Sobolev Stability of Plane Wave Solutions to the Cubic Nonlinear Schrodinger Equation on a Torus

被引：44

作者：

Faou, Erwan ^{[1
,2
,3
]}

Gauckler, Ludwig ^{[4
]}

Lubich, Christian ^{[5
]}

机构：

[1] INRIA, Bruz, France

[2] ENS Cachan Bretagne, Bruz, France

[3] Ecole Normale Super, Dept Math & Applicat, F-75231 Paris, France

[4] Tech Univ Berlin, Inst Math, D-10623 Berlin, Germany

[5] Univ Tubingen, Inst Math, Tubingen, Germany

来源：

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS | 2013年 / 38卷 / 07期

关键词：

Birkhoff normal forms; Modulated Fourier expansions; Nonlinear Schrodinger equation; Plane wave solutions; Stability; 35Q55; 37K55; 35B20; 35B35; KLEIN-GORDON EQUATIONS; BIRKHOFF NORMAL-FORM; SMALL CAUCHY DATA; PERIODIC-WAVES; EXISTENCE;

D O I：

10.1080/03605302.2013.785562

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

It is shown that plane wave solutions to the cubic nonlinear Schrodinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.

引用

页码：1123 / 1140

页数：18

共 30 条

[1]

[Anonymous], 2006, SPRINGER SERIES COMP

[2] A Birkhoff normal form theorem for some semilinear PDEs [J].

Bambusi, D. .

HAMILTONIAN DYNAMICAL SYSTEMS AND APPLICATIONS, 2008, :213-247

[3] Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on zoll manifolds [J].

Bambusi, D. ;

Delort, J.-M. ;

Grebert, B. ;

Szeftel, J. .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 2007, 60 (11) :1665-1690

[4] Birkhoff normal form for partial differential equations with tame modulus [J].

Bambusi, D. ;

Grebert, B. .

DUKE MATHEMATICAL JOURNAL, 2006, 135 (03) :507-567

[5] Birkhoff normal form for some nonlinear PDEs [J].

Bambusi, D .

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2003, 234 (02) :253-285

[6] Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrodinger equations [J].

Bourgain, J .

ANNALS OF MATHEMATICS, 1998, 148 (02) :363-439

[7]

Bourgain J, 2000, GEOM FUNCT ANAL, P32

[8] On diffusion in high-dimensional Hamiltonian systems and PDE [J].

Bourgain, J .

JOURNAL D ANALYSE MATHEMATIQUE, 2000, 80 (1) :1-35

[9]

Bourgain J., 1996, Int. Math. Res. Not., P277

[10] ENERGY CASCADES FOR NLS ON THE TORUS [J].

Carles, Remi ;

Faou, Erwan .

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 2012, 32 (06) :2063-2077

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