推广的导数非线性薛定谔方程的单/双周期背景上的呼吸子和怪波及其碰撞解

摘要/Abstract

摘要：

非线性薛定谔方程是物理和应用数学领域中一个非常重要的可积系统. 该文利用达布变换研究了推广的导数非线性薛定谔方程的单/双周期背景上的呼吸子和怪波以及呼吸子和怪波的碰撞解. 首先, 构造推广的导数非线性薛定谔方程的达布变换. 然后, 通过达布变换, 推导出周期背景和双周期背景上的呼吸子解和怪波解以及碰撞解. 最后, 借助于图示, 详细分析了有趣的新解结构. 这也为研究新型解的物理机制提供了理论依据.

关键词: 推广的导数非线性薛定谔方程, 达布变换, 周期解, 呼吸子, 怪波

Abstract:

The nonlinear Schr $\rm\ddot{o}$ dinger equation is a very important integrable system in the field of physics and applied mathematics. In this paper, the breather and rogue wave on the periodic/double periodic background and the collision solutions of breather and rogue wave for the generalized derivative nonlinear Schr $\rm\ddot{o}$ dinger equation are studied by using the Darboux transformation. Firstly, the Darboux transformation of the generalized derivative nonlinear Schr $\rm\ddot{o}$ dinger equation is constructed. Then, by using the Darboux transformation, the breather and rogue wave on the periodic/double periodic background and the collision solutions are derived. Finally, by means of the figures, the structures of interesting new solutions are analyzed in detail, which also provide a theoretical basis for studying the physical mechanism of the new solution.

Key words: The generalized derivative nonlinear Schr?dinger equation, Darboux transformation, Periodic solution, Breather, Rogue wave

中图分类号:

0175.24

娄瑜, 张翼. 推广的导数非线性薛定谔方程的单/双周期背景上的呼吸子和怪波及其碰撞解[J]. 数学物理学报, 2024, 44(6): 1511-1519.

Lou Yu, Zhang Yi. Breather and Rogue Wave on the Periodic/Double Periodic Background and Interaction Solutions of the Generalized Derivative Nonlinear Schr $\rm\ddot{o}$ dinger Equation[J]. Acta mathematica scientia,Series A, 2024, 44(6): 1511-1519.

图/表 7

图1

周期背景上的怪波解, 参数为 $a=1,c=1,\alpha=0,\beta=1,\alpha_3=0,$ (a) $\beta_3=\frac{1}{10};$ (b) $\beta_3=\frac{3}{10};$ (c) $\beta_3=\frac{3}{5}.$ "

图1

图2

周期背景上的呼吸子解, 参数为 $a=1,c=1,\alpha=0,\beta=1,$ (a) $\alpha_3=\frac{3}{10},\beta_3=\frac{3}{10};$ (b) $\alpha_3=\frac{11}{20},\beta_3=\frac{11}{20};$ (c) $\alpha_3=\frac{11}{20},\beta_3=\frac{9}{20}.$ "

图2

图3

周期背景上的呼吸子解, 参数为 $a=1,\alpha=0,\beta=1,\alpha_3=0,\beta_3=\frac{1}{10},$ (a) $c=\frac{1}{10};$ (b) $c=\frac{2}{5};$ (c) $c=\frac{4}{5}.$ "

图3

图4

双周期背景上的怪波解, 参数为 $a=1,c=1,\alpha=0,\beta=1,\alpha_3=0,\alpha_4=0,\beta_4=-\frac{3}{5},$ (a) $\beta_3=\frac{1}{10};$ (b) $\beta_3=\frac{1}{2};$ (c) $\beta_3=\frac{1}{5}.$ "

图4

图5

双周期背景上的怪波解, 参数为 $a=1,\alpha=0,\beta=1,\alpha_3=0,\alpha_4=0,\beta_3=\frac{3}{5},\beta_4=-\frac{1}{2},$ (a) $c=\frac{1}{5};$ (b) $c=\frac{1}{2};$ (c) $c=\frac{4}{5}.$ "

图5

图6

呼吸子和怪波的碰撞解, 参数为 $\alpha=0,\beta=1,$ (a) $a=1,c=1,\alpha_3=\frac{3}{5},\beta_3=-\frac{3}{5};$ (b) $a=\frac{9}{4},c=\frac{3}{2},\alpha_3=\frac{1}{2},\beta_3=-\frac{1}{2};$ (c) $a=\frac{8}{5},c=\frac{8}{5},\alpha_3=\frac{1}{2},\beta_3=-\frac{3}{5}.$ "

图6

图7

呼吸子和怪波的碰撞解, 参数为 $a=1,c=1,\alpha=0,\beta=1,\alpha_3=\frac{3}{5},$ (a) $\beta_3=0;$ (b) $\beta_3=-\frac{1}{10};$ (c) $\beta_3=-\frac{1}{5}.$ "

图7

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