טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentVeksler Hagar
SubjectLocalization of the Generalized Non-Linear Schrodinger
Equation in a Random Potential
DepartmentDepartment of Physics
Supervisor Professor Emeritus Shmuel Fishman (Deceased)
Full Thesis textFull thesis text - English Version


Abstract

The Nonlinear Schrödinger Equation (NLSE) is of fundamental importance for the understanding of the competition between wave interference effects and the effects of nonlinearities. It is of experimental relevance for Bose-Einstein condensates (BEC) where the nonlinear term approximates the affect of inter-particle interactions. In this field, the NLSE is known as the Gross-Pitaevskii Equation (GPE). In some situations in classical optics, the electric field plays the role of the wavefunction, and the nonlinear term results of the dependence of the index of refraction on the electric field. In this thesis, the NLSE is studied on a one dimensional lattice in the presence of random potential. In this situation, in the absence of nonlinearities, Anderson localization is found. A wavepacket that is initially localized will remain localized. In presence of the nonlinearity but in absence of the random potential, spreading takes place. For the NLSE with a random potential, there is a competition between the effects of the disorder and the ones of the nonlinearity. The fundamental question is: will an initially localized wavepacket spread forever? The answer to this question is not known. Results of many numerical calculations that are not well controlled were published and conflicting heuristic arguments were presented.


The main objective of this thesis is to examine some of the heuristic arguments and possible mechanisms of spreading inspired by the nonlinearities.

In the first part of this thesis, a generalized NLSE is used to demonstrate that some of the heuristic arguments are invalid and some are incomplete. In the second part, it is demonstrated how resonant (double humped) states of the linear problem influence the spreading in the nonlinear case. In the third part preliminary results for finite time spreading are presented.