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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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