10.13023/ETD.2017.164Lindgren, Joseph B.Joseph B.Lindgrenhttp://orcid.org/0000-0002-8731-631XUniversity of KentuckyOrbital Stability Results for Soliton Solutions to Nonlinear Schrödinger Equations with External PotentialsUniversity of Kentucky2017DissertationAtomic, Molecular and Optical PhysicsDynamical SystemsNon-linear DynamicsDifferential EquationsPlasma and Beam PhysicsPDFCopyright © Joseph B. Lindgren 2017For 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 $V=0$.