## 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 $$\varepsilon\to 0$$.

### MSC:

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

### Keywords:

semiclassical limit; subcritical focussing NLS
Full Text: