# Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

## TL;DR

This paper proves the existence of solitary-wave solutions for a class of weakly dispersive equations with inhomogeneous nonlinearities, under minimal assumptions on the dispersion operator and the nonlinearity.

## Contribution

It establishes existence results for solitary waves with low-regularity dispersion and inhomogeneous nonlinearities, broadening the class of equations where such solutions are known.

## Key findings

- Existence of solitary-wave solutions in Sobolev spaces H^{1+s}.
- Applicable to equations with low positive order dispersion operators.
- Handles nonlinearities that are locally Lipschitz and asymptotically homogeneous.

## Abstract

We show existence of solitary-wave solutions to the equation \begin{equation*} u_t+ (Lu - n(u))_x = 0\,, \end{equation*} for weak assumptions on the dispersion $L$ and the nonlinearity $n$. The symbol $m$ of the Fourier multiplier $L$ is allowed to be of low positive order ($s > 0$), while $n$ need only be locally Lipschitz and asymptotically homogeneous at zero. We shall discover such solutions in Sobolev spaces contained in $H^{1+s}$.