no. 4
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.

We study the Cauchy problem for the focusing nonlinear Schrödinger (fNLS) equation. Using the ∂‾ 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(x1,x2,v1,v2)={(x,t)∈R2:x=x0+vt with x0∈[x1,x2],v∈[v1,v2]} 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 L2(R) moment and (weak) derivative and that it not generate any spectral singularities.

DOI : 10.1016/j.anihpc.2017.08.006
Mots clés : Focusing, Nonlinear Schrödinger, Long time asymptotics, Integrable systems, Riemann–Hilbert, Soliton resolution 
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     title = {Long time asymptotic behavior of the focusing nonlinear {Schr\"odinger} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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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. http://www.numdam.org/articles/10.1016/j.anihpc.2017.08.006/

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