掺铒光纤中高阶NLS-MB方程的孤子解与呼吸子解

Soliton and Breather Solutions of Higher Order NLS-MB Equation in Erbium-Doped Fibers

摘要: 在非线性科学中，孤子理论占有重要的地位，它的兴起开辟了非线性科学研究的新方向，成为求解非线性偏微分方程的主要手段。特别地，在流体力学、非线性光学、等离子物理等自然科学领域中涉及的非线性模型问题大部分都可以利用孤子方程来描述，但是随着光脉冲压榨技术的发展，一些高阶效应对光脉冲的扰动就会变得突出。为了得到更为复杂并明显的脉冲现象，本文运用Darboux变换的方法，求出了高阶非线性薛定谔(NLS)方程与Maxwell-Bloch系统耦合方程的孤子解及其呼吸子解。事实证明，在非线性发展方程的求解中，此方法非常简便易行，同时实现了向高阶迈进的突破。第一节，我们给出了4阶NLS-MB方程的Lax对并求出对应的达布变换。第二节，求出其单双孤子解并绘出解的图像。第三节，求出其单双呼吸子解并绘出解的图像。第四节，得出结论。

Abstract: In the nonlinear science, the soliton theory plays a significant role. The development of the soliton theory which has become the main instrument to solve nonlinear partial differential equations, has opened up a new direction for the study of nonlinear science. Specially, some nonlinear models involved in the fluid mechanics, the nonlinear optics, the plasma physics and other fields of natural science, can be described by the soliton equation. But with the development of optical pulse pressing technology, the perturbation of light pulses by some higher-order effects becomes prominent. In order to get more complex and obvious pulse phenomena, some soliton solutions and breather solutions of higher-order nonlinear Schrödinger and Maxwell-Bloch coupling equation are found by using the method of Darboux transformation. The results indicate that this approach is still simple and efficient for solving nonlinear evolution equation, and the breakthrough to higher order has been achieved at the same time. In the first section, we get the Lax pair of 4th order NLS-MB equation and find the corresponding darboux transformation. In the second section, the solutions of single- and double-solitons have been obtained and their figures are plotted. In the third section, single- and double-breathers solutions have been obtained and figures of the solutions are plotted. In the fourth section, the conclusions have been obtained.