Solitons, peakons, and periodic cuspons of a generalized Degasperis-Procesi equation

TL;DR

This paper uses bifurcation theory to find and analyze exact traveling wave solutions, including smooth solitons, peaked solitons, and cuspons, of a generalized Degasperis-Procesi equation, revealing their relationships and physical significance.

Contribution

It provides explicit forms of various wave solutions for a generalized Degasperis-Procesi equation and explores their interrelations and physical relevance.

Findings

Explicit expressions for smooth, peaked, and cuspon solutions

Relationships among different wave solutions

Physical relevance of the solutions

Abstract

We employ the bifurcation theory of planar dynamical systems to investigate the exact travelling wave solutions of a generalized Degasperis-Procesi equation. The implicit expression of smooth soliton solutions is given. The explicit expressions of peaked soliton solutions and periodic cuspon solutions are also obtained. Further, we show the relationship among the smooth soliton solutions, the peaked soliton solutions, and the periodic cuspon solutions. The physical relevance of the found solutions and the reasonwhy these solutions can exist in this equation are also given.