Line-solitons of a three-component KP equation

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发表日期：

2024年5月30日

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Zihan Zhao,
Lin Sun,
Chuanzhong Li,
Wei Liu

摘要

In this paper, our goal is to study the line-soliton solutions of the three-component Kadomtsev-Petrovashvili equation. Its solution also represents the interaction between the three fluid layers. Firstly, we obtain the analytical solutions of thex three-component KP equations, i.e., $u_0(x, y, t), u_1(x, y, t), u_2(x, y, t)$ $u_{0} (x, y, t), u_{1} (x, y, t), u_{2} (x, y, t)$ , based on the KP theory and the three-component treatment of the $\tau $ $τ$ -function. Secondly, we make the following conjectures about the solution structure of the line-soliton on the Grassmannian Gr(N, M). For $u_0$ $u_{0}$ , there are N bright soliton solutions when $y\gg 0$ $y ≫ 0$ ; there are $M-N$ $M - N$ bright soliton solutions when $y\ll 0$ $y ≪ 0$ . For $u_1$ $u_{1}$ , there are N bright soliton solutions and N dark soliton solutions when $y\gg 0$ $y ≫ 0$ ; there are $M-N$ $M - N$ bright soliton solutions and $M-N$ $M - N$ dark soliton solutions when $y\ll 0$ $y ≪ 0$ . For $u_2$ $u_{2}$ , there are 2N bright soliton solutions and N dark soliton solutions when $y\gg 0$ $y ≫ 0$ ; there are $2 (M-N)$ $2 (M - N)$ bright soliton solutions and $M- N$ $M - N$ dark soliton solutions when $y\ll 0$ $y ≪ 0$ . Again, we verified the accuracy of the conjecture by plotting the images of the soliton solutions on Gr(N, M). In addition, we analyse the rogue waves in the three-component KP equation and briefly explain why the maxima in the line soliton solutions arise.