham modulation equations for monotone increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is…

In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by…

The Whitham equation is a model for the evolution of small-amplitude, unidirectional waves of all wavelengths on shallow water. It has been shown to accurately model the evolution of waves in laboratory experiments. We compute…

In certain models of conformal gravity, the propagation of gravitational waves is governed by a fourth order scalar partial differential equation. We study the initial value problem for a generalization of this equation, and derive a…

In this short communication, we announce an algorithmic procedure for constructing non-uniqueness counter-examples of classical solutions to initial-boundary-value problems for a wide class of linear evolution partial differential…

A laminar flow of a thin layer of mud down an inclined plane under the action of gravity is considered. The instability of a film flow and the formation of finite amplitude waves are studied in the framework of both two-dimensional…

Near linear evolution in Korteweg de Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction.

A general Hamiltonian wave system with quartic resonances is considered, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. The evolution equation for the multimode characteristic…