Related papers: Final state problem for the cubic nonlinear Schr"o…
We consider the cubic nonlinear Schr\"{o}dinger equation on the star graph with the Kirchhoff boundary condition. We prove modified scattering for the final state problem and the initial value problem. Moreover, we also consider the failure…
We consider the Cauchy problem of a dissipative nonlinear Schr\"odinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a…
We construct solutions of Schr\"odinger equations which are asymptotically self-similar solutions as time goes to infinity. Also included are situations with two bubbles. These solutions are global, with non-zero $L^2$ norms, and are…
The long-time asymptotics is analyzed for finite energy solutions of the 1D Schr\"odinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to…
The focusing nonlinear Schrodinger equation possesses special non-dispersive solitary type solutions, solitons. Under certain spectral assumptions we show existence and asymptotic stability of solutions with the asymptoic profile (as time…
We establish the scattering of solutions to the focusing mass supercritical nonlinear Schr\"odinger equation with a repulsive Dirac delta potential \[ i\partial_{t}u+\partial^{2}_{x}u+\gamma\delta(x)u+|u|^{p-1}u=0, \quad (t,x)\in {\mathbb…
We study the $d$-dimensional discrete nonlinear Schr\"odinger equation with general power nonlinearity and a delta potential. Our interest lies in the interplay between two localization mechanisms. On the one hand, the attractive…
We study the defocusing nonlinear Schr\"odinger equation in the quarter plane with asymptotically periodic boundary values. By studying an associated Riemann-Hilbert problem and employing nonlinear steepest descent arguments, we construct…
We prove local existence and uniqueness of solutions for the one-dimensional nonlinear Schr\"odinger (NLS) equations $iu_t + u_{xx} \pm |u|^2 u = 0$ in classes of smooth functions that admit an asymptotic expansion at infinity in decreasing…
Using the Fredholm theory of the linear time-dependent Schr\"odinger equation set up in our previous article arXiv:2201.03140, we solve the final-state problem for the nonlinear Schr\"odinger problem $$ (D_t + \Delta + V) u = N[u], \quad…
We present a new Riemann-Hilbert problem formalism for the initial value problem for the derivative nonlinear Schr\"odinger (DNLS) equation on the line. We show that the solution of this initial value problem can be obtained from the…
We consider a Nonlinear Schr\"odinger Equation with a very general non linear term and with a trapping $\delta $ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by…
We derive asymptotic formulas for the solution of the derivative nonlinear Schr\"odinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of…
We study the Cauchy problem for the focusing nonlinear Schrodinger (NLS) equation. Using the DBAR generalization of the nonlinear steepest descent method we compute the long time asymptotic expansion of the solution in any fixed space-time…
In this article, we present the analytical solution of the radial Schr\"{o}dinger equation for the Hulth\'{e}n potential within the framework of the asymptotic iteration method by using an approximation to the centrifugal potential for any…
A superspace version of the Schr\"odinger equation with a delta potential is studied using Fourier analysis. An explicit expression for the energy of the single bound state is found as a function of the super-dimension M in case M is…
We consider a class of nonlinear Schr\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…
We consider the cubic nonlinear Schr\"odinger equation with harmonic trapping on $\mathbb{R}^D$ ($1\leq D\leq 5$). In the case when all but one directions are trapped (a.k.a "cigar-shaped" trap), following the approach of…
We establish the existence and provide explicit expressions for the stationary states of the one-dimensional Schr\"odinger equation with a repulsive delta-prime potential and a focusing nonlinearity of power type. Furthermore, we prove…
We study the asymptotic behaviour of solutions to semi-classical nonlinear Schrodinger equations with a potential, for concentrating and oscillating initial data, when the nonlinearity is repulsive and the potential is a polynomial of…