Related papers: Conservation Laws of Discrete Korteweg-de Vries Eq…
Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of…
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…
The Korteweg-de Vries equation is known to yield a valid description of surface waves for waves of small amplitude and large wavelength. The equation features a number of conserved integrals, but there is no consensus among scientists as to…
Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic…
We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation,…
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…
The paper compares computational aspects of four approaches to compute conservation laws of single differential equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer…
By exploiting the fact that conservation laws form the kernel of a discrete Euler operator, we use a recently introduced symbolic-numeric approach to construct a new class of finite difference methods for the modified Korteweg-de Vries…
In this paper, numerical and solitonic solutions of Korteweg de Vries(KdV) and Korteweg de Vries-Burger's (KdVB) equations with initial and boundary conditions are calculated by sinc-collocation method. The basis of method is sinc…
The Korteweg-de Vries equation is one of the most important nonlinear evolution equations in the mathematical sciences. In this article invariant discretization schemes are constructed for this equation both in the Lagrangian and in the…
This short survey paper is concerned with a new method to prove global well-posedness results for dispersive equations below energy spaces, namely $H^{1}$ for the Schr\"odinger equation and $L^{2}$ for the KdV equation. The main ingredient…
It is shown that equations of the Korteweg-de Vries hierarchy and their conservation laws can be expressed via the whole powers of an integro-differential operator and functions provided by them.
The conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived. For discrete models the conserved charges are constructed explicitly. The applications of the general…
We show how to descritize the Korteweg-de Vries equation in such a way as to preserve all the Lie point symmetries of the continuous differential equation. It is shown that, for a centered implicit scheme, there are at least two possible…
An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a…
English version of abstract: The dynamic optimization problems treated by the calculus of variations are usually solved with the help of the 2nd order Euler-Lagrange differential equations. These equations are, generally speaking,…
In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate…
This paper gives a general treatment and proof of the direct conservation law method presented in Part I. In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential…
Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other…
The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation…