The (N + 1)-Dimensional Burgers Equation: A Bilinear Extension, Vortex, Fusion and Rational Solutions

Hongli An; Engui Fan; Manwai Yuen

doi:10.2991/jnmp.k.200922.004

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Volume 28, Issue 1, March 2021, Pages 27 - 37

The (N + 1)-Dimensional Burgers Equation: A Bilinear Extension, Vortex, Fusion and Rational Solutions

Authors

Hongli An¹^{, *}, Engui Fan², Manwai Yuen³

¹College of Science, Nanjing Agricultural University, Nanjing 210095, People’s Republic of China

²School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China

³Department of Mathematics and Information Technology, The Education University of Hong Kong, Tai Po, New Territories, Hong Kong

^*Corresponding author. Email: kaixinguoan@163.com

Corresponding Author

Hongli An

Received 14 September 2019, Accepted 5 March 2020, Available Online 10 December 2020.

DOI: 10.2991/jnmp.k.200922.004 How to use a DOI?
Keywords: The (N + 1)-dimensional Burgers system; bilinear formulation; generalized Cole-Hopf transformation; vortex solutions; multiple fusion solutions; rational solutions
Abstract: In this paper, by introducing a fractional transformation, we obtain a bilinear formulation for the (N + 1)-dimensional Burgers equation. Such a formulation constitutes a bilinear extension to the (N + 1)-dimensional Cole-Hopf transformation between the (N + 1)-dimensional Burgers system and generalized heat equation. As applications of the bilinear extension to the Cole-Hopf transformation, four types of physically interesting exact solutions are constructed, which contain vortex solutions, multiple fusions, rational solutions and triangular rational solutions. The behaviors of these solutions are analyzed and simulated. Interestingly, the type of fusion solutions has recently found applications in organic membrane, macromolecule material, even-clump DNA, nuclear physics and plasmas physics et al.
Copyright: © 2020 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Exact solutions to Partial Differential Equations (PDEs) are important. Not only because they can serve as useful tools to test the effectiveness of numerical algorithms of the PDEs, but also they can help us to better understand various phenomena in nature described by the PDEs and then lead to further applications. Therefore, to study solutions of PDEs has always been an interesting and important work. Up to now, lots of effective methods have been developed, including inverse scattering approach [1], Darboux and Bäcklund Transformation [26,28], Hirota bilinear method [14,18], Lie symmetry method [5], Wronskian and Casoratian technique [31,33], variable separation method [27,28], various tanh methods [11,44] and so on [9,23–25]. Among them, the Hirato bilinear method has been considered as the most simple and direct method to construct solutions.

It is noticed that among the PDEs, the Burgers equation is among the simplest models:

ut+uux-μuxx=0, (1.1)

where u denotes the fluid velocity and μ is the viscosity coefficient. This equation was originally introduced by Bateman [4] and subsequently investigated by Burgers [6,7]. Investigation shows that the Burgers equation has occurred in many branches of physics, such as fluid, gas dynamics, acoustics waves, traffic flow, population growth, density and electromagnetic waves et al (see [2,3,10,12,13,16,20,21,37]). It is known that under the Cole-Hopf transformation

u=-2μ(logf)x, (1.2)

the Burgers equation is mapped into the heat equation [17]

ft-μfxx=0, (1.3)

which is an important physical model used to solve Black–Scholes equation, to study random walks and Brownian motion.

Recently, in turbulence, Frisch and Burgulence introduced a general (N + 1)-Dimensional vector Burgers (NDB) system [22]:

ut+(u⋅▽)u-μΔu=0. (1.4)

In the above, u = (u₁, u₂, ⋯, u_N)^T is the velocity vector of fluid and μ denotes the viscosity coefficient. While ▽=(∂x1,⋯,∂xN)T represents a Hamilton gradient operator and Δ=∑j=1N∂xj2 is the Laplacian operator. It is noted that when N = 1, the system (1.4) is reducible to the classical Burgers equation (1.1), which admits the soliton fusion solutions [43]. When N = 2, it becomes the coupled Burgers equation

{u1t+u1u1x+u2u1y=u1xx+u1yy,u2t+u1u2x+u2u2y=u2xx+u2yy, (1.5)

which admits travelling wave solutions [19,29,39,42]. It is also noticed that the coupled integrable Burgers equations have been studied earlier in Ma and Zhou [34] wherein some rational solutions were given. Interestingly, a kind of similar rational solutions called lumps has been analyzed recently pretty systematically in Ma and Zhou [35].

