Soliton molecules and dynamics of the smooth positon for the Gerdjikov–Ivanov equation

(a) Soliton molecule consisting of two solitons with parameter selections ${\lambda }_1={\displaystyle \frac{2}{3}+{\displaystyle \frac{\sqrt{7}\mathrm{i}}{6}}}$ , ${\lambda }_3={\displaystyle \frac{3}{4}+{\displaystyle \frac{\sqrt{5}\mathrm{i}}{4}}}$ , ξ = 40. (b) Soliton molecule consisting of three solitons with parameter selections ${\lambda }_1={\displaystyle \frac{2}{3}+{\displaystyle \frac{\sqrt{7}\mathrm{i}}{6}}}$ , ${\lambda }_3=1+{\displaystyle \frac{\sqrt{3}\mathrm{i}}{2}}$ , ${\lambda }_5={\displaystyle \frac{3}{4}+{\displaystyle \frac{\sqrt{5}\mathrm{i}}{4}}}$ , ξ = 40. (c) Soliton molecule consisting of four solitons with parameter selections ${\lambda }_1={\displaystyle \frac{2}{3}+{\displaystyle \frac{\sqrt{7}\mathrm{i}}{6}}}$ , ${\lambda }_3=1+{\displaystyle \frac{\sqrt{3}\mathrm{i}}{2}}$ , ${\lambda }_5={\displaystyle \frac{3}{4}+{\displaystyle \frac{\sqrt{5}\mathrm{i}}{4}}}$ , ${\lambda }_7={\displaystyle \frac{4}{3}+{\displaystyle \frac{\sqrt{55}\mathrm{i}}{6}}}$ , ξ = 10.

The evolution of a two-positon |q_{2 − 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 $H\pm {\displaystyle \frac{\mathrm{l}\mathrm{n}(4096{\alpha }_1^4{\beta }_1^4{t}^2)}{8{\alpha }_1{\beta }_1}=0}$ , which compared with density plot are shown consistence; (c) 2D plot of two-positon solution |q_{2 − p}| at t = −100, t = 0, t = 100.

The evolution of a three-positon |q_{3 − 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 $H\pm {\displaystyle \frac{\mathrm{l}\mathrm{n}(4194304{\alpha }_1^8{\beta }_1^8{t}^4)}{8{\alpha }_1{\beta }_1}}$ 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 |q_{3 − p}| at t = −20, t = 0, t = 20.

The evolution of a four-positon |q_{4 − p}| with α₁ = 1 / 2, β₁ = 1 / 2 of the GI equation on (x, t)-plane: (a) the 3D plot, (b) the density plot.

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.

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.

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.