New types of solitons and multiwave solutions for two higher-dimensional nonlinear evolution equations with time-dependent coefficients

doi:10.3969/j.issn.1000-5641.2023.04.001

Abstract

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

CLC Number:

O175.4

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.

Figures/Tables 6

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Table 1

Parameters of Figure 5"

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 $

Table 1

References 28

1	MA W X, ZHANG Y, TANG Y N. Symbolic computation of lump solutions to a combined equation involving three types of nonlinear terms. East Asian Journal on Applied Mathematics, 2020, 10 (4): 732- 745.
2	LYU X, CHEN S J. Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: One-lump-multi-stripe and one-lump-multi-soliton types. Nonlinear Dynamics, 2021, 103, 947- 977.
3	GUO H D, XIA T C, HU B B. Dynamics of abundant solutions to the (3 + 1)-dimensional generalized Yu-Toda-Sasa-Fukuyama equation. Applied Mathematics Letters, 2020, 105, 106301.
4	MOON B. Nonexistence of the periodic peaked traveling wave solutions for rotation-Camassa-Holm equation. Nonlinear Analysis: Real World Applications, 2021, 59, 103244.
5	ZHA Q L. Dynamics of localized wave solutions for the coupled Higgs field equation. Nonlinear Dynamics, 2020, 101, 1181- 1198.
6	KANG Z Z, XIA T C. Construction of abundant solutions of the (2 + 1)-dimensional time-dependent Date-Jimbo-Kashiwara-Miwa equation. Applied Mathematics Letters, 2020, 103, 106163.
7	WANG W, YAO R X, LOU S Y. Abundant traveling wave structures of (1 + 1)-dimensional Sawada-Kotera equation: Few cycle solitons and soliton molecules. Chinese Physics Letters, 2020, 37 (10): 100501.
8	QIN Y X, LIU Y P. Multiwave interaction solutions for a (3 + 1)-dimensional generalized BKP equation. International Journal of Computer Mathematics, 2021, 98 (11): 2268- 2281.
9	DINVAY E, NILSSON D. Solitary wave solutions of a Whitham-Boussinesq system. Nonlinear Analysis: Real World Applications, 2021, 60, 103280.
10	RIZVI S T R, SEADAWY A R, ASHRAF F, et al. Lump and interaction solutions of a geophysical Korteweg-de Vries equation. Results in Physics, 2020, 19, 103661.
11	ULLAH M S, ROSHID H O, MA W X, et al. Interaction phenomena among lump, periodic and kink wave solutions to a (3 + 1)-dimensional Sharma-Tasso-Olver-like equation. Chinese Journal of Physics, 2020, 68, 699- 711.
12	ZHA Q L. A pair of modified short pulse equations and its two-component system in nonlinear media. Wave Motion, 2020, 96, 102553.
13	CUI W Y, LI W, LIU Y P. Multiwave interaction solutions for a (3 + 1)-dimensional nonlinear evolution equation. Nonlinear Dynamics, 2020, 101 (2): 1119- 1129.
14	QIN Y X, LIU Y P. Multiwave interaction solutions for a (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation. Chinese Journal of Physics, 2021, 71, 561- 573.
15	GUO Y R, CHEN A H. Hybrid exact solutions of the (3 + 1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles. Results in Physics, 2021, 23, 103926.
16	WANG Y H. Nonautonomous lump solutions for a variable-coefficient Kadomtsev-Petviashvili equation. Applied Mathematics Letters, 2021, 119, 107201.
17	WAZWAZ A M. Two new Painlevé-integrable (2 + 1) and (3 + 1)-dimensional KdV equations with constant and time-dependent coefficients. Nuclear Physics B, 2020, 954, 115009.
18	ALI M R, SADAT R. Lie symmetry analysis, new group invariant for the (3 + 1)-dimensional and variable coefficients for liquids with gas bubbles models. Chinese Journal of Physics, 2021, 71, 539- 547.
19	DARVISHI M T, NAJAFI M, KAVITHA L, et al. Stair and step soliton solutions of the integrable (2 + 1) and (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equations. Communications in Theoretical Physics, 2012, 58 (6): 785- 794.
20	MABROUK S M, RASHED A S. Analysis of (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation via Lax pair investigation and group transformation method. Computers and Mathematics with Applications, 2017, 74 (10): 2546- 2556.
21	LIU J G, DU J Q, ZENG Z F, et al. New three-wave solutions for the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Nonlinear Dynamics, 2017, 88 (1): 655- 661.
22	PENG W Q, TIAN S F, ZHANG T T. Breather waves and rational solutions in the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Computers and Mathematics with Applications, 2019, 77 (3): 715- 723.
23	XU G Q. Painlevé analysis, lump-kink solutions and localized excitation solutions for the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Applied Mathematics Letters, 2019, 97, 81- 87.
24	LIU J G, TIAN Y, HU J G. New non-traveling wave solutions for the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Applied Mathematics Letters, 2018, 79, 162- 168.
25	WAZWAZ A M. Painlevé analysis for Boiti-Leon-Manna-Pempinelli equation of higher dimensions with time-dependent coefficients: Multiple soliton solutions. Physics Letters A, 2020, 384 (16): 126310.
26	LI W, ZHANG Y, LIU Y P. Exact wave solutions for a (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. Computers and Mathematics with Applications, 2019, 77 (12): 3087- 3101.
27	WANG Y L, ZHOU Z X, JIANG X Q, et al. Cylindrical Kadomtsev-Petviashvili equation for relativistically magnetosonic solitary wave in the collisionless plasma. Physics Letters A, 2006, 355 (4/5): 386- 389.
28	XIA J W, ZHAO Y W, LYU X. Predictability, fast calculation and simulation for the interaction solutions to the cylindrical Kadomtsev-Petviashvili equation. Communications in Nonlinear Science and Numerical Simulation, 2020, 90, 105260.

[1]	Baiping OUYANG. Blow-up of solutions to a class of weakly coupled semilinear double-wave systems with nonlinear terms of derivative type [J]. Journal of East China Normal University(Natural Science), 2023, 2023(4): 24-34.
[2]	CAI Jiang-tao. H-oscillation of a class of second-order vector neutral partial differential equations with damped terms [J]. Journal of East China Normal University(Natural Sc, 2011, 2011(5): 73-78.
[3]	ZHU Zhen-bo;LIN Wu-zhong;. Spike-like solutions of nonlinear singularly perturbed equations (Chinese) [J]. Journal of East China Normal University(Natural Sc, 2009, 2009(1): 37-42.
[4]	HE Mei;. Invariant Solutions of Gauss Equation on Jet Bundle R³(Chinese) [J]. Journal of East China Normal University(Natural Sc, 2007, 2007(3): 16-22.