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非线性波作为非线性动力学研究中的重要课题之一, 普遍存在于各种复杂物理系统中. 理解非线性波的产生机制、确定它们的激发条件对于非线性波的实验实现、动力学特征的探测和应用是至关重要的. 本文简要综述了近年来非线性波的实验和理论研究进展, 回顾了非线性波的产生机制. 基于非线性可积模型中的严格解和线性稳定分析结果, 系统讨论了如何建立关系. 详细介绍了近来发现的扰动能量和相对相位在确定非线性波激发条件中的重要作用, 并提议了一组能够确定非线性波激发条件的完备参数. 基于完备的激发参数, 给出了多种基本非线性波的激发条件和相图. 这些结果有望用于实现多种局域波的可控激发, 并可以推广到更多非线性系统中的激发相图研究.Nonlinear waves are ubiquitous in various physical systems, and they have become one of the research hotspots in nonlinear physics. For the experimental realization, observation and application of nonlinear waves, it is very important to understand the generation mechanism, and determine the essential excitation conditions of various nonlinear waves. In this paper, we first briefly review the experimental and theoretical research progress of nonlinear waves in recent years. Based on the exact nonlinear wave solutions and linear stability analysis results, we systemically discuss how to establish the quantitative relations between fundamental nonlinear waves and modulation instability. These relations would deepen our understanding on the mechanism of nonlinear waves. To solve the excitation conditions degenerations problem for some nonlinear waves, we further introduce the perturbation energy and relative phase to determine the excitation conditions of nonlinear waves. Finally, we present a set of complete parameters that can determine the excitation conditions of nonlinear waves, and give the excitation conditions and phase diagrams of the fundamental nonlinear waves. These results can be used to realize controllable excitation of nonlinear waves, and could be extended to many other nonlinear systems.
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激发条件 非线性波类型 $\varOmega$ $\omega$ $\varepsilon$ $\varphi$ 0 $\omega^{2}-\alpha\neq0 $ 0 $\varphi\in \left(\dfrac{{\text{π}}}{2}, \dfrac{3{\text{π}}}{2}\right)+2 n{\text{π}}$ 怪波 $\omega^{2}-\alpha=0$, $\alpha\geqslant 0$ 有理W形孤子 0 $\omega^{2}-\dfrac{\varepsilon^{2}}{24}-\alpha\neq0$, $\varepsilon>0$ $\varphi\in\mathbb{R}$ Kuznetsov-Ma呼吸子 $\omega^{2}-\dfrac{\varepsilon^{2}}{24}-\alpha=0$, $\varepsilon>0$ $\varphi\in\left(\dfrac{{\text{π}}}{2},\right. \left.\dfrac{3{\text{π}}}{2}\right]+2 n{\text{π}}$ 非有理W形孤子 $\omega^{2}-\dfrac{\varepsilon^{2}}{24}-\alpha=0$, $\varepsilon > 0$ $\varphi\in \left(-\dfrac{{\text{π}}}{2},\right. \left.\dfrac{{\text{π}}}{2}\right]+2 n{\text{π}}$ 反暗孤子 $\omega^{2}+\dfrac{\varOmega^{2} }{6}-\alpha\neq0, \varOmega\in(0, 2)$ 0 $\varphi\in \left(\dfrac{{\text{π}}}{2},\dfrac{3{\text{π}}}{2}\right)+2 n{\text{π}}$ Akhmediev呼吸子 $\omega^{2}+\dfrac{\varOmega^{2} }{6}-\alpha=0$ $0<|\varOmega|<\dfrac{\sqrt{3}}{|\sec\varphi|}$ W形孤子链 $\omega^{2}+\dfrac{\varOmega^{2} }{6}-\alpha=0$ $\dfrac{\sqrt{3}}{|\sec\varphi|}<|\varOmega|<\dfrac{2}{|\sec\varphi|}$ 周期波 $1+2\beta\left(\pm\sqrt{\varDelta}-3\omega\right)+2\gamma\nabla\neq0$ $\varphi\in \mathbb{\rm R}$ Tajiri-Watanabe呼吸子 $1+2\beta\left(\pm\sqrt{\varDelta}-3\omega\right)+2\gamma\nabla=0$ 多峰孤子 注1: $\omega$, $\varOmega$, $\varepsilon$和$\varphi$分别为背景频率、扰动频率、扰动能量和相对相位. 参数$\alpha=\dfrac{\beta^{2}}{16\gamma^{2}}+\dfrac{1}{12\gamma}+a^{2}$, $\varDelta = {\bigg[ {\dfrac{ {\sqrt { { {({\varepsilon ^2} - 4{\varOmega ^2} + 16{a^2})}^2} + 16{\varepsilon ^2}{\varOmega ^2} } - ({\varepsilon ^2} - 4{\varOmega ^2} + 16{a^2})} }{8} } \bigg]^{1/2} }$, $\nabla=-2\varDelta\pm8\omega\sqrt{\varDelta}-6\omega^{2}+6 a^{2}+\dfrac{1}{4}\varepsilon^{2}-\varOmega^{2}$. 
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