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Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equations. (English) Zbl 1178.34002
The author considers the existence of travelling wave solutions to the following six partial differential equations: Vakhnenko, Ostrovsky, Camasso-Holm (generalized), Rosenau-Hyman, KdV (“more realistic”), perturbed Boussinesq. The usual ansatz \(u(x,t)=\varphi(x+ct)\) leads to boundary value problems for ordinary differential equations. All these problems can be reduced to the study of the phase portrait of planar autonomous systems. By exploiting special properties of these systems, the author is able to present exact explicit parametric representations of the traveling wave solutions.

MSC:
34A05 Explicit solutions, first integrals of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
35L05 Wave equation
35Q51 Soliton equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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