呼吸子是非线性薛定谔方程(NLSE)中的一类重要的解,在量子力学、光学和数学物理中具有重要应用。方程的非线性项使得其解析求解非常复杂,因此数值求解方法常用来模拟方程的行为。Splitting方法是一种对非线性薛定谔方程具有很高效率的算法,本文简单介绍splitting方法,然后用其求解了一种常见的呼吸子。通过splitting方法求解呼吸子,可以深入理解其在不同参数条件下的演化行为,为相关实验和应用提供理论支持。The Nonlinear Schrödinger Equation (NLSE) plays a crucial role in quantum mechanics, optics, and mathematical physics. Breather is one type of soliton solutions of NLSE. The nonlinearity of the equation makes it hard to obtain the analytic solution of the breather. Numerical methods are usually adopted to simulate the behavior of the solitons, one of which is the splitting method. In this paper, we use the splitting method to simulate one kind of breather solutions. By this way, we can gain a deeper understanding of their evolutionary behavior under varying parameter conditions, thereby providing theoretical support for related experiments and applications.
基于同伦分析方法研究了含自相位调制的非线性薛定谔方程。该方程可以用来描述光信号在光纤传输过程中因损耗、色散等导致的体系非线性效应。求出了方程的孤子解和周期解,并讨论了体系的二维和三维演化行为。This paper investigates the nonlinear Schrödinger equation with self-phase modulation based on the homotopy analysis method. This equation can be used to describe the nonlinear effects of the system caused by loss, dispersion, and other factors during optical signal transmission in optical fibers. The soliton solutions and periodic solutions of the equation are obtained, and the two-dimensional and three-dimensional evolution behaviors of the system are presented.