Based on the fact that the Burgers equation (1.1) admits the Cole-Hopf transformation, the authors in Chen et al. [8] extended the transformation to an (N + 1)-dimensional Cole-Hopf transformation

uj=-2μ(logf)xj, j=1,⋯,N (1.6)

which lead to the (N + 1)-dimensional heat equation

ft-μΔf=0. (1.7)

However, they did not discuss the bilinear formulation for the NDB equation. It is known that for most nonlinear equations, the bilinear forms can be obtained under a logarithmic type (log-type) transformation. For example, by using the transformation

u=-2(logf)xx,

the KdV equation

ut-6uux+uxxx=0

can be written into a bilinear formulation. Since the log-type transformation exists in the NDB equation (1.4), a natural question comes whether a bilinear form or a generalized bilinear operator introduced by Ma [32] exists in this NDB system, which constitutes a primary motivation to undertake the present work. Another strong motivation of our work arises from the significance of the fusion solution, which has found their applications in many physical models, such as in organic membrane, macromolecule material [40], in even-clump DNA [15] and in nuclear physics et al. [41]. Wang et al. [43] showed that the (1 + 1)-dimensional Burgers equation admits the fusion solution via the Painléve expansion method. Recently, they also have found that the Sharma–Tasso–Olver–Burgers equation admits the fusion solutions via introducing a velocity resonance mechanics [45]. Therefore, it motives us to consider whether the NDB system, like the (1 + 1)-dimensional Burgers equation, admits fusion solutions. If so, it would be interesting to use these solutions to explain/predict some physical phenomenon. The third motivation arises from the work of Ma [30], in which the author showed that the higher-dimensional integrable systems admit the general Darboux transformations. This inspires us to consider whether the general Darboux transformations exist in the NDB equation.

With the above questions bearing in mind, we expand the investigation on the NDB equation. By introducing a fractal transformation, we establish a bilinear formulation for the NDB equation, which can be regarded as a “quasi-linearization” of NDB system or a bilinear extension to the Cole-Hopf transformation. As its applications, we construct several types of physically interesting exact solutions, including the vortex solutions, multiple fusions, rational solutions and triangular rational solutions.

This paper is arranged as follows. In Section 2, the bilinear extension is given for the NDB equation by using a rational transformation. In Section 3, special reductions of the bilinear extension are discussed and several types of exact physically interesting solutions are obtained. Numerical analysis is made on the behaviors of the solution given. Finally a short conclusion is attached.

2. A BILINEAR FORMULATION TO THE NDB EQUATION

It is shown in section that the NDB equation admits a bilinear formulation by using the (N + 1)-dimensional fractional transformation. To show its effectiveness, two examples are given.

In order to construct the bilinear formulation of the NDB equation (1.4), we first write into a scalar form

ui,t+∑j=1Nujui,xj=μΔui, i=1,2,⋯,N. (2.1)

Now, to bilinearize the Eq. (2.1), a fractional transformation is introduced via

ui=gif, i=1,⋯,N, (2.2)

where g_i and f are functions to be determined.

Direct calculation shows that

∂ui∂t=gi,tf-ftgif2, (2.3)

∑j=1Nuj∂ui∂xj=∑j=1Nfgjgi,xj-gigjfxjf3, (2.4)

μΔui=μ∑j=1Ngi,xjxjf-2fxjgi,xj+gifxjxjf2+μ∑j=1N2fxj2gi-2fgifxjxjf3. (2.5)

Substituting (2.3)–(2.5) into (2.1) yields

ui,t+∑j=1Nujui,xj-μΔui=1f2{(gi,tf-ftgi)-μ∑j=1N(gi,xjxjf-2fxjgi,xj+gifxjxj)+∑j=1N(gi,xjgj-gigj,xj)}+gif3∑j=1N[f(gj+2μfxj)xj-fxj(gj+2μfxj)]. (2.6)

Let

(gi,tf-ftgi)-μ∑j=1N(gi,xjxjf-2fxjgi,xj+gifxjxj)+∑j=1N(gi,xjgj-gigj,xj)=0, (2.7)

∑j=1N[f(gj+2μfxj)xj-fxj(gj+2μfxj)]=0, (2.8)

which is equivalent to the bilinear form

(Dt-μ∑j=1NDxj2)gi⋅f+∑j=1NDxjgi⋅gj=0, (2.9)

∑j=1NDxjf⋅(gj+2μfxj)=0, i=1,⋯,N, (2.10)

where Dxj and D_t are the bilinear derivative operators defined via

DxαDtβ(f⋅g)=(∂x-∂x′)α(∂t-∂t′)βf(x,t)g(x′,t′)|x′=x, y′=y, t′=t.

