Ill-posedness and the nonexistence of standing-waves solutions for the nonlocal nonlinear Schrödinger equation

Abstract

We establish some properties for the Cauchy problem associated with the nonlocal nonlinear Schrödinger equation $\partial_{t}u=-i\alpha\partial^{2}_{x}u+\beta u\partial_{x}(\,|u|^{2})-i\beta u\mathcal{T}_{h}\partial_{x}(\,|u|^{2}) +i\gamma \left| u\right| ^{2}u,$ where $x, t\;\in\mathbb R,$ $\mathcal{T}_{h}$ is the nonlocal operator $$ \mathcal{T}_{h}u(x)=\frac{1}{2h}p.v. \int^{\infty}_{-\infty}\coth \Big (\frac{\pi(y-x)}{2h} \Big ) u(y)dy, $$ with $\alpha>0$, $\beta\geq 0$, $\gamma \geq 0$, and $h\in(0,+\infty)$. Here $\mathcal{T}_{h}\longrightarrow\mathcal{H}$ when $h\longrightarrow +\infty,$ where $\mathcal{H}$ is the Hilbert transform. We prove rigorously that a Picard interaction scheme can not be applied for solving the Cauchy problem associated with that equation in both the cases $0 < h < \infty$ and $h\rightarrow +\infty,$ with initial data in Sobolev spaces of negative index. Elsewhere, we study the asymptotic behavior of the solution in relation to a spatial variable, and we also establish the nonexistence of a standing-waves solution for the above equation in several cases.