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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.

35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations
35D05Existence of generalized solutions of PDE (MSC2000)
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K40Soliton theory, asymptotic behavior of solutions
76B25Solitary waves (inviscid fluids)
Full Text: DOI
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