Quantitative relations between fundamental nonlinear waves and modulation instability

Duan Liang; Liu Chong; Zhao Li-Chen; Yang Zhan-Ying

doi:10.7498/aps.69.20191385

Abstract

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.

Keywords:

Authors and contacts

1.
School of Physics, Northwest University, Xi’an 710127, China
2.
Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China

Corresponding author: Yang Zhan-Ying, zyyang@nwu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11875220, 11775176)

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图 1 自散焦的两组分耦合非线性薛定谔系统的调制不稳定增益的分布　(a)调制不稳定增益在$(\varOmega, \omega)$平面的分布, 绿色点状曲线表示调制不稳定区的边界; (b)调制不稳定性在$(\varOmega, a_{1})$平面的分布

Figure 1. Modulation instability distributions of the defocusing two component coupled nonlinear Schrödinger system: (a) Modulation instability distribution in the $(\varOmega, \omega)$ plane, green dot curves are the boundary of the modulation instability regime; (b) modulation instability distribution in the $(\varOmega, a_{1})$ plane.

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图 2 标准非线性薛定谔系统的调制不稳定增益分布和基本非线性波激发的相图　(a1)和(b1)分别为调制不稳定增益在$(\omega, \varOmega)$平面和$(a, \varOmega)$平面的分布. “MI”和“MS”分别表示调制不稳定性和调制稳定性, 红色虚线是共振线; (a2)和(b2)分别为基本非线性波在(a1)和(b1)中调制不稳定增益分布平面的相图. “AB”,“RW”和“KM”分别为Akhmediev呼吸子、怪波和Kuznetsov-Ma呼吸子

Figure 2. Modulation instability distributions and phase diagrams of fundamental nonlinear waves in standard nonlinear Schrödinger system: (a1) and (b1) are the distributions of the modulation instability gain in the $(\omega, \varOmega)$ plane and the $(a, \varOmega)$, respectively. “MI” and “MS” denote modulation instability and modulation stability, respectively. the red dotted line is the resonance line; (a2) and (b2) are the phase diagrams of fundamental nonlinear waves on the modulation instability gain distribution planes correspond to (a1) and (b1), respectively. "AB", "RW" and "KM" denote Akhmediev breather, rogue wave and Kuznetsov-Ma breather, respectively.

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图 3 Sasa-Satsuma系统的调制不稳定增益分布和基本非线性波激发的相图　(a) Sasa-Satsuma系统中调制不稳定增益在背景频率$\omega$和扰动频率$\varOmega$平面的分布. “MI”和“MS”分别表示调制不稳定和调制稳定, 黄颜色圆点为共振线上临界点; (b)非线性波在调制不稳定增益分布平面的相图. “AB”, “RW” 和“KM” 分别为Akhmediev呼吸子、怪波和Kuznetsov-Ma呼吸子; “WS”, “WST”, “AD”和Periodic wave分别表示W形孤子、W形孤子链、反暗孤子和周期波

Figure 3. Modulation instability distributions and phase diagrams of fundamental nonlinear waves in Sasa-Satsuma system: (a) Distributions of the modulation instability gain in the background frequency $\omega$ and perturbation frequency$\varOmega$ plane. “MI” and “MS” denote modulation instability and modulation stability, respectively. The yellow dots are the critical points on the resonance line; (b) phase diagrams of nonlinear waves in the modulation instability gain distribution planes. “AB”, “RW” and “KM” denote Akhmediev breather, rogue wave and Kuznetsov-Ma breather, respectively; “WS”, “WST” and “AD” denote the W-shaped soliton, W-shaped soliton train and anti-dark soliton, respectively.

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图 4 Hirota系统中的调制不稳定增益分布和基本非线性波激发的相图　(a) Hirota系统中调制不稳定增益在背景频率$\omega$和扰动频率$\varOmega$平面的分布. “MI”和“MS”分别表示调制不稳定和调制稳定; (b)非线性波在调制不稳定增益分布平面的相图. “AB”, “RW” 和“KM” 分别为Akhmediev呼吸子、怪波和Kuznetsov-Ma呼吸子; “WS”, “AD”, “PW”和“MPS”分别表示W形孤子、反暗孤子、周期波和多峰孤子

Figure 4. Modulation instability distributions and phase diagrams of fundamental nonlinear waves in Hirota system; (a) Distributions of the modulation instability gain in the background frequency $\omega$ and perturbation frequency$\varOmega$ plane. “MI” and “MS” denote modulation instability and modulation stability, respectively; (b) phase diagrams of nonlinear waves in the modulation instability gain distribution planes. “AB”, “RW” and “KM” denote Akhmediev breather, rogue wave and Kuznetsov-Ma breather, respectively; “WS”, “AD”, “PW” and “MPS” denote the W-shaped soliton, anti-dark soliton, periodic wave and multi-peak soliton, respectively.

