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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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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166]
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