Numerical Analysis Seminar

Friday April 11, 2008

Speaker: Michail D. Todorov,
Department of Mathematics, The University of Texas at Arlington, TX and Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Bulgaria
Title: Collision Dynamics of Coupled Nonlinear Schrodinger Equations
Time: 2:00-3:00 pm
Place: Blocker 628
Note: This is a joint ISC/NA seminar. Note the unusual time and place.

Abstract

The investigation of soliton supporting systems is of great importance both for the applications and for the fundamental understanding of the phenomena associated with propagation of solitons. Recently, elaborate models such as Coupled Nonlinear Schrodinger Equations (CNLSE) appeared in the literature. They involve more parameters and possess richer phenomenology but are not fully integrable and require numerical approaches. The non-fully-integrable models possess as a rule three conservation laws: for (wave) ``mass'', (wave) momentum, and energy.

We construct a conservative fully implicit scheme with internal iterations in complex arithmetic, which allows us to reduce the computational time fourfold compared to the real arithmetic. The scheme faithfully represents the conservation laws within the round-off error.

We investigate collisions of solitary waves (quasi-particles) with linear polarization in initial configuration. For various rates of cross-modulation parameter we elucidate numerically the role of nonlinear coupling on the QP dynamics and find that the initially linear polarizations of the QPs change after the collision to elliptic polarization. For large values of cross-modulation parameter, an additional QP is created during the collision. We establish that although the total energy is positive and conserved, the energy only of the system of identifiable after the collision QPs is negative, i.e., the different smaller excitations and radiation carry away part of the energy. The results about the phase speeds, momenta and masses of the QPs after the collision are compiled and discussed thoroughly. The effects found seem to be novel and enrich the knowledge about the intimate mechanisms of interaction of QPs.

Last revised: 03/31/08 By: abnersg@math