Four lectures on KAM for the non-linear Schrödinger equation

2008 ◽  
pp. 179-212 ◽  
Author(s):  
L. H. Eliasson ◽  
Sergei B. Kuksin
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Non Linear

scholarly journals Soliton emission in the forced non-linear Schrödinger equation

1986 ◽  
Vol 4 (3-4) ◽  
pp. 545-553 ◽  
Author(s):  
O. Larroche ◽  
M. Casanova ◽  
D. Pesme ◽  
M. N. Bussac
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Reasonable Agreement ◽  
Critical Density ◽  
Plasma Waves ◽  
Resonant Absorption ◽  
Direct Numerical Simulations ◽  
Absorption Of Light ◽  
Non Linear ◽  
Soliton Generation

The plasma waves generated by resonant absorption of light in the vicinity of the critical density of laser-produced plasmas, are modelled by a non-linear Schrödinger equation with additional terms accounting for the presence of a source and the inhomogeneity of the medium.We use an average lagrangian method to describe the behaviour of the solutions of this equation in the range of parameters where periodic soliton generation occurs. An iterating scheme describing the successive emission of solitons yields values for this range of parameters which are in reasonable agreement with those found from direct numerical simulations of the non-linear Schrödinger equation.

scholarly journals ON GLOBAL ROUGH SOLUTIONS TO A NON-LINEAR SCHRÖDINGER SYSTEM

2009 ◽  
Vol 51 (3) ◽  
pp. 499-511 ◽  
Author(s):  
LI MA ◽  
XIANFA SONG ◽  
LIN ZHAO
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Nonlinear Schrödinger ◽  
Non Linear ◽  
Schrödinger Systems ◽  
Schrödinger System

AbstractThe non-linear Schrödinger systems arise from many important physical branches. In this paper, employing the ‘I-method’, we prove the global-in-time well-posedness for a coupled non-linear Schrödinger system in Hs(n) when n = 2, s > 4/7 and n = 3, s > 5/6, respectively, which extends the results of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao (Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math Res. Lett. 9, 2002, 659–682) to the system.

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