Chin. Phys. B, 2020, Vol. 29(10): 100501    DOI: 10.1088/1674-1056/ab9de0
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# Soliton molecules and dynamics of the smooth positon for the Gerdjikov–Ivanov equation

Xiangyu Yang(杨翔宇), Zhao Zhang(张钊), and Biao Li(李彪)†
1 School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
Abstract

Soliton molecules are firstly obtained by velocity resonance for the Gerdjikov–Ivanov equation, and n-order smooth positon solutions for the Gerdjikov–Ivanov equation are generated by means of the general determinant expression of n-soliton solution. The dynamics of the smooth positons of the Gerdjikov–Ivanov equation are discussed using the decomposition of the modulus square, the trajectories and time-dependent “phase shifts” of positons after the collision can be described approximately. Additionally, some novel hybrid solutions consisting solitons and positons are presented and their rather complicated dynamics are revealed.

Keywords:  soliton molecules      degenerate Darboux transformation      positons      phase shift      Gerdjikov-Ivanov equation
Received:  13 February 2020      Revised:  05 June 2020      Accepted manuscript online:  18 June 2020
 PACS: 05.45.Yv (Solitons) 02.30.Ik (Integrable systems)
Corresponding Authors:  Corresponding author. E-mail: libiao@nbu.edu.cn
 Fig. 1.  (a) Soliton molecule consisting of two solitons with parameter selections ${\lambda }_{1}=\displaystyle \frac{2}{3}+\displaystyle \frac{\sqrt{7}{\rm{i}}}{6}$ , ${\lambda }_{3}=\displaystyle \frac{3}{4}+\displaystyle \frac{\sqrt{5}{\rm{i}}}{4}$ , ξ = 40. (b) Soliton molecule consisting of three solitons with parameter selections ${\lambda }_{1}=\displaystyle \frac{2}{3}+\displaystyle \frac{\sqrt{7}{\rm{i}}}{6}$ , ${\lambda }_{3}=1+\displaystyle \frac{\sqrt{3}{\rm{i}}}{2}$ , ${\lambda }_{5}=\displaystyle \frac{3}{4}+\displaystyle \frac{\sqrt{5}{\rm{i}}}{4}$ , ξ = 40. (c) Soliton molecule consisting of four solitons with parameter selections ${\lambda }_{1}=\displaystyle \frac{2}{3}+\displaystyle \frac{\sqrt{7}{\rm{i}}}{6}$ , ${\lambda }_{3}=1+\displaystyle \frac{\sqrt{3}{\rm{i}}}{2}$ , ${\lambda }_{5}=\displaystyle \frac{3}{4}+\displaystyle \frac{\sqrt{5}{\rm{i}}}{4}$ , ${\lambda }_{7}=\displaystyle \frac{4}{3}+\displaystyle \frac{\sqrt{55}{\rm{i}}}{6}$ , ξ = 10. Fig. 2.  The evolution of a two-positon |q2 − p| with α1 = 1 / 3, β1 = 2 / 5 of the GI equation: (a) 3D plot, (b) density plot, where two red curves are approximate trajectories defined by $H\pm \displaystyle \frac{\mathrm{ln}(4096{\alpha }_{1}^{4}{\beta }_{1}^{4}{t}^{2})}{8{\alpha }_{1}{\beta }_{1}}=0$ , which compared with density plot are shown consistence; (c) 2D plot of two-positon solution |q2 − p| at t = −100, t = 0, t = 100. Fig. 3.  The evolution of a three-positon |q3 − p| with α1 = 1 / 3, β1 = 2 / 5 of the GI equation: (a) 3D plot, (b) density plot, where two red curves are approximate trajectories defined by $H\pm \displaystyle \frac{\mathrm{ln}(4194304{\alpha }_{1}^{8}{\beta }_{1}^{8}{t}^{4})}{8{\alpha }_{1}{\beta }_{1}}$ and the middle white curve is trajectory without phase shift, which compared with density plot are shown consistence; (c) 2D plot of three-positon solution |q3 − p| at t = −20, t = 0, t = 20. Fig. 4.  The evolution of a four-positon |q4 − p| with α1 = 1 / 2, β1 = 1 / 2 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot. Fig. 5.  The evolution of hybrid solution consisting of a soliton and two-positon with α1 = 2 / 5, α1 = 1 / 5, α3 = 1 / 5, β3 = 2 / 5 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot. Fig. 6.  The evolution of hybrid solution consisting of a soliton and three-positon with α1 = 2 / 5, β1 = 1 / 5, α3 = 1 / 5, β3 = 2 / 5 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot. Fig. 7.  The evolution of hybrid solution consisting of two solitons and two-positon with α1 = 1 / 2, β1 = 1 / 2, α3 = 1 / 2, β3 = 1 / 3, α5 = 1 / 5, β5 = 2 / 5 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot.