It is well known that some dispersive non--linear equations admit traveling wave solutions
$$
 u(t,x)=u\_0(x-ct)\in L^2\_x\,, \qquad (t,x)\in \mathbb{R}\times \mathbb{R}\,\text{ or }\, \mathbb{R}\times\mathbb{T}\,,
$$
where $u\_0$ is the profile and $c$ is a real constant.
Sometimes these traveling waves can be obtained as ground states (minimizers) of the energy, mass... functionals of the equation, leading to say that there exists an $L^2$--threshold below (or above) which the existence of traveling waves cannot occur.


**My question :** Do there exist nonlinear dispersive PDEs, which have traveling waves with large *and* small $L^2$ norms ?



Does there exist always an $L^2$ threshold below (or above) which a traveling waves of a nonlinear dispersive PDE cannot exist?
I recently came across a [paper on arXiv](https://arxiv.org/abs/2307.01592) that addresses the question. I am sharing it here if it may be of interest to someone else.


The author seems to consider a nonlocal nonlinear schrödinger equation, referred to as *the Calogero-Sutherland DNLS equation*. And she finds periodic (in the space variable) traveling waves with small and large $L^2$-norms.


See Remark 1.3 (1) of [page 5](https://arxiv.org/pdf/2307.01592.pdf).


