Abstract
Currently, experimental and theoretical studies of solitons have been conducted in the context of several areas of science, from applied mathematics and physics to chemistry and biology. A significant amount of soliton research is concentrated on nonlinear optics (light waves) and Bose-Einstein condensates (matter waves) in optical lattices, which are often described by the Gross-Pitaevskii equation or the nonlinear Schrodinger equation with a periodic potential.
We studied soliton dynamics in both time-independent and time-dependent potentials. We used a variational approach to simplify the problem by reducing it to the study of the dynamics of a particle in an effective potential. We saw good agreement between the numerical and approximate variational solutions, indicat- ing the variational method is a simple yet powerful method for the study of soliton dynamics in a frequency-modulated potential. We studied several scenarios, includ- ing soliton trapping by a periodic potential, â jumpingâ between adjacent wells
and parametric resonance. We also investigated the soliton dynamics by using Poincar´e sections
and histograms to examine the velocity distribution of the driven solitons for different initial conditions.