Conservativity of symplectic schemes for the nonlinear Schrödinger equation

Yifa Tang, Hua Guan, Quandong Feng

LSEC, ICMSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, P.R. China

tyf@lsec.cc.ac.cn

Contributed talk

We consider the nonlinear cubic Schrödinger equation (NLSE) with initial conditions:

$$\left\{ \begin{array}{l} iW_t+W_{xx}+a\vert W\vert^2W=0\\ W(x,0)=W_0(x) \end{array} \right. \eqno (1)$$

where $x\in \mathbb{R}$, $a$ is a constant and $W(x,t)$ is a complex function. Different initial conditions $W_0(x)$ decide different motions. For example, some kind of $W_0(x)$ with $W_0(\pm\infty)=0$ will produce bright solitons motion. It's known that NLSE (1) has infinite conserved quantities such as the energy, charge, momentum. For equation (1), two popular spatial discretizations are

$$iW_t^{(l)}+\frac{W^{(l+1)}-2W^{(l)}+W^{(l-1)}}{h^2} +\fra... ...W^{(l)}\vert^2\left(W^{(l+1)}+W^{(l+1)}\right)=0. \eqno (2a)$$

and

$$iW_t^{(l)}+\frac{W^{(l+1)}-2W^{(l)}+W^{(l-1)}}{h^2} +a\vert W^{(l)}\vert^2W^{(l)}=0, \eqno (2b)$$

where $h$ is the spatial step size and $W^{(l)}(t)=W(lh,t)$, $l=\cdots,-1,0,1,\cdots$. These two discrete models are nonstandard and standard hamiltonian systems respectively. (2a) is the well-known Ablowitz-Ladik model. Analogously, (2a) has also infinite conserved quantities, while (2b) has only two invariants: the discrete energy and discrete charge. For both models, we use some symplectic schemes to simulate the solitons motions, and test the evolution of six discrete invariants or approximations. And for (2a), two kind of standardizing transformations suggested by Hairer, Lubich and Wanner, and by Tang, Pérez-García and Vázquez are used. The nice performance and conservativity of the symplectic schemes is shown.