Symplectic Runge-Kutta Fourier Spectral Method And Its Application

We pay great attention to nonlinear Schrodinger equations(NLSEs) and its wide use in the physical field such as water waves, plasma physics, nonlinear transmission, and condensed matter physics.In this paper, we primarily study nonlinear Schrodinger equations of Bose-Einstein condensates, because they are the theoretical basis of the Bose-Einstein condensates experiments. Nonlinear Schrodinger equations are hard to solve exactly at most cases. We turn to numerically solve them. We present a novel numerical method—2-stage 4-order implicit Symplectic Runge-Kutta Fourier spectral method for nonlinear Schrodinger equations, and compare it with the numerical results by the other Algorithms in explaining the physical meaning of the numerical results.The main ideas cover the followed three parts:(1)At first we introduce several classical examples of nonlinear issues and nonlinear equations in physical field, secondly we study nonlinear equations in nonlinear optics. At last we mainly care for Bose-Einstein condensates'emerging, development and latest progress. At the same time we study the Gross-Pitaevskii equation, it is the specific nonlinear equations Bose-Einstein condensates abeying to.(2)we present the 2-stage 4-order implicit symplectic Runge-Kutta Fourier spectral method in detail for nonlinear Schrodinger equations with varying coefficients. First of all, we concatenate 2-stage 4-order implicit symplectic Runge-Kutta method in temporal direction and fast Fourier transform method in spatial direction, and then the numerical method is used to solve three nonlinear Schrodinger equations with varying coefficients. We study the norm, the error of norm, the energy, the error of energy of the wave function for the physical systems. Numerical results show that the algorithm is very effective in that it can preserve the global energy conservation and the norm conservation well, it takes a great advantages over the other Algorithms, especially the difference+4-order Runge-Kutta method.(3)We demonstrate the evolution of soliton of Bose-Einstein condensates with the different strength of common trap and localized impurity, we explaining the corresponding physical meanings at the same time.