On the standing wave solutions to a nonlocal, nonlinear Schrödinger equation occurring in plasma physics (Q2366487)

The following Cauchy problem is considered \[ i\varphi_ t+ \sum^ 3_{i,j=1} a_{ij} {{\partial^ 2\varphi} \over {\partial x_ i \partial x_ j}}=- \text{div} (|\nabla\psi|^ \sigma \nabla\psi), \quad \Delta\psi=\varphi, \quad \varphi(x,0)=\varphi_ 0(x) \] where the matrix \(a_{ij}\) is real, symmetric, nonsingular. The existence of standing wave solutions to this problem is proved, some stability and instability results are provided.