Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

Abstract

We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces Hs, s>0, to a class of nonlinear, dispersive evolution equations of the formut+(Lu+n(u))x=0,where the dispersion L is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse K and the nonlinearity n is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves and Wahlén on a class of equations which includes Whitham’s model equation for surface gravity water waves featuring the exact linear dispersion relation. Tools involve constrained variational methods, Lions’ concentration-compactness principle, a strong fractional chain rule for composition operators of low relative regularity, and a cut-off argument for n which enables us to go below the typical s>12 regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when K is nonnegative, and provide a nonexistence result when n is too strong.