FROM THE KLEIN–GORDON–ZAKHAROV SYSTEM TO THE NONLINEAR SCHRÖDINGER EQUATION

2005 ◽  
Vol 02 (04) ◽  
pp. 975-1008 ◽  
Author(s):  
NADER MASMOUDI ◽  
KENJI NAKANISHI
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Zakharov System ◽  
Nonlinear Schrödinger ◽  
Bilinear Estimates ◽  
Resonant Interactions ◽  
Klein Gordon ◽  
Nonlinear Energy

In this paper, we study the convergence of solutions in the limit from the Klein–Gordon–Zakharov system to the nonlinear Schrödinger equation. The major difficulties are resonant bilinear interactions whose frequency are going to infinity, and the diverging total energy. We overcome them by combining bilinear estimates for non-resonant interactions and a modified nonlinear energy at the resonant frequency.

scholarly journals Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

2020 ◽  
Vol 120 (1-2) ◽  
pp. 73-86 ◽  
Author(s):  
Yuslenita Muda ◽  
Fiki T. Akbar ◽  
Rudy Kusdiantara ◽  
Bobby E. Gunara ◽  
Hadi Susanto
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Gordon Equation ◽  
Nonlinear Schrodinger Equation ◽  
Numerical Comparison ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Nonlinear Schrödinger ◽  
Klein Gordon ◽  
Klein Gordon Equation

We consider a discrete nonlinear Klein–Gordon equation with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein–Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.

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