Journal of Nonlinear Science

, Volume 22, Issue 3, pp 351–370 | Cite as

Effective Dynamics for N-Solitons of the Gross–Pitaevskii Equation

Article

Abstract

We consider several solitons moving in a slowly varying external field. We present results of numerical computations which indicate that the effective dynamics obtained by restricting the full Hamiltonian to the finite-dimensional manifold of N-solitons (constructed when no external field is present) provides a remarkably good approximation to the actual soliton dynamics. This is quantified as an error of size h2 where h is the parameter describing the slowly varying nature of the potential. This also indicates that previous mathematical results of Holmer and Zworski (Int. Math. Res. Not. 2008: Art. ID runn026, 2008) for one soliton are optimal. For potentials with unstable equilibria, the Ehrenfest time, log(1/h)/h, appears to be the natural limiting time for these effective dynamics. We also show that the results of Holmer et al. (arXiv:0912.5122, 2009) for two mKdV solitons apply numerically to a larger number of interacting solitons. We illustrate the results by applying the method with the external potentials used in the Bose–Einstein soliton train experiments of Strecker et al. (Nature 417:150–153, 2002).

Keywords

Effective dynamics Ehrenfest time Multiple soliton interaction Bose–Einstein condensates Spectral methods 

Mathematics Subject Classification (2000)

35Q51 65M70 35K05 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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