Effective Dynamics for N-Solitons of the Gross–Pitaevskii Equation
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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).
KeywordsEffective dynamics Ehrenfest time Multiple soliton interaction Bose–Einstein condensates Spectral methods
Mathematics Subject Classification (2000)35Q51 65M70 35K05
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- Holmer, J., Zworski, M.: Soliton interaction with slowly varying potentials. Int. Math. Res. Not. 2008, Art. ID runn026 (2008), 36 pp. Google Scholar
- Holmer, J., Perelman, G., Zworski, M.: Soliton interaction with slowly varying potentials. Preprint arXiv:0912.5122 (2009)
- Robert, D.: On the Herman–Kluk semiclassical approximation. arXiv:0908.0847 (2009)