A High-order Conservative Scheme For The Nonlinear Fractional Schr(?)dinger Equation

Schr?dinger equation is a partial differential equation, which is to describe the evolution of the quantum state of a physical system over time, it is one of the basic equation of quantum mechanics. Schr?dinger equation is widely used in the field of atomic, molecular, solid state physics, nuclear physics, chemistry and so on. Moreover,the fractional Schr?dinger equation is generalized by the classical Schr?dinger equation.As a result, the fractional Schr?dinger equation has important theoretical significance.We first give a simple introduction to the research status of the Schr?dinger equation and fractional Schr?dinger equation, summarize and analyze the method to approximate the fractional operator, and review the high-order approximation format of the Riemann-Liouville fractional derivative, which is exported by the weighted and shifted Lubich difference operator.Subsequently, we consider the nonlinear fractional Schr?dinger equation with Riesz space fractional derivative. For this equation, in spatial, considering the equivalent relation between Riesz space fractional derivative and Riemann-Liouville fractional derivative,then we use the high-order approximation format of the Riemann-Liouville fractional derivative to acquire the approximation of the former fractional derivative; in temporal,we adopt the Crank-Nicolson scheme. Consequently, we obtain a high-order and conservative difference scheme, and the truncation error of this scheme is ? ?42?hO ?,where? and h is the time and space step, respectively.Afterwards, we analyze the conservative properties rigorously, including the mass conservation and energy conservation, we get the unconditional stability with taking advantage of the mass conservation. In view of the Brouwder fixed point theorem, we prove the existence of the difference solution. Moreover, based on the Gronwall inequality and the Cauchy-Schwarz inequality, we show the convergence of our difference scheme at the order of ? ?42?hO ? in L2-norm, without any restriction on the grid ratio.Finally, some numerical tests are implemented to testify the result of theoretical analysis and efficiency of the scheme. Besides, we point out the application of the parameters in the equation.