Discrete Nonlinear Schrodinger Equation

Abstract

This dissertation addresses the Initial Value Problem (IVP) of the Discrete Nonlinear Schro ̈dinger Equations (DNLS) with a complex potential and general nonlinearity on multidimensional lattices. The research builds on and extends previous work by exploring the global well-posedness, the existence of global solutions, and the behavior of solutions in weighted lp spaces. Through the use of semigroup theory, we establish comprehensive frameworks and provide in- sights into the evolution and stability of DNLS solutions. In the first paper, we utilized the integral equation defining the mild solution of the DNLS to prove the existence of global solutions for the DNLS with a weighted lp initial value for 1 ≤ p < 2 by leveraging the existing l2 global solutions. In the second paper, we investigated the global solutions in the weighted lp space for 2 < p < ∞. In the third paper, we improved the results on global solutions for the DNLS with a weighted lp initial value for two cases 1 < p < 2, and 2 < p < ∞ with optimal estimates and minimum assumptions and proved the existence of a global attrac- tor for lp solutions to the initial value problem. This comprehensive approach enhances the mathematical understanding of DNLS in higher dimensions, providing significant insights into the behavior and stability of solutions in complex, multidimensional discrete systems.