Soliton dynamics in a potential. (English) Zbl 0955.35067

Summary: We study the semiclassical limit of subcritical focussing NLS with a potential \[ iu^\varepsilon_t+ {\varepsilon\over 2} \Delta u^\varepsilon+{1\over\varepsilon} |u^\varepsilon|^{p- 1} u^\varepsilon- {1\over\varepsilon} V(x) u^\varepsilon= 0 \] for initial data of the form \(s({x- x_0\over \varepsilon}) e^{i{v_0\cdot x\over \varepsilon}}\), where \(s\) is the ground state of an associated unscaled problem. We show that in the semiclassical limit, the solution has roughly the form \(s({x- x^\varepsilon\over \varepsilon}) e^{i{v^\varepsilon(t)\cdot x\over\varepsilon}}\), and we show that the approximate center of mass \(x^\varepsilon(\cdot)\) converges to a solution of the equation \(x''= -DV(x)\), \(x(0)= x_0\), \(x'(0)= v_0\) as \(\varepsilon\to 0\).


35Q55 NLS equations (nonlinear Schrödinger equations)
81Q15 Perturbation theories for operators and differential equations in quantum theory
35B20 Perturbations in context of PDEs
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