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

Abstract

<p style='text-indent:20px;'>In this paper, we are concerned with the standing waves for the following nonlinear Schrödinger equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ 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, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0<p<4 $\end{document}</tex-math></inline-formula>. This equation arises as an effective single particle model in X-ray Free Electron Lasers. We mainly study the existence and stability/instability properties of standing waves for this equation, in two cases: the first one is that no magnetic potential is involved (i.e. <inline-formula><tex-math id="M2">\begin{document}$ b = 0 $\end{document}</tex-math></inline-formula> in the equation), and the second one is that <inline-formula><tex-math id="M3">\begin{document}$ b\neq 0 $\end{document}</tex-math></inline-formula>. To be precise, in the first case, when <inline-formula><tex-math id="M4">\begin{document}$ p\in [\frac{4}{3},4) $\end{document}</tex-math></inline-formula>, by considering a minimization problem on a suitable Pohozaev manifold, we prove the existence of radial ground states, and show further that the corresponding standing waves are strongly unstable by blow-up in a finite time. Moreover, by making use of the ideas of these proofs, we are able to prove the existence of normalized solutions, whose proof seems to be new, compared with the studies of normalized solutions in the existing literature. This study also indicates that there is a close connection between the study of the strong instability and the one of the existence of normalized solutions. In the second case, the situation is more difficult to be treated, due to the additional term of the partial harmonic potential. We manage to prove the existence of stable standing waves for <inline-formula><tex-math id="M5">\begin{document}$ p\in (0,4) $\end{document}</tex-math></inline-formula>, where solutions are obtained as global minimizers when <inline-formula><tex-math id="M6">\begin{document}$ p\in (0,\frac{4}{3}] $\end{document}</tex-math></inline-formula>, and as local minimizers when <inline-formula><tex-math id="M7">\begin{document}$ p\in [\frac{4}{3}, 4) $\end{document}</tex-math></inline-formula>. In the mass-critical and supercritical cases <inline-formula><tex-math id="M8">\begin{document}$ p\in [\frac{4}{3}, 4) $\end{document}</tex-math></inline-formula>, we also establish the variational characterization of ground state solutions on a new manifold which is neither of the Nehari type nor of the Pohozaev type, and then prove the existence of ground states. Finally under some assumptions on the coefficients, we prove that the ground state standing waves are strongly unstable.</p>

On the ground states for the X‐ray free electron lasers Schrödinger equation

“…Although huge amount of studies have been made in physics, it seems that only very few mathematical results could be found in the literatures. In mathematics, it seems that the earliest time that the Cauchy problems for the XFEL equations with $$$ V(x)=0 $$$ were considered in Antonelli et al 6,7 For some recent developments, we refer to previous works 8–12 and the references therein. Especially, Feng and Zhao 9–11 are devoted to study of the time‐dependent problems; Cao et al 8 studied the existence and stability properties of standing waves under the assumption that $$$ A $$$ depends on time $$$ t $$$ only; Han et al 12 provided the existence of ground states and normalized solutions for the XFEL Schrödinger equation with harmonic potential.…”