Exact quasi-periodic solutions to the sine(sinh)-Gordon equations: The method for computation and analysis

TL;DR

This paper develops a method to compute and analyze quasi-periodic solutions of the sine(sinh)-Gordon equations using spectral curve uniformization and Hamiltonian techniques, with explicit examples in low genera.

Contribution

It introduces a novel computational and analytical approach for quasi-periodic solutions of sine(sinh)-Gordon equations based on spectral methods and Hamiltonian analysis.

Findings

Explicit quasi-periodic solutions expressed via $ ext{wp}$-functions.

Revised reality conditions for solutions.

Application of the method to genera one and two cases.

Abstract

The sine(sinh)-Gordon hierarchy of integrable Hamiltonian systems is described in detail, and all dynamic variables are expressed in terms of the $\wp$-functions that uniformize the associated spectral curve. Quasi-periodic solutions to the sine(sinh)-Gordon equations are obtained in terms of the function $\wp_{1,2g-1}$, reality conditions are revised, and a method of computation and analysis is presented. The proposed method is designed to analyze solutions by means of the Hamiltonian technique, which is illustrated in genera one and two.