On the standing waves for the X-ray free electron laser Schrödinger equation

Abstract: In this paper, we are concerned with the standing waves for the following nonlinear Schr\"{o}dinger equation $$i\partial_{t}\psi=-\Delta \psi+b^2(x_1^2+x_2^2)\psi+\frac{\lambda_1}{|x|}\psi+ \lambda_2(|\cdot|^{-1}\ast |\psi|^2)\psi- \lambda_3|\psi|^p \psi,~~~ (t,x)\in \mathbb{R}^+\times \mathbb{R}^3,$$ where $0<p<4$. We mainly study the existence and stability/instability properties of standing waves for this equation, in two cases: the first one is that no magnetic potential is involved, (i.e. $b=0$ in the equation) and the second one is that $b\neq 0$. To be precise, in the first case, by considering a minimization problem on a suitable Pohozaev manifold we prove the existence of ground states, and show further that all ground state standing waves are strongly unstable by blow-up in finite time. Moreover, by making use of the ideas of their proofs, we are able to prove the existence and instability of normalized solutions, whose proofs seem to be new, compared with the studies of normalized solutions in the existing literature. In the second case, the situation is more difficult to be treated, due to the additional term of the partial harmonic potential. We manage to prove the existence of stable standing waves for $p\in (0,4)$ and with some assumptions on the coefficients, where solutions are obtained as global minimizers if $p\in (0,\frac{4}{3}]$, and as local minimizers if $p\in [\frac{4}{3}, 4)$. In the mass-critical and supercritical cases $p\in [\frac{4}{3}, 4)$, we establish the variational characterization of the ground states on a suitable manifold which is different from the one neither of the Nehari type nor of the Pohozaev type, and then prove the existence of ground states. Finally under some assumptions on $\omega$ and $p$, we prove that the ground state standing waves are strongly unstable.