A note to the integrable discretization of the mKdV and Schrödinger equations

Abstract

An approach has been proposed to the integrable discretization of nonlinear evolution equations. Based on the bilinear formalism, we choose appropriate substitution from hyperbolic operator into continuous Hirota operators and obtain several new kinds of integrable system through seeking their 3-soliton solutions, such as the mKdV equation, the nonlinear Schrödinger equation and so on. By applying Adomian decompose method, we discuss the numerical analysis property to the discrete mKdV equation. In addition, we also point out the relations between the above discreted equations and some well-known equations.

Keywords:

• discrete mKdV equation
• discrete NLS equation
• bilinear operator
• hyperbolic operator
• ACM

2000 AMS Subject Classifications :

• 35Q53
• 35Q55
• 34L99
• 39A99

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 10771196 and 10831003), the Natural Science Foundation of Zhejiang Province (No. Y7080198) and Zhejiang Innovation Project (Grant No. T200905).