Modulation theory for the sine-Gordon equation

TL;DR

This paper solves the Whitham modulation equations for sine-Gordon pulses, interpreting them as relativistic hydrodynamics, and demonstrates their application through an example of nonlinear wave packet evolution.

Contribution

It provides an explicit solution method for the modulation equations of sine-Gordon pulses using the hodograph technique, linking them to relativistic hydrodynamics.

Findings

Solution of Whitham equations via hodograph method

Interpretation of modulation equations as relativistic hydrodynamics

Illustration with nonlinear wave packet evolution example

Abstract

We give the solution of the Whitham modulation equations for envelopes of pulses evolving according to the sine-Gordon equation. The Whitham equations are interpreted as the equations of relativistic hydrodynamics and their solving is reduced by the hodograph method to solving a linear partial differential equation. We describe the class of solutions of this equation with separation of variables and illustrate the theory by an example of the nonlinear wave packet evolution accompanied by its shrinking and decrease of the number of oscillations in the Whitham nonlinear region.