Abstract
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 =A ≠0, 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 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.
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Zhang, G., Qiao, Z. Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition. Math Phys Anal Geom 10, 205–225 (2007). https://doi.org/10.1007/s11040-007-9027-2
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DOI: https://doi.org/10.1007/s11040-007-9027-2