At this stage, the bilinear formulation of the NDB equation has been constructed by using the N-dimensional fractal transformation. This result is quite novel. In the following, we shall take N = 1 and N = 2, namely the classical Burgers (1.1) and coupled Burgers equations (1.5), as examples to show the effectiveness and feasibility.

Example 2.1.

For the (1 + 1)-dimensional Burgers equation (1.1), by using the formula

u=g1f,

and noticing that D_xg₁ · g₁ = 0, we can obtain the bilinear formula of the Burgers system (1.1), which is

(Dt-μDx2)f⋅g1=0,Dxf⋅(g1+2μfx)=0.

Example 2.2.

For the coupled Burgers equation (1.5), via the formula (2.2), (2.9) and (2.10), we get that under the transformation

u1=g1f, u2=g2f,

the coupled Burgers system (1.5) admits the bilinear formula:

(Dt-μDx12-μDx22)g1⋅f+Dx1g1⋅g2=0,(Dt-μDx12-μDx22)g2⋅f+Dx2g2⋅g1=0,Dx1f⋅(g1+2μfx1)+Dx2f⋅(g2+2μfx2)=0.

3. REDUCTIONS OF THE BILINEAR FORMULATION

In this section, special reductions of bilinear equations (2.9) and (2.10) are considered and thereby a generalized (N + 1)-dimensional heat equation is derived.

It is noticed that the bilinear equations (2.9) and (2.10) are equivalent to (2.7) and (2.8). For convenience, we shall go with the latter two equations. For Eq. (2.8), if we take

f(gj+2μfxj)xj-fxj(gj+2μfxj)=0, j=1,⋯,N, (3.1)

which can be written in a form of

(gj+2μfxj)xjgj+2μfxj=fxjf. (3.2)

Integrating with respect to x_j produces

gj=-2μfxj-2μcjf, j=1,⋯,N, (3.3)

where c_j is an arbitrary constant. On substituting it into (2.2), we obtain that

uj=-2μ(logf)xj-2μcj, j=1,⋯,N. (3.4)

Interestingly, the expression of u_j readily satisfies the irrotational condition

∇×u=∑i,j=1N(uj,xi-ui,xj)ei∧ej=0, (3.5)

where e_i (i = 1, 2, ⋯, N) are the basis of the Euclidean space R^N.

Now we turn back to consider Eq. (2.9). Inserting (3.3) into (2.9) leads to a vector equation

f(ft-μΔf-2μc⋅▽f)xi-fxi(ft-μΔf-2μc⋅▽f)=0, (3.6)

where c = (c₁, ⋯, c_N) is an N-dimensional vector. It is found that (3.6) gives a generalized heat equation

ft-μΔf-2μc⋅▽f-c0f=0, (3.7)

where c₀ is an constant of integration.

Remark 3.1.

What needs to point out is that Eq. (3.7) is the model describing the 1-dimensional unsteady convective mass transfer with a first-order volume chemical reaction in a continuous medium that moves with a constant velocity. A similar equation is used to analyze the corresponding 1-dimensional thermal processes in a moving medium with volume heat release proportional to temperature. See p. 283, in Polyanin [36].

Without loss of generality, we may take c₀ = 0. Since under the transformation f→ec0tf , we have

ft-μΔf-2μc⋅▽f=0. (3.8)

Remark 3.2.

This Eq. (3.8) is called a convective heat equation, which is encountered in 1-dimensional nonstationary problems of convective mass transfer in a continuous medium that moves with a constant velocity with no absorption or release of substance. See, p. 280 in Polyanin [36].

In the case of c = 0, Eq. (3.6) becomes

ft-μΔf-c0f=0.

This equation comes from the heat equation with an additional term to account for radiative loss of heat, which depends upon the excess temperature at a given point compared with the surroundings.

It is seen that the formula (3.4), compared with (1.2) and (1.6), is indeed a more general (N + 1)-dimensional Cole-Hopf transformation. And Eq. (3.8) is a more general linear equation than the heat equation (1.7).

Example 3.1.

