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 construct 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.

In this chapter, we derive a convenient representation for the integral kernel of the Schrodinger evolution operator, e-itH / n . This representation, the "Feynman path integral" , will provide us with a heuristic but effective tool for investigating the connection between quantum and classical mechanics. This investigation will be undertaken in the next section. The Feynman Path IntegralConsider a particle in IRd described by a self-adjoint Schrodinger operatorRecall that the dynamics of such a particle is given by the Schrodinger equation . 8'IjJRecall also that the solution to this equation, with the initial condition is given in terms of the evolution operator U(t):= e-iHt / n asOur goal in this section is to understand the evolution operator U(t) = e-iHt / n by finding a convenient representation of its integral kernel. We denote the integral kernel of U (t) by Ut (y, x) (also called the propagator from x to y). A representation of the exponential of a sum of operators is provided by the Trotter product formula (Theorem 10.2) which is explained in Section 10.3 at the end of this chapter. The Trotter product formula says that S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics

We study the global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schrödinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence.

Abstract. Nonlinear Schrödinger (NLS) equations 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 longtime dynamics of solutions of (NLS). We study these spectra in detail, both analytically and numerically.

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schrödinger flow as special cases) for degree m equivariant maps from R 2 to S 2 . If m ≥ 3, we prove that near-minimal energy solutions converge to a harmonic map as t → ∞ (asymptotic stability), extending previous work [11] down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m = 3, involving (among other tools) a "normal form" for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schrödinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m = 2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even "eternal oscillation".

We investigate the asymptotic behavior at time infinity of solutions close to a nonzero constant equilibrium for the Gross-Pitaevskii (or Ginzburg-Landau-Schrödinger) equation. We prove that, in dimensions larger than 3, small perturbations can be approximated at time infinity by the linearized evolution, and the wave operators are homeomorphic around 0 in certain Sobolev spaces.

For Schrödinger maps from R 2 × R + to the 2-sphere S 2 , it is not known if finite energy solutions can form singularities ("blowup") 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 Schrödinger 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.

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.