Physics Letters A 380 (2016) 9–14

Contents lists available at ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

The Hopf–Cole transformation, topological solitons and multiple fusion solutions for the n-dimensional Burgers system

Yang Chen a, Engui Fan b,∗, Manwai Yuen ca Department of Mathematics, University of Macau, Macau, PR Chinab School of Mathematics and Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, PR Chinac Department of Mathematics and Information Technology, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 April 2015Received in revised form 19 September 2015Accepted 21 September 2015Available online 25 September 2015Communicated by R. Wu

Keywords:n-Dimensional Burgers systemsn-Dimensional Hopf–Cole transformationTopological solitonsMultiple fusion solutions

We show that, under an irrotational condition, there exists an n-dimensional Hopf–Cole transformation between the n-dimensional Burgers system and an n-dimensional heat equation. Further, as application of the Hopf–Cole transformation, two kinds of physically interesting exact solutions for the n-dimensional Burgers equations are found. In the first kind of solutions, the velocity fields are topological solitons. In the second kind of solutions, velocity fields are all multiple fusion soliton solutions.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

The well-known Burgers equation [1],

ut + uux − μuxx = 0, (1.1)is the simplest nonlinear model in fluid dynamics. Over the decades, the Burgers equation has been widely used to model a large class of physical systems, such as the surface perturbations, acoustical waves, electromagnetic waves, density waves, popula-tion growth, magnetohydrodynamic waves, etc. [2–12]. It is well known that the Burgers equation (1.1) can be transformed into the heat equation [13]

wt − μwxx = 0, (1.2)by the Hopf–Cole transformation

u = −2μ∂x ln w.Motivated by this idea, we intend to extend the Hopf–Cole trans-formation to linearize the following general n-dimensional vector Burgers equations [14]

* Corresponding author.E-mail addresses: yayangchen@umac.mo (Y. Chen), faneg@fudan.edu.cn (E. Fan),

nevetsyuen@hotmail.com (M. Yuen).

http://dx.doi.org/10.1016/j.physleta.2015.09.0330375-9601/© 2015 Elsevier B.V. All rights reserved.

Ut + (U · ∇)U = μ�U , (1.3)where U = (u1, u2, . . . , un)T is the fluid velocity fields, μ is the kinematic viscosity of the fluid. t is time, the sign

� = ∂2

∂x21+ ∂

2

∂x22+ · · · + ∂

2

∂x2n

denotes the Laplace operator, and

∇ = ( ∂∂x1

,∂

∂x2, . . . ,

∂

∂xn)T

is the Hamilton gradient operator. In the case n = 1, the system (1.3) becomes well-known Burgers equation (1.1), For n = 2, the (1.3) is two-dimensional coupled Burgers equation,

ut + uux + vu y = uxx + u yy,vt + uvx + v v y = vxx + v yywhose some exact traveling wave solutions are obtained [15–19].

The purpose of this paper is to present a necessary condition that general n-dimensional Burgers system (1.3) can be linearized into a heat equation. That is, we hope to establish n-dimensional Hopf–Cole transformation between n-dimensional Burgers system and a n-dimensional heat equation.

This paper is arranged as follows. In Section 2, we establish a n-dimensional Hopf–Cole transformation between the n-dimen-

http://dx.doi.org/10.1016/j.physleta.2015.09.033http://www.ScienceDirect.com/http://www.elsevier.com/locate/plamailto:yayangchen@umac.momailto:faneg@fudan.edu.cnmailto:nevetsyuen@hotmail.comhttp://dx.doi.org/10.1016/j.physleta.2015.09.033http://crossmark.crossref.org/dialog/?doi=10.1016/j.physleta.2015.09.033&domain=pdf

10 Y. Chen et al. / Physics Letters A 380 (2016) 9–14

sional Burgers equations and n-dimensional heat equation. In Sec-tion 3, as applications of the Hopf–Cole transformation, we present two kinds of novel exact solutions for the n-dimensional Burg-ers equations. In the first kind of solutions, the velocity fields are topological solitons. In the second of solutions, velocity fields are multiple fusion soliton solutions.

2. The Hopf–Cole transformation

In this section, we show that there exists a n-dimensional Hopf–Cole transformation between n-dimensional Burgers system and a n-dimensional heat equation with being assumed to have the irrotational condition

∇ × U =n∑

i, j=1

(∂u j∂xi

− ∂ui∂x j

)ei ∧ e j = 0, (2.1)

where ei, i = 1, 2, . . . , n are basis of n-dimensional Euclidean space Rn .

For convenience to discuss in the context, we write the vector Burgers system (1.3) into the following scalar form

∂ui∂t

+n∑

j=1u j

∂ui∂x j

= μ�ui, i = 1,2, . . . ,n. (2.2)

The irrotational condition (2.1) is also expressed in the scalar form

∂ui∂x j

= ∂u j∂xi

, i, j = 1,2, . . . ,n, and i �= j, (2.3)

which inspire us to introduce the following transformation

ui = ∂ϕ∂xi

, i = 1,2, . . . ,n. (2.4)

In this way, the system (2.3) automatically holds, and the system of partial differential equations (2.2) are changed to

∂2ϕ

∂xi∂t+

n∑j=1

∂ϕ

∂x j

∂2ϕ

∂xi∂x j− μ∂�ϕ

∂xi= 0, i = 1,2, . . . ,n. (2.5)

Noticing the following relations

n∑j=1

∂ϕ

∂x j

∂2ϕ

∂xi∂x j= 1

2

∂

∂xi

n∑j=1

(∂ϕ

∂x j

)2, (2.6)

then the system (2.5) can be written on the form

∂

∂xi

⎡⎣∂ϕ

∂t+ 1

2

n∑j=1

(∂ϕ

∂x j

)2− μ�ϕ

⎤⎦ = 0, i = 1,2, . . . ,n. (2.7)

These equations are satisfied if ϕ satisfy the following potential Burgers equation

∂ϕ

∂t+ 1

2

n∑j=1

(∂ϕ

∂x j

)2− μ�ϕ = 0. (2.8)

In the following, we show that the system (2.8) can be lin-earized into the following n-dimensional heat equation

∂ F

∂t− μ�F = 0, (2.9)

under well-known Hopf–Cole transformation

ϕ = −2μ ln F . (2.10)

In fact, introducing transformation

F (x1, x2, . . . , xn, t) = F (ϕ) (2.11)and substituting it into the equation (2.9) yields

∂ϕ

∂t−

(μ

∂2 F

∂ϕ2/∂ F

∂ϕ

) n∑j=1

(∂ϕ

∂x j

)2− μ�ϕ = 0. (2.12)

Comparing the equation (2.12) with (2.8) gives

2μ∂2 F

∂ϕ2+ ∂ F

∂ϕ= 0.

This linear ordinary differential equation has a solution

F (ϕ) = exp(− 12μ

ϕ),

which implies the relation (2.10).Substituting (2.10) into (2.4), we finally obtain the following re-

lations

ui = −2μ ∂∂xi

ln F , i = 1,2, . . . ,n. (2.13)

It is obvious that if F (x1, x2, . . . , xn, t) is a solution of the heat equation (2.9), then the system (2.13) provides a solution of the system (1.3). Therefore the system (2.13) make up a Hopf–Cole transformation for the n-dimensional Burgers system.

3. Application to construct exact solutions

In this section, we mainly discuss application of the n-dimen-sional Hopf–Cole transformation (2.13) based on the solutions of the n-dimensional heat equation. Two kinds of physically inter-esting solutions for the n-dimensional Burgers equations are con-structed as follows.

3.1. The topological solitons

By using Fourier transformation, it can be show that heat equa-tion (2.9) has a general solution

F = 1 + 14πt

exp

[− x

21 + x22 + · · · + x2n

4πμt

]≡ 1 + exp(ξ), (3.1)

where we have used the notation

ξ = − x21 + x22 + · · · + x2n

4πμt− ln(4πt). (3.2)

Substituting (3.1) into the Hopf–Cole transformation (2.13) leads to a solution for the Burgers system

u j = − x j2πt [1 + tanh(ξ/2)], j = 1,2, . . . ,n, (3.3)where ξ is given by (3.2). We can find some interesting physi-cal properties for this kind of solutions. The velocity fields u and v describe a non-vortex motion as shown in Figs. 1–2, where these figures are plotted for special case when n = 2, that is, two-dimensional Burgers system. We also denote u1 = u, u2 = v , x1 = x, x2 = y. In addition, for fixed t , the velocity fields u and vare all topological double peak solutions. For small change of t , the velocity field u is double peak alone x-direction and single peak alone y-direction. While the velocity field v reverse result with u.

Y. Chen et al. / Physics Letters A 380 (2016) 9–14 11

Fig. 1. Figure of u in the solution (3.3). (a) Perspective view of the wave. (b) Overhead view of the wave, with contour plot shown. The bright lines are crests and the dark lines are troughs. (c) and (d) The propagation along x direction. (e) and (f) The propagation along y direction.

12 Y. Chen et al. / Physics Letters A 380 (2016) 9–14

Fig. 2. Figure of v in the solution (3.3). (a) Perspective view of the wave. (b) Overhead view of the wave, with contour plot shown. The bright lines are crests and the dark lines are troughs. (c) and (d) The propagation along x direction. (e) and (f) The propagation along y direction.

Y. Chen et al. / Physics Letters A 380 (2016) 9–14 13

Fig. 3. Plot of two solitary wave fusion for u in the solutions (3.7)–(3.8). (a) Perspective view of the wave. (b) Overhead view of the wave, with contour plot shown.

Fig. 4. Plot of two solitary wave fusion for v in the solutions (3.7)–(3.8). (a) Perspective view of the wave. (b) Overhead view of the wave, with contour plot shown.

3.2. The multiple fusion soliton solutions

We denote

ξi =n∑

j=1αi jx j + μt

n∑j=1

α2i j, i = 1,2, . . . , N, (3.4)

where αi j, i = 1, 2, . . . , N; j = 1, 2, . . . , n are arbitrary constants.It is easy to verify that

1, exp (ξi) , i = 1,2, . . . , N (3.5)are all solutions of the linear heat equation (2.9). By using super-position formula, the heat equation admits the following solution

F = 1 +N∑

i=1exp(ξi). (3.6)

Substituting (3.6) into (2.13) and (2.13) yields the following N-soliton solutions for the nD Burgers system

u j =∑N

i=1 αi j exp(ξi)1 + ∑Ni=1 exp(ξi) , j = 1,2, . . . ,n, (3.7)

p = (2μ − 1)∑n

j=1(∑N

i=1 αi j exp(ξi))2

2[1 + ∑Ni=1 exp(ξi)]2 . (3.8)Specially taking n = 2, N = 2, we get two-soliton solution for

the 2D Burgers equations

u = α11eξ1 + α21eξ2

1 + eξ1 + eξ2 , (3.9)

v = α12eξ1 + α22eξ2

1 + eξ1 + eξ2 . (3.10)

The two soliton solutions display fusion phenomenon. We can clearly see that two single solitary waves fusion to one solitary wave at a specific time (see Figs. 3, 4).

4. Conclusion remarks

In this paper, we show that, under irrotational condition, there exists a n-dimensional Hopf–Cole transformation between the n-dimensional Burgers system n-dimensional heat equation. So a natural question is that whether there exists appropriate Hopf–Cole transformation for the n-dimensional incompressible magne-tohydrodynamics equations or other equations? This question is difficult and worth to be considered.

Acknowledgements

We would like to thank the Macao Science and Technology De-velopment Fund for generous support: FDCT 077/2012/A3, FDCT 130-2014-A; and the University of Macau for generous support: MYRG2014-00011-FST, MYRG2014-00004-FST; the National Natu-ral Science Foundation of China (Grant No. 11271079) and Tin Ka Ping Foundation from Hong Kong (Grant No. TKPF2015003).

14 Y. Chen et al. / Physics Letters A 380 (2016) 9–14

References

[1] J.M. Burgers, Mathematical examples illustrating relations occurring in the the-ory of turbulent fluid motion, Verhand. Kon. Neder. Akad. Wetenschappen, Afd. Natuurkunde, Eerste Sect. 17 (1939) 1–53.

[2] R.A. Kraenkel, J.G. Pereira, M.A. Manna, Nonlinear surface-wave ex-citations in the Bernard–Marangoni system, Phys. Rev. A 46 (1992) 4786–4790.

[3] J. Doyle, M.J. Englefield, Similarity solutions of a generalized Burgers equation, IMA J. Appl. Math. 44 (1990) 145–153.

[4] J.G. Kingston, C. Sophocleous, On point transformations of a generalised Burgers equation, Phys. Lett. A 155 (1991) 15–19.

[5] C.Z. Qu, Allowed transformations and symmetry classes of variable coefficient Burgers equations, IMA J. Appl. Math. 54 (1995) 203–225.

[6] W.P. Hong, On Bäcklund transformation for a generalized Burgers equation and solitonic solutions, Phys. Lett. A 268 (2000) 81–84.

[7] Y.T. Gao, X.G. Xu, B. Tian, Variable-coefficient forced Burgers system in nonlin-ear fluid mechanics and its possibly observable effects, Int. J. Mod. Phys. C 14 (2003) 1207–1222.

[8] M.J. Ablowitz, S. De Lillo, The Burgers equation under deterministic and stochastic forcing, Physica D 92 (1996) 245–259.

[9] V. Gurarie, A. Migdal, Instantons in the Burgers equation, Phys. Rev. E 54 (1996) 4908–4914.

[10] A. Balogh, D.S. Gilliam, V.I. Shubov, Stationary solutions for a boundary controlled Burgers’ equation, Math. Comput. Model. 33 (2001) 21–37.

[11] A.J. Muhammad-Jawad, M. Petkovic, A. Biswas, Soliton solutions of Burgers equations and perturbed Burgers equations, Appl. Math. Comput. 216 (2010) 3370–3377.

[12] A. Biswas, H. Triki, T. Hayat, O.M. Aldossary, 1-Soliton solutions of generalized Burgers equation with generalized evolution, Appl. Math. Comput. 217 (2011) 10289–10294.

[13] E. Hopf, The partial differential equation Ut + U Ux = Uxx , Commun. Pure Appl. Math. 3 (1950) 201–230.

[14] U. Frisch, J. Bec, Burgulence, in: M. Lesieur, A. Yaglom, F. David (Eds.), New Trends in Turbulence, Springer, 2001, pp. 341–383.

[15] G.W. Wang, T.Z. Xu, A. Biswas, Topological solitons and conservation laws of the coupled Burgers equations, Rom. Rep. Phys. 66 (2014) 251–261.

[16] K. Doan, Y. Asif, A decomposition method for finding solitary and periodic solutions for a coupled higher-dimensional Burgers equations, Appl. Math. Comput. 164 (2005) 857–864.

[17] Z.S. Lu, H.Q. Zhang, New applications of a further extended tanh method, Phys. Lett. A 324 (2004) 293–298.

[18] K.M. Tamizhmani, P. Punithavathi, Similarity reductions and Painleve property of the coupled higher dimensional Burgers equation, Int. J. Non-Linear Mech. 26 (1991) 427–438.

[19] M. Salerna, On the phase manifold geometry of the two-dimensional Burgers equations, Phys. Lett. A 121 (1987) 15–18.

http://refhub.elsevier.com/S0375-9601(15)00820-8/bib62757267657273s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib62757267657273s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib62757267657273s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6B72s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6B72s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6B72s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib646Fs1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib646Fs1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6B69s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6B69s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib7175s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib7175s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib686F6E67s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib686F6E67s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib67616Fs1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib67616Fs1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib67616Fs1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6162s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6162s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6775s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6775s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6261s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib6261s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib42697377617331s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib42697377617331s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib42697377617331s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib42697377617332s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib42697377617332s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib42697377617332s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib52686F7066s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib52686F7066s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib4265s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib4265s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib42697377617333s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib42697377617333s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib446Fs1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib446Fs1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib446Fs1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib3137s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib3137s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib3138s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib3138s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib3138s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib5361s1http://refhub.elsevier.com/S0375-9601(15)00820-8/bib5361s1

The Hopf-Cole transformation, topological solitons and multiple fusion solutions for the n-dimensional Burgers system1 Introduction2 The Hopf-Cole transformation3 Application to construct exact solutions3.1 The topological solitons3.2 The multiple fusion soliton solutions

4 Conclusion remarksAcknowledgementsReferences