Solitary wave dynamics in an external potential

• Published in: Communications in mathematical physics Vol. 250; no. 3; pp. 613 - 642
• Main Authors: FRÖHLICH, J, GUSTAFSON, S, JONSSON, B. L. G, SIGAL, I. M
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.10.2004
• Subjects: elliptic equation; Exact sciences and technology; existence; Fluid dynamics; Fundamental areas of phenomenology (including applications); Global analysis, analysis on manifolds; ground-states; hartree equation; Hydrodynamic waves; Mathematical analysis; Mathematics; nonintegrable equations; nonlinear schrodinger-equations; Partial differential equations; Physics; positive radial solutions; scalar field-equations; Sciences and techniques of general use; stability theory; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; uniqueness; Damping; Non linear equation; Solitary wave; Lyapunov functional; Initial condition; Hamiltonian; Schrödinger equation; Mathematical physics
• ISSN: 0010-3616; 1432-0916; 1432-0916
• DOI: 10.1007/s00220-004-1128-1

We study the behavior of solitary-wave solutions of some generalized nonlinear Schrodinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We consider solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.