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 ?