Related papers: Conservation Laws and Web-Solutions for the Benney…
We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions…
In nonlinear physics, the interactions among solitons are well studied thanks to the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of…
Some turbulent flows self-organize into large-scale structures, rather than breaking up into ever-smaller scales. Underpinning this phenomenon is the existence of two sign-definite quantities which are conserved by the dynamics.…
We consider the Cauchy problem for a model of non-linear acoustics, named the Kuznetsov equation, describing sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation,…
In this paper we study the long time behavior for a semilinear wave equation with space-dependent and nonlinear damping term. After rewriting the equation as a first order system, we define a class of approximate solutions that employ…
Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich's f-plane extension of the Miyata-Maltseva-Choi-Camassa (MMCC) model. Within the scope of this model, we develop an asymptotic procedure which…
All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted…
Hydrodynamic modulation of short ocean surface waves by longer ambient waves significantly influences remote sensing, interpretation of in situ wave measurements, and numerical wave forecasting. This paper revisits the wave crest and action…
We consider the modulationally stable version of the Kaup-Boussinesq system which models propagation of nonlinear waves in various physical systems. It is shown that the Whitham modulation equations for this model have a new type of…
The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free…
We consider a mechanism of generation of huge waves by multi-soliton resonant interactions. A non-stationary wave amplification phenomenon is found in some exact solutions of the Kadomtsev-Petviashvili (KP) equation. The mechanism proposed…
We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation…
We consider the nonlinear equations obtained from soliton equations by adding self-consistent sources. We demonstrate by using as an example the Kadomtsev-Petviashvili equation that such equations on periodic functions are not isospectral.…
In the dynamics generated by the suspension bridge equation, traveling waves are an essential feature. The existing literature focuses primarily on the idealized one-dimensional case, while traveling structures in two spatial dimensions…
A model for the wave motion of an internal wave in the presence of current in the case of intermediate long wave approximation is studied. The lower layer is considerably deeper, with a higher density than the upper layer. The flat surface…
In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the…
We study possible Lie and non-classical reductions of multidimensional wave equations and the special classes of possible reduced equations - their symmetries and equivalence classes. Such investigation allows to find many new conditional…
A condition of reduction of multidimensional wave equations to the two-dimensional equation is studied, and the necessary conditions of compatibility and exact solutions of the resulting d'Alembert-Hamilton system are obtained.
Assuming that a formal approximation of multiple waves has been obtained by matched asymptotic methods, we derive a {\em Spatial Shadowing lemma} to construct exact solutions near the formal approximation. In Part I, we consider a general…
The manuscript presents a new technique for computing the exponential of skew-Hermitian operators. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many…