We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces H^s, s > 0, to a class of nonlinear, dispersive evolution equations of the form

u_t + (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. Notably, this class includes Whitham's model equation for surface gravity water waves featuring the exact linear dispersion relation.

Our work is a generalization of [1]. Tools involve constrained variational methods, Lions' concentration-compactness principle, a strong fractional chain rule for composition operators of low regularity, and a cut-off argument for n which enables us to go below the typical s > 1/2 regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when K is nonnegative.

[1] M. Ehrnström, M. D. Groves, and E. Wahlén, “On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type,” Nonlinearity, vol. 25, no. 10, pp. 2903– 2936, 2012.