For the (1 + 1)-dimensional Burgers equation (1.1), by using formula (3.4) and (3.8), we obtain that under the Cole-Hopf transformation

u=-2μ(logf)x+c1,

the Burgers equation (1.1) is linearized into linear heat equation

ft-μfxx-2μc1fx=0,

which is related to the classical heat equation

wt-μwxx=0

under transformation f=e-c1x-μc12tw(x,t) .

4. APPLICATIONS FOR EXACT SOLUTIONS TO THE NDB EQUATION

This section is devoted to using the more general (N + 1)-dimensional Cole-Hopf transformation (3.4) to construct exact solutions of the NDB equation.

4.1. The Vortex Solutions

To solve Eq. (3.8), the Fourier transformation is introduced via

f^(ξ)=∫eiξ⋅xf(x)dx, (4.1)

where ξ · x = ξ₁x₁ + ⋯ + ξ_Nx_N . Its Fourier inverse transformation takes a form of

f(x)=1(2π)N∫e-iξ⋅xf^(ξ)dξ, (4.2)

When applying the Fourier transform (4.1) to (3.8), it produces

f^t+μ|ξ|2f^+2iμc⋅ξf^=0, (4.3)

where c = (c₁, ⋯, c_N) is a constant vector. It is shown that (4.3) admits a solution

f^(ξ)=exp[-μt(|ξ|2+2ic⋅ξ)]. (4.4)

Taking the Fourier inverse transformation to the above equation and then we obtain a special solution of the heat equation (3.8), which is

f(x,t)=1(4πμt)N/2exp[-∑j=1N(xj+2μcjt)24μt], (4.5)

It is noticed that the heat equation (3.8) is linear and 1 is also its solution. According to the superposition principle, the combination of 1 and (4.5), namely

f(x,t)=1+1(4πμt)N/2exp[-∑j=1N(xj+2μcjt)24μt]≡1+exp(η), (4.6)

is also a solution of (3.8). In the above

η=-∑j=1N(xj+2μcjt)24μt-N2ln(4πμt). (4.7)

Therefore, substituting (4.6) into the Cole-Hopf transformation (3.4), leads to the solution for the NDB system

uj=xj+2μcjtt[1+tanh(η/2)]-2μcj, j=1,2,⋯,N. (4.8)

Surprisingly, such a kind of solutions (4.8) exhibits an interesting physical properties. Here we take N = 2 and the solution for the coupled Burgers system is

u1=x1+2μc1tt[1+tanh(η/2)]-2μc1,u2=x2+2μc2tt[1+tanh(η/2)]-2μc2. (4.9)

Since the structure of u₁ and u₂ takes the same form, here we only take u₁ as illustrative example to draw the pictures. The behaviors of the solution u₁ are exhibited in Figure 1. From figures, we can see that such a solution exhibits a localized wave propagation, being a soliton form in x₁-axis but a vortex form in x₂-axis.

4.2. The Multiple Fusion Solutions

For constructing the multiple fusion solutions, we introduce the variable transformation via

ηi=∑j=1Nαijxj+μt∑j=1N(αij2+2cjαij), i=1,2,⋯,N, (4.10)

where α_ij (i, j = 1, 2, ⋯, N) are arbitrary constants.

One may readily verify that

1, exp(ηi), i=1,2,⋯,N (4.11)

are all solutions of the linear heat equation (3.8). Therefore, according to the superposition formula, their combination is also the solution of the heat equation, which is

f=1+∑i=1Nexp(ηi). (4.12)

Insertion of (4.12) into (3.4) produces a travelling wave solution to the NDB equation (1.4), which is in the form of

uj=-2μ∑i=1Nαijexp(ηi)1+∑i=1Nexp(ηi)-2μcj, j=1,2,⋯,N. (4.13)

Interestingly, unlike the first type solution (4.8), such a solution (4.13) shows N-fusion phenomenon. For convenience, we take N = 2 as example to show the results, wherein the two-travelling wave solution to the (2 + 1)-dimensional Burgers system is given by

uj=α1jeη1+α2jeη21+eη1+eη2-2μcj, j=1,2. (4.14)

The 2-soliton fusion phenomenon of u₁ can be clearly seen in Figure 2. From the pictures, we can see that there are two single kink-soliton collide and fusion to one at certain time. Such fusion phenomenon are very important, which have been observed in many physical models, like in organic membrane, macromolecule material, even-clump DNA, nuclear physics and Sr-Ba-Ni oxidation crystal (see [15,40,41]). Deep analysis shows that for all the ranges of the parameters a₁_j and a₂_j, only fusion can occur in the 2-solitons expressed by (4.14). Both elastic scattering and fission can not happen.

