On a parabolic sine-Gordon model

TL;DR

This paper studies a parabolic sine-Gordon model, establishing maximum principles, classifying steady states in 1D, analyzing numerical schemes' stability, and highlighting its similarities with Allen-Cahn equations.

Contribution

It provides a maximum principle, classifies steady states in 1D, and proves energy stability of numerical schemes for the parabolic sine-Gordon model.

Findings

Maximum principle for the model

Explicit solutions and classification of steady states in 1D

Energy stability of numerical discretizations

Abstract

We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.