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图 5 四阶非线性薛定谔系统调制不稳定增益分布和基本非线性波激发的相图　(a) 调制不稳定增益在背景频率$\omega$和扰动频率$\varOmega$平面的分布, “MI”和“MS” 分别表示调制不稳定性和调制稳定性; (b),(c) 基本非线性波在背景频率$\omega$和扰动频率$\varOmega$平面的相图, “AB”, “RW”, “KM”、“PW”, “WST”, “WS$_{\rm r}$”, “$\rm WS_{ {nr}}$” 和“AD”分别为Akhmediev呼吸子、怪波、Kuznetsov-Ma呼吸子、周期波、W形孤子链、有理的W形孤子、非有理的W形孤子和反暗孤子

Figure 5. Modulation instability distributions and phase diagrams of fundamental nonlinear waves in fourth-order nonlinear Schrödinger system: (a) Distributions of the modulation instability gain in the background frequency $\omega$ and perturbation frequency$\varOmega$ plane. “MI” and “MS” denote modulation instability and modulation stability, respectively; (b), (c) phase diagrams of nonlinear waves in the background frequency $\omega$ and perturbation frequency $\varOmega$ plane. “AB”, “RW”, “KM”, “PW”, “WST”, “WS$_{\rm r}$”, “WS$\rm WS_{{nr}}$” and “AD” denote Akhmediev breather, rogue wave, Kuznetsov-Ma breather, periodic wave, W-shaped soliton train, rational W-shaped soliton, nonrational W-shaped soliton and anti-dark soliton, respectively.

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图 6 不同系统中平面波背景上基本非线性波在背景频率$\omega$, 扰动频率$\varOmega$, 扰动能量$\varepsilon$和相对相位$\varphi$空间的相图　(a) 四阶非线性薛定谔系统, 参数取$\beta={1}/{12}$, $\gamma=-{1}/{36}$, $a=1$; (b) Hirota系统, 参数取$\beta={1}/{12}$, $\gamma=0$, $a=1$; (c)非线性薛定谔系统, 参数取$\beta={1}/{12}$, $\gamma=0$, $a=1$; (d)反暗孤子和非有理W形孤子依赖于相对相位的相图; (e)周期波, W形孤子链和有理W形孤子在$(\varphi, \varOmega)$平面的相图. 图中“TW”, “KM”, “AB”, “RW”, “MPS”, “AD”, “WS$_{\rm {nr}}$”, “PW”, “WST”和“WS$_{\rm r}$”分别表示Tajiri-Watanabe呼吸子、Kuznetsov-Ma呼吸子、Akhmediev呼吸子、怪波、多峰孤子、反暗孤子、非有理W形孤子、周期波、W形孤子链和有理W形孤子

Figure 6. Phase diagrams of nonlinear waves in the background frequency $\omega$, perturbation frequency $\varOmega$, perturbation energy $\varepsilon$ and relative phase $\varphi$ space for different systems: (a) Fourth-order nonlinear Schrödinger system. Parameters are $\beta={1}/{12}$, $\gamma=-{1}/{36}$, $a=1$; (b) hirota system. Parameters are $\beta={1}/{12}$, $\gamma=0$, $a=1$; (c) nonlinear Schrödinger system. Parameters are $\beta=\gamma=0$, $a=1$; (d) phase diagram of anti-dark soliton and nonrational W-shaped soliton in relative phase space; (e) phase diagram of periodic wave, W-shaped soliton train and rational W-shaped soliton in the $(\varphi, \varOmega)$ plane. “TW”, “KM”, “AB”, “RW”, “MPS”, “AD”, “WS$_{\rm {nr}}$”, “PW”, “WST” and “WS$_{\rm r}$” denote Tajiri-Watanabe breather, Kuznetsov-Ma breather, Akhmediev breather, rogue wave, multi-peak soliton, anti-dark soliton, nonrational W-shaped soliton, periodic wave, W-shaped soliton train and rational W-shaped soliton.

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图 7 不同非线性波的转换关系　(a) 呼吸子和怪波之间的转换关系； (b) 孤子和周期波之间的转换关系. 图中“TW”, “KM”, “AB”, “RW”分别为Tajiri-Watanabe呼吸子、Kuznetsov-Ma呼吸子、Akhmediev呼吸子和怪波, “MPS”, “AD”, “WS$_{\rm {nr}}$”, “PW”, “WST”和“WS$_{\rm r}$” 分别表示多峰孤子、反暗孤子、非有理W形孤子、周期波、W形孤子链和有理W形孤子

Figure 7. Conversion relationship of different nonlinear waves: (a) Conversion relationship between breathers and rogue wave; (b) conversion relationship between the solitons and periodic waves. “TW”, “KM”, “AB”, “RW”, “MPS”, “AD”, “WS$_{\rm {nr}}$”, “PW”, “WST” and “WS$_{\rm r}$” denote Tajiri-Watanabe breather, Kuznetsov-Ma breather, Akhmediev breather, rogue wave, multi-peak soliton, anti-dark soliton, nonrational W-shaped soliton, periodic wave, W-shaped soliton train and rational W-shaped soliton.

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表 1 基本非线性波的激发条件

Table 1. Excitation conditions of fundamental nonlinear waves.

激发条件				非线性波类型
$\varOmega$	$\omega$	$\varepsilon$	$\varphi$	非线性波类型
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
0	$\omega^{2}-\alpha=0$, $\alpha\geqslant 0$	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
$1+2\beta\left(\pm\sqrt{\varDelta}-3\omega\right)+2\gamma\nabla=0$			$\varphi\in \mathbb{\rm R}$	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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Vol. 69, Issue 1 - 2020-01-05

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