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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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