Conservation laws of semidiscrete canonical Hamiltonian equations

TL;DR

This paper explores conservation laws in semidiscrete canonical Hamiltonian equations, deriving them from symmetries for specific equations like nonlinear wave and Schrödinger equations.

Contribution

It introduces a method to find conservation laws for semidiscrete Hamiltonian equations using symmetry analysis, extending continuous Hamiltonian theory.

Findings

Conservation laws are derived for semidiscrete nonlinear wave and Schrödinger equations.

Symmetry methods effectively identify conserved quantities in semidiscrete Hamiltonian systems.

The approach bridges continuous and discrete Hamiltonian formulations.

Abstract

There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one-parameter Hamiltonian symmetries admitted by the PDEs. The same result holds for semidiscrete Hamiltonian equations. In this paper we consider semidiscrete canonical Hamiltonian equations. Using symmetries, we find conservation laws for the semidiscretized nonlinear wave equation and Schrodinger equation.