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Weakly nonlinear time-adiabatic theory. (English) Zbl 1337.81056
Author’s abstract: We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schrödinger equations in which either the nonlinear coupling constant or, equivalently, the solution is asymptotically small. To this end, a notion of criticality is introduced at which the linear bound states stay adiabatically stable, but nonlinear effects start to show up at leading order in the form of a slowly varying nonlinear phase modulation. In addition, we prove that in the same regime a class of nonlinear bound states also stays adiabatically stable, at least in terms of spectral projections.
Reviewer: Ma Wen-Xiu (Tampa)

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q55 NLS equations (nonlinear Schrödinger equations)
70H11 Adiabatic invariants for problems in Hamiltonian and Lagrangian mechanics
Full Text: DOI arXiv
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