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Full-Text Articles in Physical Sciences and Mathematics

On Osgood's Criterion For Classical Wave Equations And Nonlinear Shallow Water Wave Equations, Timothy Smith, Greg Spradlin Nov 2017

On Osgood's Criterion For Classical Wave Equations And Nonlinear Shallow Water Wave Equations, Timothy Smith, Greg Spradlin

Timothy Smith

The problem on classical solutions for the wave equation and the BBM equation is considered. The equations are considered with a forcing term and sufficient conditions of solvability, existence and uniqueness are established.


Orbital Stability Results For Soliton Solutions To Nonlinear Schrödinger Equations With External Potentials, Joseph B. Lindgren Jan 2017

Orbital Stability Results For Soliton Solutions To Nonlinear Schrödinger Equations With External Potentials, Joseph B. Lindgren

Theses and Dissertations--Mathematics

For certain nonlinear Schroedinger equations there exist solutions which are called solitary waves. Addition of a potential $V$ changes the dynamics, but for small enough $||V||_{L^\infty}$ we can still obtain stability (and approximately Newtonian motion of the solitary wave's center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential $V$. Our method is an adaptation of the well-known Lyapunov method.

For the sake of completeness, we also prove long-time stability of traveling solitons in the case $V=0$.