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- W1970110634 abstract "We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic (ae2iωt). The goal of this Note is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann–Hilbert problem. We show that the solution of the IBV problem has different asymptotic behaviors in different regions. In the region x>4bt (b=(a2−ω)/2>0) the solution has the form of a Zakharov–Manakov vanishing asymptotics. In the region 4bt−12aNlogt<x<4bt, where N is an integer, the solution behaves as a finite train of asymptotic solitons. In the region 4(b−a2)t<x<4bt the solution is a modulated elliptic wave. Finally, in the sector 0<x<4(b−a2)t the solution is a plane wave. To cite this article: A. Boutet de Monvel et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007). Nous considérons l'équation de Schrödinger non linéaire focalisante sur le premier quadrant (x>0, t>0), avec donnée initiale décroissante à l'infini et donnée au bord t-périodique, de la forme ae2iωt. Notre objectif est d'étudier le comportement asymptotique de la solution de ce problème aux limites. Notre méthode consiste à faire l'analyse asymptotique d'un problème de Riemann–Hilbert matriciel associé. Nous montrons que la solution du problème aux limites a des comportements asymptotiques différents suivant la région dans laquelle (x,t) tend vers l'infini. Dans le secteur x>4bt, pour b=(a2−ω)/2>0, le comportement asymptotique est décrit par des formules du type de Zakharov–Manakov. Dans la région 4bt−12aNlogt<x<4bt, où N est un entier, la solution se comporte comme un train fini de solitons asymptotiques. Dans le secteur 4(b−a2)t<x<4bt la solution est comme une onde elliptique modulée. Enfin, dans le secteur 0<x<4(b−a2)t la solution se comporte comme une onde plane. Pour citer cet article : A. Boutet de Monvel et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007)." @default.
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- W1970110634 date "2007-12-01" @default.
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- W1970110634 title "Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition" @default.
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- W1970110634 doi "https://doi.org/10.1016/j.crma.2007.10.018" @default.
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