Mathematical Physics, Analysis and Geometry

, Volume 10, Issue 3, pp 205–225 | Cite as

Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition

  • Guoping Zhang
  • Zhijun Qiao


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.


Soliton Integrable system Analysis Traveling wave 

Mathematics Subject Classifications (2000)

35D05 35G30 35Q53 37K10 37K40 76B15 76B25 


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© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas – Pan AmericanEdinburgUSA
  2. 2.Department of MathematicsMorgan State UniversityBaltimoreUSA

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