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Integrable third order equations of the KdV type. (English) Zbl 1205.35266
Summary: We study two nonlinear equations that represent second and third order approximations of long wavelength, small amplitude waves of inviscid, and incompressible fluids. We apply a procedure, which is based on the application of hodograph transformations. In this way, we reveal two integrable cases for the third-order approximation. The first was recently identified, as been equivalent to an integrable equation recently appeared in the literature, while the second appears to be new. Our study fails to reveal any integrable cases for the second-order approximation. We also discuss the existence of additional integrable cases.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35Q35PDEs in connection with fluid mechanics
35A30Geometric theory for PDE, characteristics, transformations
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
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