Soliton solutions to the coupled Sasa-Satsuma equation under mixed boundary conditions

Abstract

In this paper, we derive general bright-dark soliton solutions to the coupled Sasa-Satsuma (CSS) equation using the Kadomtsev-Petviashvili (KP) reduction method. Since the CSS equation is a special case of the four-component Hirota equation, our approach begins with the construction of two-bright-two-dark soliton solutions for the four-component Hirota equation. By imposing specific parameter constraints, these solutions are subsequently reduced to the bright-dark soliton solutions of the CSS equation. Finally, the dynamical behaviors of the one- and two-bright-dark soliton solutions are thoroughly analyzed and illustrated.

Figures (8)

• Figure 1: One bright-dark soliton solution to the coupled Sasa-Satsuma equation under parameters $N=1, \alpha=2, \rho=1, \epsilon_1=-1, \epsilon_2=1, p_1=1, C_1=1,\xi_{1,0}=0$. (b) and (d) are the corresponding density plots of (a) and (c), respectively
• Figure 2: One bright-dark soliton solution to the coupled Sasa-Satsuma equation under parameters $N=1, \alpha=1/2, \rho=1, p_1=1, C_1=1,\xi_{1,0}=0$ with (a) and (b): $\epsilon_1=\epsilon_2=1$; (c) and (d): $\epsilon_1= \epsilon_2=-1$.
• Figure 3: One bright-dark soliton solution to the coupled Sasa-Satsuma equation under parameters $N=2, \alpha=-2, \rho=1, \epsilon_1=-1, \epsilon_2=1, p_1=2+0.2\mathrm{i}$ with the first row $C_1=1,C_2=0,\xi_{1,0}=\xi_{2,0}=0$ and the second row $C_1=C_2=1$. The second column and the forth column show the density plots of the first and third columns, respectively.
• Figure 4: One bright-dark solutions to the coupled Sasa-Satsuma equation under parameters $N = 2$, $\alpha = 2$, $\rho = 1$, $\epsilon_1 = -1$, $\epsilon_2 = 1$, $p_1 = 1 + \mathrm{i},\xi_{1,0}=\xi_{2,0}=0$ and $C_1 = 2$ with the first row $C_2=0$ and the second row $C_2=1$. (b) and (d) are the corresponding density plots of (a) and (c), respectively. (f) and (h) are the corresponding density plots of (e) and (g), respectively.
• Figure 5: Two bright-dark soliton solutions to the coupled Sasa-Satsuma equation under parameters $N=3, \alpha=1, \rho=1, \epsilon_1=-1, \epsilon_2=1, p_1=2+\mathrm{i}, p_2=2$, $C_1=C_2=1,\xi_{1,0}=\xi_{2,0}=\xi_{3,0}=0$ with (a)-(b) $C_3=0$, and (c)-(d) $C_3=1$.

Theorems & Definitions (11)

• Lemma 1
• Theorem 1
• Theorem 2
• Remark 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5