Exponential decay and symmetry of solitary waves to the Degasperis-Procesi equation

TL;DR

This paper enhances decay analysis techniques for solitary waves in dispersive equations, demonstrating exponential decay and symmetry properties for solutions to the Degasperis-Procesi equation, with broader implications for related equations.

Contribution

It introduces improved decay arguments and a novel method to confirm symmetry of solitary waves, linking symmetry to the steady structure of solutions in the Degasperis-Procesi equation.

Findings

Exponential decay of solitary waves proved.

Symmetry of maximum height solitary waves confirmed.

Connection between symmetry and steady structure established.

Abstract

We improve the decay argument by [Bona and Li, J. Math. Pures Appl., 1997] for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady Degasperis-Procesi equation in the nonlocal formulation. In addition, we give a method which confirms the symmetry of solitary waves of the maximum height. Finally, we discover how the symmetric structure is connected to the steady structure of solutions to the Degasperis-Procesi equation, and give a more intuitive proof for symmetric solutions to be traveling waves. The improved argument and new methods above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations.