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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+3mu 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| u=A0, or possesses the regular peakon solutions ce -|x-ct| H 1 (c is the wave speed) only when lim |x| u=0 (see Theorem 4.1). In particular, we obtain the stationary cuspon solution u=1-e -2|x| W loc 1,1 of the DP equation. Moreover we present new cusp solitons (in the space W loc 1,1 ) 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:
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)
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