Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition

Abstract

This paper is contributed to explore all possible single peakon solutions for the Degasperis–Procesi (DP) equation mt + mxu + 3mux = 0, m = u − uxx. Our procedure shows that the DP equation either has cusp soliton and smooth soliton solutions only under the inhomogeneous boundary condition lim|x|→ ∞ u =A ≠0, or possesses the regular peakon solutions ce − |x − ct| ∈ H1 (c is the wave speed) only when lim|x|→ ∞ u = 0 (see Theorem 4.1). In particular, we first time obtain the stationary cuspon solution \(u = {\sqrt {1 - e^{{ - 2{\left| x \right|}}} } } \in W^{{1,1}}_{{loc}} \) of the DP equation. Moreover we present new cusp solitons (in the space of \(W^{{1,1}}_{{loc}} \)) and smooth soliton solutions in an explicit form. Asymptotic analysis and numerical simulations are provided for smooth solitons and cusp solitons of the DP equation.