非近轴光学介质中非克尔效应驱动的相空间重组与行波涌现
Phase Space Reorganization and Traveling Wave Emergence Driven by Non-Kerr Effects in Nonparaxial Optical Media
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中文总结 AI 辅助
该研究分析非近轴光学介质中非克尔效应对相空间与行波的影响，通过行波变换将非线性亥姆霍兹方程约化为哈密顿系统，揭示自陡峭与自频移的不同作用，明确受迫下会涌现准周期与混沌响应。
中文摘要 AI 辅助
本文考虑包含自陡峭和自频移等非克尔非线性项的非线性亥姆霍兹方程，应用行波变换将扩展的非线性亥姆霍兹方程约化为哈密顿动力学系统，随后通过平衡点分类、相空间分析及精确波解构造对该约化哈密顿系统进行分析，并通过参数空间分析进一步建立约化动力学系数与原始物理参数的关系。研究表明，自陡峭直接改变约化动力学，而自频移通过实行波约化的相容性条件发挥作用，这些非克尔效应共同重塑相空间几何结构与行波结构，获得局域和周期行波，其存在由色散、非近轴性、克尔非线性及非克尔效应之间的平衡决定。此外，研究还考察了约化系统的周期受迫版本以探究从规则到非规则动力学的转变，观察到外部受迫可诱导复杂振荡行为，分岔分析、时间序列演化、相空间分析、最大李雅普诺夫指数及庞加莱截面均表明，在足够强的受迫下会出现准周期和混沌响应。所有解析分支均通过全方程残差评估得到验证，部分选定分支还通过直接数值传播及复高斯扰动下的鲁棒性测试进行检验，结果显示自陡峭直接重整化有效非线性动力学，而自频移限制了可容许的实包络行波流形。
英文摘要
In this article, the nonlinear Helmholtz equation with non-Kerr nonlinearity, such as self steepening and self frequency shift, is considered. A traveling wave transformation is applied, and the extended nonlinear Helmholtz equation is reduced to a Hamiltonian dynamical system. Then, the reduced Hamiltonian system is analyzed by classification of equilibrium points, phase space analysis, and the construction of exact wave solutions. The relationship between the reduced dynamical coefficients and the original physical parameters is further established through a parameter space analysis. It is shown that self steepening directly modifies the reduced dynamics, whereas self frequency shift acts through the compatibility condition for the real traveling wave reduction. Together, these non-Kerr effects reshape the phase space geometry and traveling wave structure. Localized and periodic traveling waves are obtained, with their existence determined by the balance among dispersion, nonparaxiality, Kerr nonlinearity, and non-Kerr effects. Furthermore, a periodically forced version of the reduced system is examined to study the transition from regular to irregular dynamics. It has been observed that external forcing can induce complex oscillatory behavior. Bifurcation analysis, time series evolution, phase space analysis, largest Lyapunov exponent, and Poincaré section demonstrate the emergence of quasiperiodic and chaotic responses under sufficiently strong forcing. All analytical branches are verified through full-equation residual evaluation, while a few selected branches are additionally examined through direct numerical propagation and robustness tests under complex Gaussian perturbations. The results show that self steepening directly renormalizes the effective nonlinear dynamics, whereas self frequency shift restricts the admissible real-envelope traveling wave manifold.
发表机构
- Dayananda Sagar University（戴扬丹达萨加尔大学）
- Kaliyaganj College（卡利亚甘杰学院）
- Raiganj University（赖根杰大学）
- Indian Institute of Technology Gandhinagar（甘地纳加尔印度理工学院）
- SRM Institute of Science & Technology（SRM科学与技术研究所）
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