Modulational instabilities of perturbed nonlinear Schroedinger-type equations

• Main Author: Rapti, Zoi
• Language: ENG
• Published: ScholarWorks@UMass Amherst 2004
• Subjects: Mathematics
• Online Access: https://scholarworks.umass.edu/dissertations/AAI3152738

In this thesis we examine the stability thresholds for nonlinear Schrödinger-type equations. We use variational techniques to rederive the modulational instability criterion in the case with cubic nonlinearity, considering the equation as a finite dimensional dynamical system. We proceed to the analysis of the case where we have a potential; which is either linear, V( x,t) = −αx, using a Tappert transformation, or quadratic, V(x,t) = −k (t)x2, using a lens-type transformation to eliminate the potential. Also, the cases of time dependent coefficient of the dispersive term and strength of the nonlinearity are considered for the continuous and discrete equation. To analyze these cases, we use Floquet's theory and multiple-scale analysis, respectively. A case with higher order dissipation is also examined, motivated by the resent work of Köhler (2003), who studied the three-body problem in Bose-Einstein Condensates. The results that we obtain are both analytical and numerical. Finally, we study a system of two discrete nonlinear Schrödinger equations coupled by both linear and nonlinear terms, and we examine domain wall solutions in the particular case with linear coupling.