A deformation of the method of characteristics and the Cauchy problem for Hamiltonian PDEs in the small dispersion limit

TL;DR

This paper develops a deformation of the method of characteristics for Hamiltonian PDEs in the small dispersion limit, introducing a variational string equation to analyze perturbations and compute solutions.

Contribution

It introduces a new deformation of the method of characteristics using a variational string equation, providing explicit perturbative solutions for Hamiltonian PDEs.

Findings

Constructed the string equation up to fourth order in perturbation theory.

Proved the existence of a quasi-triviality transformation for the KdV equation.

Computed the first two perturbative corrections for general Hamiltonian PDEs.

Abstract

We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the 'variational string equation', a functional-differential relation originally introduced by Dubrovin in a particular case, of which we lay the mathematical foundation. Starting from first principles, we construct the string equation explicitly up to the fourth order in perturbation theory, and we show that the solution to the Cauchy problem of the Hamiltonian PDE satisfies the appropriate string equation in the small dispersion limit. We apply our construction to explicitly compute the first two perturbative corrections of the solution to the general Hamiltonian PDE. In the KdV case, we prove the existence of a quasi-triviality transformation at any order and for arbitrary initial data.