Effective Dynamics of the Nonlinear Schrödinger Equation on Large Domains

T. Buckmaster, Pierre Germain, Z. Hani, Jalal Shatah

Research output: Contribution to journalArticle

Abstract

We consider the nonlinear Schrödinger (NLS) equation posed on the box [0,L]d with periodic boundary conditions. The aim is to describe the long-time dynamics by deriving effective equations for it when L is large and the characteristic size e(open) of the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in the theory of statistical physics of dispersive waves, which goes by the name of "wave turbulence." Our main result is deriving a new equation, the continuous resonant (CR) equation, which describes the effective dynamics for large L and small e(open) over very large timescales. Such timescales are well beyond the (a) nonlinear timescale of the equation, and (b) the euclidean timescale at which the effective dynamics are given by (NLS) on ℝd. The proof relies heavily on tools from analytic number theory, such as a relatively modern version of the Hardy-Littlewood circle method, which are modified and extended to be applicable in a PDE setting.

Original languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - Jan 1 2018

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Nonlinear equations
Nonlinear Equations
Time Scales
Number theory
Hardy-Littlewood Method
Water waves
Circle Method
Dispersive Equations
Statistical Physics
Water Waves
Turbulence
Physics
Periodic Boundary Conditions
Boundary conditions
Ocean
Euclidean

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Effective Dynamics of the Nonlinear Schrödinger Equation on Large Domains. / Buckmaster, T.; Germain, Pierre; Hani, Z.; Shatah, Jalal.

In: Communications on Pure and Applied Mathematics, 01.01.2018.

Research output: Contribution to journalArticle

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