| Abstract: |
| We are interested in traveling wave solutions of the nonlocal derivative nonlinear Schr\{o}dinger (nonlocal DNLS) equation
$$
i u_t - u_{xx} - b |u|^2 u
+ i \alpha |u|^2 u_x
+ i \beta u^2 \bar{u}_x
+ \gamma u\, \partial_x \mathcal{H}(|u|^2) = 0,
$$
where $b, \alpha, \beta, \gamma \in \mathbb{R}$ are parameters, and $\mathcal{H}$ denotes the Hilbert transform
$$
\mathcal{H}u(x,t) = \frac{1}{\pi}\,\mathrm{p.v.}\int_{\mathbb{R}} \frac{u(y,t)}{x-y}\,dy.
$$
Depending on the sign of the parameters, we consider subcritical and critical minimization problems. Our main techniques are the Nehari manifold method and the concentration-compactness principle of P.L. Lions. |
|