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A remark on the Cauchy problem for the generalized Benney-Luke equation. (English) Zbl 1224.35263

Summary: In this article we address the well posedness of the Cauchy problem associated with the generalized Benney-Luke equation

Φ tt -ΔΦ+aΔ 2 Φ-bΔΦ tt +θΦ t x [( x Φ) p ]+ y [( y Φ) p ]+2( x Φ) p Φ xt +( y Φ) p Φ yt +β·(|Φ| m Φ)=0

in 1+2 under a reasonable “physical” initial condition, which is imposed from the formal derivation of the Benney-Luke water wave model.

35L30Higher order hyperbolic equations, initial value problems
35L76Semilinear higher-order hyperbolic equations
35Q35PDEs in connection with fluid mechanics
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction