Scholarly article on topic 'Analytical multi-soliton solutions of a (2+1)-dimensional breaking soliton equation'

Analytical multi-soliton solutions of a (2+1)-dimensional breaking soliton equation Academic research paper on "Nano-technology"

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Academic research paper on topic "Analytical multi-soliton solutions of a (2+1)-dimensional breaking soliton equation"

Wang SpringerPlus (2016)5:891 DOI 10.1186/S40064-016-2403-2

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Analytical multi-soliton solutions of a (2+1)-dimensional breaking soliton

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equation

Shao-fu Wang*

'Correspondence: nctsg09@163.com College of Electrical and Electronic Engineering, Department of Electronics, Anhui Science and Technology University, Bengbu, Fengyang 233100, Anhui, People's Republic of China

Springer Open

Abstract

The analytical solutions for a (2+1)-dimensional breaking solution equation is proposed in this paper by using mapping and projective method darboux transformation, and Some exact propagating solutions are constructed for this Breaking equation, and the M x N multi-soliton could be obtained by using Weierstrassp function and setting the perfect parameters. The potential application of breaking Soliton equation will be of great interest in future research.

Keywords: Breaking equation, Darboux transformation, Weierstrassp function, Multi-soliton

Background

In recent years, many research have studied soliton and its evolvements in nonlinear equations via kinds of method (Zarmi 2014; Chen and Ma 2013; Ashorman 2014; Zhang and Chen 2016; Meng and Gao 2014; Mohamed 2016; Liu and Liu 2016; Jiang and Ma 2012; Guo and Hao 2013; Dou et al. 2007). Vertex dynamics in multi-soliton solutions and some new exact solution of breaking equation are studied in Zarmi (2014) and Chen and Ma (2013), the methods of Multi-soliton Solutions are given in Ashorman (2014), Zhang and Chen (2016), Meng and Gao (2014), Mohamed (2016), Liu and Liu (2016), Jiang and Ma (2012), Guo and Hao (2013), Dou et al. (2007), Zuo and Gao (2014), Huang (2013), Liu and Luo (2013), Cotea and Munoza (2014), Xu and Chen (2014), Hua and Chen (2014) and Zhang and Cai (2014), complex solutions for the [BLP System are proposed in Ma and Xu (2014)], and these so-called new solutions is identical to the universal formula in Doungmo Goufo (2016), Atangana and Doungmo Goufo (2015), Gao (2015a, b, c, d), Xie and Tian (2015), Sun and Tian (2015) and Zhen et al. (2015).

The aim of this paper is to investigate the analytical solutions of the (2+1)-dimensional breaking equation by the mapping and Darboux transformation method. And the dynamical behaviours of (2+1)-dimensional breaking equation will be discussed in detail.

The structure of this paper is as follows: In second section, the (2+1) dimensional breaking equation is studied and its exact solutions are derived. And properties of this breaking equation will be investigated. In third section, influence of the parameters

© 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

which are related to the analytical solution will also be discussed. Finally, the conclusion is drawn in fourth section.

(2+1)-dimensional breaking equation and exact solution

Consider (2+1)-dimensional breaking equation as follows (Dou et al. 2007):

ut + buxxy + 4 bvx + 4 buxV = 0 (1)

In which, the functions of u(x, y, t) and v(x, y, t) are corresponding physical fields and set

Uy = Vx (2)

In which b is the arbitrary function, substituting Eq. (2) into Eq. (1), the analytical solution of Eq. (1) could be gotten as follows:

u = 3 (Inf)xx + Uo

V = 3 (Inf )xy + Vo

f = ao + aip(x) + a2q(y, t) + a3p(x)q(y, t)

In which, u(x,y, t) is seed solution of equation, u0 = a(x), v0 = 0, the parameters a0, ai, a2, a3 are constants.

Substituting Eqs. (3) and (4) into Eq. (1), the analytical solution of (2+1) dimensional breaking equation could be gotten as follows (Dou et al. 2007)

u = — 2

(ai + «3q)2p

2 „2 x

(ai + a3q)px

(ao + aip + a2q + a3pq)2 (ao + aip + a2q + a3pq)

bfxxx + çfx

v = -2

(«3«0 - «1«2)Pxqy

(ao + aip + a,2q + a.3pq)2

M x N multi-soliton

Define the Weierstrassp function as following Xn+1 = Xn + an cos(2nbkx)

Yn+1 = Yn + an cos(2nbny)

In which 0 < a < 1, ab > 1, and when the parameters a = 0.5, b = 3, n = 10, its response is shown in Fig. 1 and we set

<p(x) = Xn+1;

<p(y) = Yn+i;

when p = < (kx, k2, k3), q = p(y + ct, k2, k3), The Eq. (7) can be given as

v = -2

in which

(«3«o - a,ia2)$xVy

(ao + aiv + a2<p + a^tyy)2

= \J 4^3 - /<2<P - h

Fig. 1 The response of Weierstrassp function

Py = V4P3 - k2P - k3 (13)

When the parameters k, c, k2, k3, n are selected as different constants, the

N x M multi-soliton could be achieved according to Eq. (11), when the

parameters a0 = 4; a1 = 1; a2 = 1; a3 = 1; k = 0.6; k2 = 1; k3 = 1; c = 1; n = 5, and

x e [—3,3],y e [—3,3], the 5 x 6 multi-soliton are shown in Fig. 2, and x e [—5,5],y e [—5,5], the 10 x 10 multi-soliton structure are shown in Fig. 3.

