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

We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces ${\mathrm{H}^s}$, ${ s > 0 }$, to a class of nonlinear, dispersive evolution equations of the form \begin{equation*} u_t + \left(Lu+ n(u)\right)_x = 0, \end{equation*} 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 Ehrnstr\"om, Groves & Wahl\'en 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 ${s > \frac{1}{2}}$ 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.