A new proof of long-range scattering for critical nonlinear Schrödinger equations

By J. Kato, F. Pusateri. Differential and Integral Equations, 2010

132 citations recorded by the source corpus.

We present a new proof of global existence and long range scattering, from small initial data, for the one-dimensional cubic gauge invariant nonlinear Schr\"odinger equation, and for Hartree equations in dimension $n \geq 2$. The proof relies on an analysis in Fourier space, related to the recent works of Germain, Masmoudi and Shatah on space-time resonances. An interesting feature of our approach is that we are able to identify the long range phase correction term through a very natural stationary phase argument.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0d024dce74934786efcbcb01b9004269073dd0ff

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Citations in the corpus

Nonlinear fractional Schrödinger equations in one dimension (2012, methodology): “As shown in [15], a “stationary phase” type argument, inspired by the Fourier analysis of [6], can serve as a substitute for such a factorization.”
Asymptotic stability of solitons for mKdV (2015, unknown): “As observed in [29], this will give a very simple means of computing the large-time correction to scattering, due to the long-range effects of the critically dispersing nonlinear term.”
Long time behavior of solutions to the mKdV (2014, methodology): “In the related case of the cubic nonlinear Schrödinger equation onR, modi ed asymptotics have been proved without inverse scattering techniques using both spatial methods [30] and Fourier methods [13, 23].”
MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, methodology): “Previous modified scattering results that we are aware of, only concerned equations (or systems) posed on Rd, quasilinear or semilinear [1, 26, 33, 34, 54, 55, 56, 64, 65, 68, 76, 91] and had an integrable asymptotic system for (1.”
Global solutions for the gravity water waves system in 2d (2013, methodology): “obal existence and pointwise decay, one still needs to identify an appropriate logarithmic correction, and prove modiﬁed scattering usin g a norm similar to the Z norm. A similar argument was used in [KP11] in the case of scattering critical semilinear Schrödinger equations (see also [Oza91, HN98]).”