From Figs. 2 and 3, it can be seen that multi-soliton could be obtained by selecting the Weierstrassp function.

Conclusion

To summarize, we have constructed M x N multi-solutions of a (2+1)-dimensional breaking equation using Darboux transformation and mapping method. The analytical solutions which changes the shape of the solutions are explored and the derived analytical expressions of (2+1)-dimensional breaking equation can be used in communication system, and it is highly anticipated that this investigation on (2+1)-dimensional breaking equation may have wider application in various physical models. Competing interests

The authors declare that they have no competing interests.

Received: 11 April 2016 Accepted: 24 May 2016 Published online: 27 June 2016

References

Ashorman R (2014) Multi-soliton solutions for a class of fifth-order evolution equations. Int J Hybrid Inf Technol 7(4):11-18 Atangana A, Doungmo Goufo EF (2015) A model of the groundwater flowing within a leaky aquifer using the concept of

local variable order derivative. J Nonlinear Sci Appl 8(5):763-775 Chen Y-M, Ma S-H (2013) New exact solution of a (3+1)-dimensional Jimbo-Miwa system. Chin Phys B 22:5101-5105 Cotea R, Munoza C (2014) Multi-solitons for nonlinear Klein-Gordon equations. Forum Math Sigma 2:e15 Dou F-Q, Sun J-A et al (2007) The new periodic wave solutions and localized excitations for (2+1)-dimensional Breaking

Soliton equation. J Northwest Norm Univ 43(1):34-38 Doungmo Goufo EF (2016) Application of the caputo-fabrizio fractional derivative without singular Kernel to Korteweg-

de Vries-Bergers equation. Math Model Anal 21(2):188-198 Gao X-Y (2015a) Comment on "Solitons, Backlund transformation, and Lax pair for the (2+1)-dimensional Boiti-Leon-

Pempinelli equation for the water waves". J Math Phys 56(1):014101 Gao X-Y (2015b) Backlund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-

coefficient B-type Kadomtsev-Petviashvili equation in fluid mechanics. Ocean Eng 96:245-247 Gao X-Y (2015c) Incompressible-fluid symbolic computation and backlund transformation: (3+1)-dimensional variable-

coefficient Boiti-Leon-Manna-Pempinelli model. Z Naturforschung A 70(1):59-61 Gao X-Y (2015d) Variety of the cosmic plasmas: general variable-coefficient Korteweg-de Vries-Burgers equation with

experimental/observational support. Europhys Lett 110(1):15002-15007 Guo R, Hao H-Q (2013) Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrodinger

equation. Commun Nonlinear Sci Numer Simul 18(9):2426-2435 Hua Y-J, Chen H-L (2014) New kink multi-soliton solutions for the (3+1)-dimensional potential—Yu-Toda-Sasa-Fukuy-

ama equation. Appl Math Comput 234:548-556 Huang Y (2013) Explicit multi-soliton solutions for the KdV equation by Darboux transformation. Int J Appl Math 43(3):135-137

Jiang L-H, Ma S-H (2012) New soliton solutions and soliton evolvements for the (3+1)-dimensional Burgers system. Acta Phys Sin 61:0510-0514

Liu N, Liu Y-S (2016) New multi-soliton solutions of a (3+1)-dimensional nonlinear evolution equation. Comput Math Appl 71(8):1645-1654

Liu J, Luo H-Y (2013) New multi-soliton solutions for generalized burgers-huxley equation. Therm Sci 17(5):1486-1489 Ma S-H, Xu G-H (2014) Complex solutions and novel complex wave localized excitations for the (2+1)-dimenaional Boiti—

Leon-Pempinelli System. Chin Phys B 23:051 1-0515 Meng G-Q, Gao Y-T (2014) Multi-soliton and double Wronskian solutions of a (2+1)-dimensional modified Heisenberg

ferromagnetic system. Comput Math Appl 66(12):2559-2569 Mohamed S (2016) Osman Multi-soliton rational solutions for some nonlinear evolution equations. Open Phys 14(1):26-36

Sun W-R, Tian B (2015) Nonautonomous matter-wave solitons in a Bose-Einstein condensate with an external potential. J Phys Soc Jpn 84:074003

Xie X-Y, Tian B (2015) Solitary wave and multi-front wave collisions for the BKP equation in physics, biology and electrical

networks. Mod Phys Lett B 29:1550192 Xu Z-H, Chen H-L (2014) Resonance and deflection of multi-soliton to the (2+1)-dimensional Kadomtsev-Petviashvili

equation. Nonlinear Dyn 78:461-466 Zarmi Y (2014) Vertex dynamics in multi-soliton solutions of Kadomtsev-Petviashvili II equation. Nonlinear Sci 27:1499-1523

Zhang S, Cai B (2014) Multi-soliton solutions of a variable-coefficient KdV hierarchy. Nonlinear Dyn 78(3):1593-1600 Zhang C-C, Chen A-H (2016) Bilinear form and new multi-soliton solutions of the classical Boussinesq-Burgers system. Appl Math Lett 58:133-139

Zhen H-L, Tian B et al (2015) Soliton solutions and chaotic motions of the Zakharov equations for the Langmuir wave in

the plasma. Phys Plasmas 22(3):1-9 Zuo D-W, Gao Y-T (2014) Multi-soliton solutions for the three-coupled KdV equations engendered by the Neumann system. Nonlinear Dyn 75(4):701-708

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