Related papers: Evolution equations: Frobenius integrability, cons…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability…
Following conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ along forward and backward characteristics, we identify criteria, which guarantee that wave breaking either occurs in the nearby future…
This article produces wave equations and constructs traveling wave solutions that are intimately related to Newton's equations of celestial mechanics. The traveling wave solutions are expressed in ``closed form'' in terms of elementary…
We establish near-optimal quantitative uniqueness of continuation for solutions of evolution equations vanishing on the lateral boundary. These results were obtained simply by combining existing observability inequalities and energy…
The paper is devoted to the group analysis of equations of motion of two-dimensional uniformly stratified rotating fluids used as a basic model in geophysical fluid dynamics. It is shown that the nonlinear equations in question have a…
We provide a novel existence result for energy-variational solutions to a general class of evolutionary partial differential equations. Compared to previous works on this solution concept, the generalization is mainly twofold: a relaxation…
Stability of a set of travelling wave solutions to the hyperbolic generalization of the convection-reaction-diffusion equation is studied by means of the qualitative methods and numerical simulation.
This paper is concerned with the traveling wave solutions for integro-difference systems of higher order. By using Schauder fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and…
We provide a complete classification of generalized and formal symmetries and local conservation laws for an evolution equation which generalizes the Kawahara equation having important applications in the study of plasma waves and…
In this letter we present an analytic evidence of the non-integrability of the discrete nonlinear Schroedinger equation, a well-known discrete evolution equation which has been obtained in various contexts of physics and biology. We use a…
It is well known that the existence of traveling wave solutions (TWS) for many partial differential equations (PDE) is a consequence of the fact that an associated planar ordinary differential equation (ODE) has certain types of solutions…
We use a simple method that leads to the integrals involved in obtaining the traveling wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in…
Stability of the traveling wave solution to a general class of one-dimensional nonlocal evolution equations is studied in $L^2$-spaces, thereby providing an alternative approach to the usual spectral analysis with respect to the supremum…
Ostrovsky's equation with time- and space- dependent forcing is studied. This equation is model for long waves in a rotating fluid with a non-constant depth (topography). A classification of Lie point symmetries and low-order conservation…
In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions…
This paper presents recent results concerning the existence and qualitative properties of travelling wave solutions to the Gross-Pitaevskii equation posed on the whole space R^N. Unlike the defocusing nonlinear Schr\"odinger equations with…
Conservation laws are formulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not…
A complete classification of low-order conservation laws is obtained for time-dependent generalized Korteweg-de Vries equations. Through the Hamiltonian structure of these equations, a corresponding classification of Hamiltonian symmetries…
It is shown that the `conservation law' for metric fluctuations with long wavelengths is indeed applicable for growing modes of perturbations, which are of interest in cosmology, in spite of a recent criticism [L. P. Grishchuk, Preprint…