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

Abstract

Abstract We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces , , to a class of nonlinear, dispersive evolution equations of the form where the dispersion is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse and the nonlinearity 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 which enables us to go below the typical regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when is nonnegative, and provide a nonexistence result when is too strong.