相关论文: Special Polynomials and Exact Solutions of the Dis…
The Korteweg-de Vries (KdV) equation is known as a universal equation describing various long waves in dispersive systems. In this article, we prove that in a certain scaling regime, a large class of rough solutions to the Boussinesq…
Rational solutions of the fourth order analogue to the Painlev'e equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulas for their coefficients…
Historically the finite volume methods have been developed for the numerical integration of conservation laws. In this study we present some recent results on the application of such schemes to dispersive PDEs. Namely, we solve numerically…
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and…
We study rational solutions of the Boussinesq equation, which is a soliton equation solvable by the inverse scattering method. These rational solutions, which are algebraically decaying and depend on two arbitrary parameters, are expressed…
We study the behavior of shallow water waves propagating over bathymetry that varies periodically in one direction and is constant in the other. Plane waves traveling along the constant direction are known to evolve into solitary waves, due…
Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.
It is shown that asymptotically consistent modifications of (Boussinesq-like) shallow water approximations, in order to improve their dispersive properties, can fail for uneven bottoms (i.e., the dispersion is actually not improved). It is…
In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et. al. [Stud. Appl. Math., 53 (1974) 249--315] and one by Hirota and…
A fifth--order nonlinear partial differential equation for the description of nonlinear waves in a liquid with gas bubbles is considered. Special solutions of this equation are studied. Some elliptic and simple periodic traveling waves…
Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean…
Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems.…
This article concludes the study of (2+1)-dimensional nonlinear wave equations that can be derived in a model of an ideal fluid with irrotational motion. In the considered case of identical scaling of the $x,y$ variables, obtaining a…
Some higher-order quasilinear parabolic, hyperbolic, and nonlinear dispersion equations are shown to admit various blow-up, extinction, and travelling wave solutions, which reduce to variational problems admitting countable families of…
Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems.…
In this paper, we investigate the well-posedness of a nonlinear dispersive model with variable coefficients that describes the evolution of surface waves propagating through a one-dimensional shallow water channel of finite length with…
We study solitary-wave and kink-wave solutions of a modified Boussinesq equation through a multiple-time reductive perturbation method. We use appropriated modified Korteweg-de Vries hierarchies to eliminate secular producing terms in each…
Given a nonlinear dispersive equation which admits a scaling invariance, there may exist self-similar solutions. In this work, we present a systematic approach for the construction of small-amplitude self-similar solutions, together with…
New exact solutions to the KdV2 equation (known also as the extended KdV equation) are constructed. The KdV2 equation is a second order approximation of the set of Boussinesq's equations for shallow water waves which in first order…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…