Inverse problem of variational calculus for nonlinear evolution equations

TL;DR

This paper develops a variational framework for nonlinear evolution equations, deriving a Lagrangian and Hamiltonian formulation that reveals bi-Hamiltonian structures in KdV-type systems, with potential physical applications.

Contribution

It introduces a novel inverse variational approach coupling evolution equations with an associated system, expressing the Lagrangian in original variables, and demonstrating bi-Hamiltonian structures in key examples.

Findings

Derivation of action principle for coupled nonlinear evolution equations

Expression of Lagrangian density in original field variables

Identification of bi-Hamiltonian structure in KdV and mKdV systems

Abstract

We couple a nonlinear evolution equation with an associated one and derive the action principle. This allows us to write the Lagrangian density of the system in terms of the original field variables rather than Casimir potentials. We find that the corresponding Hamiltonian density provides a natural basis to recast the pair of equations in the canonical form. Amongst the case studies presented the KdV and modified KdV pairs exhibit bi-Hamiltonian structure and allow one to realize the associated fields in physical terms