UVM Theses and Dissertations
Format:
Print
Author:
Roma, Eric
Dept./Program:
Mathematics
Year:
2013
Degree:
MS
Abstract:
A soliton is a type of solitary wave that maintains its structure while it travels at a constant speed. They arise in the solutions of nonlinear and dispersive partial differential equations (PDEs). One of the most unique properties of solitons is that they are not destroyed when they collide with one another. N-soliton solutions in the Korleweg-de Vries (KdV) and focusing nonlinear Schrödinger (NLS) equations are derived by the bilinear method. These solutions are given in terms of Wronskian and Gram determinants.
In addition, higher-order rogue waves in the focusing NLS are derived by bilinear methods. Rogue waves are now a current phenomenon in· nonlinear wave theory that occur in both, water and optical waves. It is shown that their general solutions can be simplified to algebraic expressions using Gram determinants and Schur polynomials where the N-th order solutions have N -1 free parameters. Applying numeriCal methods yield rich and symmetrical 3-dimensional graphs, and contour plots, whose amplitudes can be maximized by varying the free parameters.
In addition, higher-order rogue waves in the focusing NLS are derived by bilinear methods. Rogue waves are now a current phenomenon in· nonlinear wave theory that occur in both, water and optical waves. It is shown that their general solutions can be simplified to algebraic expressions using Gram determinants and Schur polynomials where the N-th order solutions have N -1 free parameters. Applying numeriCal methods yield rich and symmetrical 3-dimensional graphs, and contour plots, whose amplitudes can be maximized by varying the free parameters.