4.3. The Rational Solutions

Here, we aim to seek to the polynomial solution of heat equation (3.8) in the form of

f(x,t)=∑j=1N(xj+Ajt)2+Bt, (4.15)

where A_j and B are constants to be determined.

Substituting (4.15) into (3.8) produces

2∑j=1NAj(xj+Ajt)+B-2nμ+4μ∑j=1Ncj(xj+Ajt)=0. (4.16)

which leads to

Aj=2μcj, B=2Nμ.

So the solution of the heat equation (3.8) is

f(x,t)=∑j=1N(xj+2μcjt)2+2Nμt. (4.17)

Inserting it into the Cole-Hopf transformation (3.4), the rational solution for the NDB system is obtained as follows

uj=-4μ∑j=1N(xj+2μcjt)∑j=1N(xj+2μcjt)2+2Nμt-2μcj, j=1,⋯,N. (4.18)

For t > 0 and x → ±∞, it shows that the solution u_j tends to −2μc_j.

For example, setting N = 1, the Burgers equation (1.1) admits a solution of

u=-4μ(x+2μct)(x+2μct)2+2μt-2μc, (4.19)

whose structure can be seen in Figure 3.

4.4. The Triangular Rational Solutions

Now, we search for a kind of triangular function solutions for the heat equation (3.8) in the form of

f(x,t)=1+∑j=1NeAjtcos(xj+Bjt), (4.20)

where A_j and B_j are constants to be determined.

By substituting (4.20) into (3.8), we have

∑j=1N(Aj+μ)cos(xj+Bjt)+∑j=1N(2μcj-Bj)sin(xj+Bjt)=0, (4.21)

which shows

Aj=-μ, Bj=2μcj.

So the generalized heat equation (3.8) admits a solution

f(x,t)=1+e-μt∑j=1Ncos(xj+2μcjt). (4.22)

By using the Cole-Hopf transformation (3.4), we then find a kind of triangular rational solutions for NDB system

uj=2μe-μt∑j=1Nsin(xj+2μcjt)1+e-μt∑j=1Ncos(xj+2μcjt)-2μcj, j=1,⋯,N. (4.23)

When N = 1, we get a triangular rational solution for the Burger equation (1.1)

u=2μe-μtsin(x+2μct)1+e-μtcos(x+2μct)-2μc, (4.24)

which is bounded for t > 0 and μ > 0, which can be clearly seen in Figure 4.

5. CONCLUSION

In this paper, the bilinear formulation for the (N + 1)-dimensional Burgers equation is established by using a fractal transformation. A special reduction of the bilinear formulation is introduced and thereby the generalized Cole-Hopf transformation is obtained, which leads to the general heat equation. When N = 1, the reduced general heat equation coincides with the some physical models (see Remarks 3.1 and 3.2). As applications of the N-dimensional Cole-Hopf transformation, some physically interesting solutions, such as vortex solutions, multiple fusion and two different types of rational solutions are given. Numerical analysis shows that these solutions exhibit physically interesting behaviors. However, there are some interesting problems that deserve further investigation. For example, since the momentum equation of the NDB system shares some similarities to that of the N-dimensional incompressible/compressible Euler equation and Navier–Stokes (NS) equation. Therefore, a natural question is that whether there exists appropriate bilinear formulations for the Euler and NS equations. If the answer is positive, then it means that we can quasi-linearize the Euler and NS equations. Based on the importance and wide applications of the two equations, these questions are worth deep investigations in the future.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

ACKNOWLEDGMENTS

The authors would like to express their sincere thanks to the referees for their kind comments and valuable suggestions. This work is supported by the National Science Foundation of China under grant No. 11775116, No. 11671095, No. 51879045, Jiangsu Qinglan high-level talent Project and Dean’s Research Fund from the Education University of Hong Kong under grant No. 2018-19, FLASS/DRF/IRS-5.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 28 - 1
Pages: 27 - 37
Publication Date: 2020/12/10
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.2991/jnmp.k.200922.004 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Hongli An
AU  - Engui Fan
AU  - Manwai Yuen
PY  - 2020
DA  - 2020/12/10
TI  - The (N + 1)-Dimensional Burgers Equation: A Bilinear Extension, Vortex, Fusion and Rational Solutions
JO  - Journal of Nonlinear Mathematical Physics
SP  - 27
EP  - 37
VL  - 28
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.200922.004
DO  - 10.2991/jnmp.k.200922.004
ID  - An2020
ER  -

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