## Soliton interaction with slowly varying potentials.(English)Zbl 1147.35084

Summary: We study the Gross-Pitaevskii equation with a slowly varying smooth potential, $$V(x)= W(hx)$$. We show that up to time $$\log(1/h)/h$$ and errors of size $$h^2$$ in $$H^1$$, the solution is a soliton, evolving according to the classical dynamics of a natural effective Hamiltonian, $$(\xi^2+ \text{sech}^2* V(x))/2$$. This provides an improvement $$(h\to h^2)$$ compared to previous works, and is strikingly confirmed by numerical simulations.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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