Nonlinear Schrödinger Equation: Simulation and Application

Great experimental breakthroughs in the physics of ultracold quantum gases are marked by the first realization of Bose-Einstein condensation in 1995, which was first conceived by Bose and then by Einstein. At temperatures much smaller than the critical condensation temperature the dynamics of a Bose-Einstein condensate is very well modelled by the nonlinear Schrödinger equation with cubic nonlinearity and harmonic potential. The project is dedicated to this nonlinear Schrödinger equation. There are four main issues of the project. The first part of the project we shall dedicate to numerical simulations with varying coefficient of the nonlinearity. In this case we are interested in the adiabatic approximation and in experiments with oscillating coefficient of the nonlinearity, for which we expect resonances. We intend to compare our numerical results with physical ones on varying scattering length, which should be done by the research group of R. Grimm (Innsbruck, Austria). In the next part we deal with a system of two coupled nonlinear Schrödinger equations. Firstly, we want to analyze numerically the problem where a solution component will blow-up. Alternatively we will add a damping term. Secondly, we want to investigate the case of strongly anisotropic harmonic potential, which leads us to the problem of dimension reduction. Thus we want to prove rigorously the validity of an effective nonlinear Schrödinger system obtained by the process of dimension reduction of the coupled system. Thirdly, we want to address the ground state problem of the coupled system of nonlinear Schrödinger equations with an additional term from the analytical, as well as from the numerical point of view. Finally, we intend to investigate also the case of coupled systems with more than two equations, since an increased interest is dedicated to systems with more equations. The next issue is dedicated to nonlinear Schrödinger equations with a harmonic and a periodic potential. The first step in this issue will be the numerical simulation of the equation when the strength of the periodic potential becomes large. Secondly, we will analyze the semi-classical approximation of the nonlinear Schrödinger equation with periodic potential. As an application we consider the Bose-Einstein condensate in optical lattices and perform numerical simulations in agreement with the physicists in Innsbruck. Finally, we consider an integro-differential nonlinear Schrödinger equation, which arises in Bose-Einstein condensate in a dipole configuration. Mathematically there are many open problems, e.g. wellposedness. Moreover, we want to investigate the different parameter regimes analytically and numerically.

The project "Nonlinear Schrödinger equation: Applications and Simulations" analyzed the Schrödinger equation, firstly applied in the modeling of Bose-Einstein condensates, and secondly describing solitary waves through a modification of the equation.A Bose-Einstein condensate is a state of matter existing at very low temperatures around nano Kelvin and whose main characteristic is that all particles are in the same quantum state, thus they are undistinguishable one from each other. Various properties of condensates can be analyzed. In the first part of the project we dealt with a system of two nonlinear Schrödinger equations describing a mixture of Bose-Einstein condensates consisting of two different hyperfine spin states of Rubidium atoms below the critical temperature. Here we analyzed under which conditions global solutions exist or finite-time blow-up of the system occur.Then we analyzed the nonlinear Schrödinger equation describing the exciton-polariton Bose-Einstein condensate. This condensate is produced at temperatures much higher than a few nano Kelvin. The drawback is the instability of such condensates. In order to prolongate the condensates' lifetime particles have to be pumped into the condensate continuously. This process is modeled by a nonlinear Schrödinger equation with an additional pumping and damping term. We performed numerical simulations in order to understand better the effect of the damping and pumping terms.Furthermore, we did numerical simulations in collaboration with physicists in order to optimize their experimental setup for one dimensional Bose-Einstein condensates.In another project phase we worked on solitary waves. An important question for nonlinear Schrödinger equations with turned off trapping potential is the existence and stability of standing waves solutions.The dispersive effect of the linear part of the equation may induce the solution to spread out, the nonlinearity to concentrate. However, in some cases these behaviors cancel and special solutions that neither disperse nor focus appears, the so-called solitary waves, thus waves which move and do not change their profile. In dealing with a system of Schrödinger equations we understood better the behavior in large times of solutions starting at initial time as two scalar solitary waves carried by the two different components. In all the working domains, numerical simulations were essential in proving theoretical results, on the one hand, and understanding new behavior when there is no theoretical work, on the other hand.