Stephen Gustafson

Solitary Wave Dynamics in an External Potential

We study the behavior of solitary-wave solutions of some generalized nonlinear Schrödinger 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.

Spectra of Linearized Operators for NLS Solitary Waves

Nonlinear Schrodinger equations (NLSs) with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves and to the long-time dynamics of solutions of NLSs. We study these spectra in detail, both analytically and numerically.

Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves

We study a class of nonlinear Schrodinger equations which admit families of small solitary wave solutions. We consider solutions which are small in the energy space H 1 , and decompose them into solitary wave and dispersive wave components. The goal is to establish the asymptotic stability of the solitary wave and the asymptotic completeness of the dispersive wave. That is, we show that as t → ∞, the solitary wave component converges to a fixed solitary wave, and the dispersive component converges strongly in H 1 to a solution of the free Schrodinger equation.

Long Time Motion of NLS Solitary Waves in a Confining Potential

Abstract.We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schrödinger equations with a confining, slowly varying external potential, V(x).A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval.We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential V(x) over a long time interval.Communicated by Rafael D. Benguria

Global Dispersive Solutions for the Gross–Pitaevskii Equation in Two and Three Dimensions

Abstract.We study asymptotic behaviour at time infinity of solutions close to the non-zero constant equilibrium for the Gross–Pitaevskii equation in two and three spatial dimensions. We construct a class of global solutions with prescribed dispersive asymptotic behavior, which is given in terms of the linearized evolution.

Asymptotic stability of harmonic maps under the Schrödinger flow

For Schr ¨ odinger maps from R 2 × R + to the 2-sphere S 2 , it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space–dimensional linear Schr¨ odinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map.

Statics and dynamics of magnetic vortices and of Nielsen–Olesen (Nambu) strings

We review recent works on statics and dynamics of magnetic vortices in the Ginzburg–Landau model of superconductivity and of Nielsen–Olesen (Nambu) strings in the Abelian–Higgs model of particle physics.

Dynamic stability of magnetic vortices

The well-known equivariant (radially-symmetric) vortex solutions of the Ginzburg–Landau equations of superconductivity are static solutions of several time-dependent equations. We study the nonlinear dynamic stability of these vortices. As conjectured by Jaffe and Taubes, we prove that for type I superconductors, all vortices are stable, while for type II superconductors, the vortices of topological degree ±1 are stable, and the higher-degree vortices are unstable.

Dynamic Stability and Instability of Pinned Fundamental Vortices

We study the dynamic stability and instability of pinned fundamental ±1 vortex solutions to the Ginzburg–Landau equations with external potential in ℝ2. For sufficiently small external potentials, there exists a perturbed vortex solution centered near each non-degenerate critical point of the potential. With respect to both dissipative and Hamiltonian dynamics, we show that perturbed vortex solutions which are concentrated near local maxima (resp. minima) are orbitally stable (resp. unstable). In the dissipative case, the stability is in the stronger “asymptotic” sense.

Stability in H1/2 of the sum of K solitons for the Benjamin–Ono equation

This note proves the orbital stability in the energy space H1/2 of the sum of widely spaced 1-solitons for the Benjamin–Ono equation, with speeds arranged so as to avoid collisions.