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Soliton dynamics for the nonlinear Schrödinger equation with magnetic field. (English) Zbl 1179.81066

Summary: The semiclassical regime of a nonlinear focusing Schrödinger equation in presence of non-constant electric and magnetic potentials \(V, A\) is studied by taking as initial datum the ground state solution of an associated autonomous stationary equation. The concentration curve of the solutions is a parameterization of the solutions of the second order ordinary equation \({\ddot x=-\nabla V(x)-\dot x\times B(x)}\), where \({B=\nabla\times A}\) is the magnetic field of a given magnetic potential \(A\).

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q40 PDEs in connection with quantum mechanics
35Q51 Soliton equations
35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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