Fig. 1. Modulation instability distributions of the defocusing two component coupled nonlinear Schrödinger system: (a) Modulation instability distribution in the plane, green dot curves are the boundary of the modulation instability regime; (b) modulation instability distribution in the plane. 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 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 平面的分布

Fig. 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 plane and the , 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. 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 平面和 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

Fig. 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 and perturbation frequency 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. 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 和扰动频率 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

Fig. 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 and perturbation frequency 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. 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 和扰动频率 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

Fig. 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 and perturbation frequency plane. “MI” and “MS” denote modulation instability and modulation stability, respectively; (b), (c) phase diagrams of nonlinear waves in the background frequency and perturbation frequency plane. “AB”, “RW”, “KM”, “PW”, “WST”, “WS ”, “WS ” 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. 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 和扰动频率 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 和扰动频率 平面的相图, “AB”, “RW”, “KM”、“PW”, “WST”, “WS ”, “ ” 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

Fig. 6. Phase diagrams of nonlinear waves in the background frequency , perturbation frequency , perturbation energy and relative phase space for different systems: (a) Fourth-order nonlinear Schrödinger system. Parameters are , , ; (b) hirota system. Parameters are , , ; (c) nonlinear Schrödinger system. Parameters are , ; (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 plane. “TW”, “KM”, “AB”, “RW”, “MPS”, “AD”, “WS ”, “PW”, “WST” and “WS ” 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. 不同系统中平面波背景上基本非线性波在背景频率 , 扰动频率 , 扰动能量 和相对相位 空间的相图　(a) 四阶非线性薛定谔系统, 参数取 , , ; (b) Hirota系统, 参数取 , , ; (c)非线性薛定谔系统, 参数取 , , ; (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 平面的相图. 图中“TW”, “KM”, “AB”, “RW”, “MPS”, “AD”, “WS ”, “PW”, “WST”和“WS ”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

Fig. 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 ”, “PW”, “WST” and “WS ” 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. 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 ”, “PW”, “WST”和“WS ” 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

Table 1. Excitation conditions of fundamental nonlinear waves.