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On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation. (English) Zbl 1080.76016
Summary: In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by R. Dullin, G. Gottwald and D. Holm [Phys. Rev. Lett. 87, No. 9, 4501–4504 (2001)]. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, the convergence of solutions to the DGH equation as ${\alpha }^{2}\to 0$ is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained for the DGH equation. After giving the condition of existence of a peakon for the DGH equation, it turns out that the peakon is nonlinearly stable.
##### MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B25 Solitary waves (inviscid fluids) 35Q35 PDEs in connection with fluid mechanics 35Q51 Soliton-like equations
##### Keywords:
Cauchy problem; convergence; peaked solitary wave
##### References:
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