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Traveling wave solutions of the Degasperis-Procesi equation. (English) Zbl 1068.35163

Summary: We classify all weak traveling wave solutions of the Degasperis-Procesi equation \[ u_t= u_{txx}+ 4uu_x= 3u_xu_{xx}+ uu_{xxx}, \quad x\in \mathbb R,\;t> 0. \] In addition to smooth and peaked solutions, the equation is shown to admit more exotic traveling waves such as cuspons, stumpons, and composite waves.

MSC:

35Q58 Other completely integrable PDE (MSC2000)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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