一类非线性薛定谔方程的数值解法

发布时间：2018-12-20 17:48

【摘要】：非线性薛定谔方程是现代科学中具有普遍意义的重要方程之一,它在流体力学、固体物理、非线性光学、电磁学等领域中具有广泛应用.19世纪70年代,人们用逆散射方法研究薛定谔方程时发现其解由一系列孤立子及色散波构成,随后由于其孤立子解具有较好的性质而被广泛地研究与应用.特别是立方非线性薛定谔方程所描述的光纤通信系统,由此产生的光纤孤子在光纤通信中具有重要应用.非线性薛定谔方程是一类非线性抛物型偏微分方程,对于其一般形式的解析解求解较为困难,没有普遍而有效的求解方法,通常采用数值方法研究其性质.本文利用有限差分法给出数值求解立方非线性薛定谔方程的三种差分格式,并从理论上分析了它们的局部截断误差与稳定性.在数值实验部分,针对具体的初边值问题,对立方非线性薛定谔方程的孤立子解进行数值求解,并利用三种差分格式数值求解初值受到扰动时数值解与解析解之间的误差值随时间变化的发展情况,同时数值模拟了在初始时刻完全分离的两个孤立子的碰撞情形,与理论结果相同,从而验证了所给出的三种差分格式的有效性.研究结果表明本文所给出的三种有限差分格式可以对立方非线性薛定谔方程进行有效地数值求解,三种格式各有优缺点,根据具体问题的需要,可以选取相应的数值格式进行数值求解.最后对本文内容进行总结,并对进一步的工作进行展望.
[Abstract]:Nonlinear Schrodinger equation is one of the most important equations in modern science. It is widely used in fluid mechanics, solid state physics, nonlinear optics, electromagnetism and so on. It is found that the solution of Schrodinger equation is composed of a series of solitons and dispersive waves when the inverse scattering method is used to study the Schrodinger equation. Then the soliton solution is widely studied and applied because of its good properties. Especially for the optical fiber communication system described by cubic nonlinear Schrodinger equation, the resulting optical fiber solitons have important applications in optical fiber communication. Nonlinear Schrodinger equation is a kind of nonlinear parabolic partial differential equation. It is difficult to solve its general analytical solution and there is no universal and effective method to solve it. Numerical method is usually used to study its properties. In this paper, three finite difference schemes for solving cubic nonlinear Schrodinger equation are given, and their local truncation errors and stability are analyzed theoretically. In the numerical experiment, the soliton solution of the square nonlinear Schrodinger equation is numerically solved for the specific initial-boundary value problem. Three difference schemes are used to numerically solve the evolution of the error between the numerical solution and the analytical solution when the initial value is disturbed. At the same time, the collision of two solitons completely separated at the initial time is numerically simulated. The results are the same as the theoretical results, so the validity of the three difference schemes is verified. The results show that the three finite difference schemes presented in this paper can be numerically solved effectively against the square nonlinear Schrodinger equation. Each of the three schemes has its own merits and demerits. The corresponding numerical scheme can be selected for numerical solution. Finally, the content of this paper is summarized, and the future work is prospected.
【学位授予单位】：河南大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

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