Long time asymptotic behavior of the focusing nonlinear Schrödinger equation

Borghese, Michael; Jenkins, Robert; McLaughlin, Kenneth D.T.-R.

doi:10.1016/j.anihpc.2017.08.006

Borghese, Michael ; Jenkins, Robert ; McLaughlin, Kenneth D.T.-R.

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 887-920

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Résumé

We study the Cauchy problem for the focusing nonlinear Schrödinger (fNLS) equation. Using the $\overline{\partial}$ generalization of the nonlinear steepest descent method we compute the long-time asymptotic expansion of the solution $ψ (x, t)$ in any fixed space-time cone $C (x_{1}, x_{2}, v_{1}, v_{2}) = {(x, t) \in R^{2} : x = x_{0} + v t with x_{0} \in [x_{1}, x_{2}], v \in [v_{1}, v_{2}]}$ up to an (optimal) residual error of order $O (t^{- 3 / 4})$ . In each cone C the leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton–soliton and soliton–radiation interactions as one moves through the cone. Our results require that the initial data possess one $L^{2} (R)$ moment and (weak) derivative and that it not generate any spectral singularities.

MR Zbl

DOI : 10.1016/j.anihpc.2017.08.006

Keywords: Focusing, Nonlinear Schrödinger, Long time asymptotics, Integrable systems, Riemann–Hilbert, Soliton resolution

@article{AIHPC_2018__35_4_887_0,
     author = {Borghese, Michael and Jenkins, Robert and McLaughlin, Kenneth D.T.-R.},
     title = {Long time asymptotic behavior of the focusing nonlinear {Schr\"odinger} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {887--920},
     publisher = {Elsevier},
     volume = {35},
     number = {4},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.08.006},
     mrnumber = {3795020},
     zbl = {1390.35020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.08.006/}
}

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Borghese, Michael; Jenkins, Robert; McLaughlin, Kenneth D.T.-R. Long time asymptotic behavior of the focusing nonlinear Schrödinger equation. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 887-920. doi: 10.1016/j.anihpc.2017.08.006

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