Ill-posedness and the nonexistence of standing-waves solutions for the nonlocal nonlinear Schrödinger equation. (English) Zbl 1212.35153

Summary: We establish some properties for the Cauchy problem associated with the nonlocal nonlinear Schrödinger equation \(\partial _tu=-i\alpha \partial ^2_x u+\beta u\partial _x(| u| ^2)-i\beta u\mathcal T_h\partial _x(| u| ^2)+i\gamma | u| ^2u\), where \(x,t\in \mathbb R\), \(\mathcal T_h\) is the nonlocal operator \(\mathcal T_hu(x)=1/(2h)p.v. \int _{-\infty }^\infty \coth \bigl (\pi (y-x)/(2h))u(y)dy\), with \(\alpha >0\), \(\beta \geq 0\), \(\gamma \geq 0\), and \(h\in (0,+\infty )\). Here \(\mathcal T_h\to \mathcal H\) when \(h\to +\infty \), 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 \(h\to +\infty \), 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.


35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations