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Cuspons and smooth solitons of the Degasperis-Procesi equation under inhomogeneous boundary condition. (English) Zbl 1153.35385
Summary: This paper is contributed to explore all possible single peakon solutions for the Degasperis-Procesi (DP) equation $$m_{t}+ m_{x} u +3 mu _{x}=0, m = u - u_{xx}$$. Our procedure shows that the DP equation either has cusp soliton and smooth soliton solutions only under the inhomogeneous boundary condition $$\lim_{|x|\rightarrow \infty} u = A \not= 0$$, or possesses the regular peakon solutions $$ce^{-|x -ct|}\in H^{1} (c$$ is the wave speed) only when $$\lim_{|x|\rightarrow \infty} u =0$$ (see Theorem 4.1). In particular, we obtain the stationary cuspon solution $$u = \sqrt {1- e^{-2|x|}} \in W^{1,1}_{\text{loc}}$$ of the DP equation. Moreover we present new cusp solitons (in the space $$W^{1,1}_{\text{loc}}$$) and smooth soliton solutions in explicit form. Asymptotic analysis and numerical simulations are provided for smooth solitons and cusp solitons of the DP equation.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations 35D05 Existence of generalized solutions of PDE (MSC2000) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 76B25 Solitary waves for incompressible inviscid fluids
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