Cusp solitons of the Degasperis-Procesi equation

Abstract

In this paper, we present cusp solitons for the Degasperis-Procesi (DP) equation under the inhomogeneous boundary condition $\lim_{|x|\rightarrow\infty}u=A$, where $A$ is a non-zero constant. Through mathematical analysis, a stationary cusp soliton is presented in an explicit form of $u(x,t)=\sqrt{1-e^{-2|x|}}\in W_{loc}^{1,1}, \ {\rm but}\notin H^1_{loc}$, while most cusp solitons are expressed in an implicit form in the space of $W_{loc}^{1,1}$.