两个含时变系数的高维非线性演化方程新型孤子和多波解

doi:10.3969/j.issn.1000-5641.2023.04.001

摘要/Abstract

摘要：

在构造非线性演化方程的精确解时, 通常采用的行波变换都是线性变换. 通过引入特定形式的非线性行波变换, 首次将 $N$ -孤子分解算法及继承求解策略推广应用于变系数非线性演化方程, 求解了两个含有时变系数的高维非线性演化方程: Boiti-Leon-Manna-Pempinelli (BLMP)方程和圆柱Kadomtsev-Petviashvili (cylindrical Kadomtsev-Petviashvili, cKP)方程. 应用直接代数方法和继承求解策略, 构造了BLMP方程的多种不同类型的多波相互作用解, 尤其是马蹄形孤子及它与lump波、不同周期波之间的相互作用解. 利用 $N$ -孤子分解算法构造了cKP方程的马蹄形孤子、呼吸子和lump波解之间的高阶相互作用解. 这些新型多波相互作用解在一定程度上丰富了变系数非线性演化方程的解的类型.

关键词: Boiti-Leon-Manna-Pempinelli方程, 圆柱Kadomtsev-Petviashvili方程, 多波相互作用解, 马蹄形孤子, 时变系数

Abstract:

Linear traveling-wave transformations are usually applied when constructing exact traveling-wave solutions for nonlinear evolution equations. Herein, for the first time, specific nonlinear traveling-wave transformations are introduced to extend the $N$ -soliton decomposition algorithm and an inheritance-solving strategy to a variable-coefficient nonlinear evolution equation. Two higher-dimensional nonlinear evolution equations with time-dependent coefficients, the Boiti-Leon-Manna-Pempinelli (BLMP) equation and the cylindrical Kadomtsev-Petviashvili (cKP) equation, are solved. The direct algebraic method and inheritance-solving strategy are used to construct several different types of multiwave-interaction solutions for the BLMP equation, specifically, the horseshoe-like solitons and their interaction with lump as well as different periodic waves. Using the $N$ -soliton decomposition algorithm, the higher-order interaction solutions between the horseshoe-like solitons, breathers, and lump waves of the cKP equation are established. These new multiwave-interaction solutions contribute to the existing solutions of nonlinear evolution equations with variable coefficients, enriching the repository of solutions to a certain extent.

Key words: Boiti-Leon-Manna-Pempinelli equation, cylindrical Kadomtsev-Petviashvili equation, multiwave-interaction solution, horseshoe-like soliton, time-dependent coefficient

中图分类号:

O175.4

秦宇昕, 柳银萍, 徐桂琼. 两个含时变系数的高维非线性演化方程新型孤子和多波解[J]. 华东师范大学学报（自然科学版）, 2023, 2023(4): 1-10.

Yuxin QIN, Yinping LIU, Guiqiong XU. New types of solitons and multiwave solutions for two higher-dimensional nonlinear evolution equations with time-dependent coefficients[J]. Journal of East China Normal University(Natural Science), 2023, 2023(4): 1-10.

图/表 6

图1

图2

图3

图4

图5

表1

图5的参数表"

解	参数值
0L-0B-7S	$ {k_1} = 1/10,{r_1} = 1/10,{c_1} = 0,{k_2} = 1/10,{r_2} = 3/10,{c_2} = 0,{k_3} = 1/10,{r_3} = 2/5, $ $ {c_3} = 0,{k_4} = 1/10,{r_4} = 1/5,{c_4} = 0,{k_5} = 1/10,{r_5} = 3/5,{c_5} = 0,{k_6} = 1/10, $ $ {r_6} = 1/2,{c_6} = 0,{k_7} = 1/10,{r_7} = 7/10,{c_7} = 0 $
3L-0B-1S	${k_{1R} } = 0,{k_{1I } } = 1/7,{r_{1I} } = 0,{c_{1R} } = 0,{c_{1I} } = 0,{k_{2R} } = 0,{k_{2I} } = 1/8,{r_{2I} } = 0,{c_{2R} } = 0,$ ${c_{2I} } = - 10,{k_{3R} } = 0,{k_{3I} } = 1/9,{r_{3I} } = 0,{c_{3R} } = - 10,{c_{3I} } = - 20,{\delta _7} = 5, {k_7} = 1/10,{r_7} = 1/10,{c_7} = 10$
0L-3B-1S	${k_{1R} } = 0,{k_{1I} } = 1/10,{r_{1R} } = 7/4,{r_{1I} } = 0,{c_{1R} } = 0,{c_{1I} } = 0,{k_{2R} } = 1,{k_{2I} } = 1/10,$ ${r_{2I} } = 0,{c_{2R} } = - 10,{c_{2I} } = - 10,{k_{3R} } = 0,{k_{3I} } = 1/10,{r_{3R} } = 11/8,{r_{3I} } = 0,$ ${c_{3R} } = - 5,{c_{3I} } = - 5,{\delta _7} = 8,{k_7} = 1/10,{r_7} = 1/10,{c_7} = 10,t = 1/10$
1L-1B-3S	${k_{1R} } = 0,{k_{1I} } = 1/7,{r_{1I} } = 0,{c_{1R} } = 5,{c_{1I} } = 10,{k_{2R} } = 0,{k_{2I} } = 1/10,{r_{2R} } = 3/2,$ ${r_{2I} } = 0,{c_{2R} } = 0,{c_{2I} } = 0,{\delta _5} = 10,{k_5} = 1/10,{r_5} = 1/10,{c_5} = 15,{\delta _6} = 9,$ $ {k_6} = 1/10,{r_6} = 3/10,{c_6} = 12,{\delta _7} = 9,{k_7} = 1/10,{r_7} = 1/2,{c_7} = 8,t = 1/10 $
1L-2B-1S	${k_{1R} } = 0,{k_{1I} } = 1/7,{r_{1I} } = 0,{c_{1R} } = 5,{c_{1I} } = 0,{k_{2R} } = 0,{k_{2I} } = 1/10,{r_{2R} } = 3/2,$ ${r_{2I } } = 0,{c_{2R} } = 0,{c_{2I } } = 0,{k_{3R} } = 0,{k_{3I } } = 1/10,{c_{3R} } = - 10,{r_{3I } } = 0,{c_{3I} } = - 10,$ $ {\delta _7} = 10,{k_7} = 1/10,{r_7} = 1/10,{c_7} = 10,t = 1/10 $
2L-1B-1S	${k_{1R} } = 0,{k_{1I} } = 1/7,{r_{1I} } = 0,{c_{1R} } = 10,{c_{1I} } = 22,{k_{2R} } = 0,{k_{2I} } = 1/8,{c_{2R} } = 10,$ ${r_{2I } } = 0,{c_{2I } } = 12,{k_{3R} } = 0,{k_{3I } } = 1/10,{r_{3R} } = 3/2,{r_{3I } } = 0,{c_{3R} } = 0,{c_{3I} } = 0,$ $ {\delta _7} = 10,{k_7} = 1/10,{r_7} = 1/10,{c_7} = 8,t = 1/10 $

表1

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