On the integrability of the sine-Gordon equation

TL;DR

This paper explores the mathematical structure of the sine-Gordon equation, showing its connection to the mKdV equation near equilibrium and identifying initial data that lead to non-existence of classical solutions.

Contribution

It demonstrates the Hamiltonian's relation to the Poisson algebra of mKdV and identifies initial data sets where classical solutions do not exist.

Findings

Hamiltonian near equilibrium can be viewed within the Poisson algebra of mKdV

Existence of a large set of initial data with no classical solutions

Connection between sine-Gordon and mKdV equations established

Abstract

Among other results we show that near the equilibrium point, the Hamiltonian of the sine-Gordon (SG) equation on the circle can be viewed as an element in the Poisson algebra of the modified Korteweg-de Vries (mKdV) equation and hence by well established properties of the latter equation admits Birkhoff coordinates. On the other hand we prove that there exists a large set of smooth initial data, away from the equilibrium point and lying on the ramification locus of a double cover, for which the initial value problem of the SG equation has no classical solution.