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Special solutions of a new class of water wave equations. (English) Zbl 1084.76511
Summary: We consider a nonlinear partial differential equation (PDE) which represents physically a more realistic description of uni-directional wave motion in a one-dimensional inviscid and incompressible fluid, than the one provided by the well-known KdV equation. Even though this more “physical” PDE is not integrable, it can be transformed (by a nonlinear change of variables) to a PDE) which is completely integrable to the same order of approximation. We use Painlevé analysis to obtain two different types of solutions for the “physical” (nonintegrable) PDE: the first type are solitary waves, and the other have unbounded amplitude. We examine numerically the stability of solitary wave solutions under time evolution and observe similar phenomena as exhibited by soliton solutions of the KdV equation.

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25 Solitary waves for incompressible inviscid fluids
35Q51 Soliton equations