ﻻ يوجد ملخص باللغة العربية
In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schr{o}dinger equation begin{equation} ipartial_{t}psi=(-Delta)^{s}psi-f(psi), qquad (0.1)end{equation} where $0<s<1$, $f(psi)=|psi|^{p}psi$ with $frac{4s}{N}<p<frac{4s}{N-2s}$ or $f(psi)=(|x|^{-gamma}ast|psi|^2)psi$ with $2s<gamma<min{N,4s}$. To this end, we look for normalized solutions of the associated stationary equation begin{equation} (-Delta)^s u+omega u-f(u)=0. qquad (0.2) end{equation} Firstly, by constructing a suitable submanifold of a $L^2$-sphere, we prove the existence of a normalized solution for (0.2) with least energy in the $L^2$-sphere, which corresponds to a normalized ground state standing wave of(0.1). Then, we show that each normalized ground state of (0.2) coincides a ground state of (0.2) in the usual sense. Finally, we obtain the sharp threshold of global existence and blow-up for (0.1). Moreover, we can use this sharp threshold to show that all normalized ground state standing waves are strongly unstable by blow-up.
We study the inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scatt
In this paper, we study important Schr{o}dinger systems with linear and nonlinear couplings begin{equation}label{eq:diricichlet} begin{cases} -Delta u_1-lambda_1 u_1=mu_1 |u_1|^{p_1-2}u_1+r_1beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+kappa (x)u_2~hbox{in}~math
In this paper, we study the nonlinear Schr{o}dinger equation $ -Delta u+V(x)u=f(x,u) $on the lattice graph $ mathbb{Z}^{N}$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$ is periodic o
In this paper, we consider an optimal bilinear control problem for the nonlinear Schr{o}dinger equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first order optimality syst
In this paper, we give a complete study on the existence and non-existence of normalized solutions for Schr{o}dinger system with quadratic and cubic interactions. In the one dimension case, the energy functional is bounded from below on the product o