📝 Abstract

N-dimensional B"acklund transformation (BT), Cole-Hopf transformation and Auto-B"acklund transformation (Auto-BT) of n-dimensional Burgers system are derived by using simplified homogeneous balance (SHB). By the Auto-BT, another solution of the n-dimensional Burgers system can be obtained provided that a particular solution of the Burgers system is given. Since the particular solution of n-dimensional Burgers system can be given easily by the Cole-Hopf transformation, then using the Auto-BT repeatedly, more solutions of n-dimensional Burgers system can be obtained successively

💡 Analysis

N-dimensional B"acklund transformation (BT), Cole-Hopf transformation and Auto-B"acklund transformation (Auto-BT) of n-dimensional Burgers system are derived by using simplified homogeneous balance (SHB). By the Auto-BT, another solution of the n-dimensional Burgers system can be obtained provided that a particular solution of the Burgers system is given. Since the particular solution of n-dimensional Burgers system can be given easily by the Cole-Hopf transformation, then using the Auto-BT repeatedly, more solutions of n-dimensional Burgers system can be obtained successively

📄 Content

N-dimensional Auto-Bäcklund Transformation and Exact Solutions to n-dimensional Burgers System Mingliang Wang1, , Jinliang Zhang 1* & Xiangzheng Li 1 2

1. School of Mathematics & Statistcs, Henan University of Science & Technology, Luoyang, 471023, PR China
2. School of Mathematics & Statistcs, Lanzhou University, Lanzhou, 730000, PR China

• Corresponding author. E-mail: zhangjin6602@163.com Abstract: N-dimensional Bäcklund transformation (BT), Cole-Hopf transformation and Auto-Bäcklund transformation (Auto-BT) of n-dimensional Burgers system are derived by using simplified homogeneous balance (SHB). By the Auto-BT, another solution of the n-dimensional Burgers system can be obtained provided that a particular solution of the Burgers system is given. Since the particular solution of n-dimensional Burgers system can be given easily by the Cole-Hopf transformation, then using the Auto-BT repeatedly, more solutions of n-dimensional Burgers system can be obtained successively. Keywords: n-dimensional Burgers system; n-dimensional BT; n-dimensional Cole-Hopf transformation; n-dimensional Auto-BT; SHB; exact solution AMS(2000) Subject Classification: 35Q20; 35Q53

1. Introduction The classical Burgers equation is the simplest nonlinear model in fluid dynamics[1] and has been widely used in the surface perturbations, acoustical waves, electromagnetic waves, density waves, population growth, magnetohydrodynamic waves[2-4], etc. The n-dimensional Burgers system in the form[5-7]

0 1

Δ − ∂ ∂ + ∂ ∂ ∑

i n j j i j i u x u u t u μ , n i , ,2,1 "

, ∑ = ∂ ∂ ≡ Δ n j jx 1 2 2 , (1) with an irrotational condition:
i j j i x u x u ∂ ∂

∂ ∂ , , n j i , ,2,1 "

， j i ≠ , is an important generalization of the Burgers equation and was investigated recently by Yang Chen etc. In paper [5], where the authors have shown that the n-dimensional Burgers system (1) can be transformed into n-dimensional linear heat equation by a n-dimensional Cole-Hopf transformation. In the present paper we will study further n-dimensional Burgers system (1) by using the idea of SHB[8], which means that an undetermined function ) (ϕ f f = and its derivatives x x f f ϕ ϕ) (′

,…, that appearing in the original homogeneous balance(HB)[9-12], are replaced by a

1

logarithmic function ) ln(ϕ A and its derivatives ( ) ϕ ϕ ϕ x x A A

ln ,…, respectively, where constant A and the function ) , ( t x ϕ ϕ = are to be determined. The aim of this paper is to derive the n-dimensional BT for n-dimensional Burgers system (1), and from the BT to reason out the n-dimensional Cole-Hopf transformation and the n-dimensional Auto-BT for the n-dimensional Burgers system (1). By the Auto-BT,another exact solution of the Burgers system can be obtained provided a particular solution of it is given. Hence, once the particular solution is given, using the Auto-BT repeatedly, more exact solutions of the Burgers system can be obtained successively. 2. Derivation of n-dimensional Auto-BT Considering the homogeneous balance between i j i x u u ∂ ∂ and j i u x2 2 ∂ ∂ in system (1) ( ), according to SHB, we can suppose that the solution of the n-dimensional Burgers system (1) is of the form 1 2 1

→ +

m n n m ) 0 ( ) 0 ( ) (ln i x i x i u A u A u i i +

= ϕ ϕ ϕ , n i , ,2,1 "

                       (2) 

where , ( )t x x x u u n i i , , , , 2 1 ) 0 ( ) 0 ( "

n i , ,2,1 "

, be a given particular solution of n-dimensional Burgers system (1), i.e. 0 ) 0 ( 1 ) 0 ( ) 0 ( ) 0 (

Δ − ∂ ∂ + ∂ ∂ ∑

i n j j i j i u x u u t u μ , n i , ,2,1 "

                       (3) 

with the irrotational condition j i i j x u x u ∂ ∂

∂ ∂ ) 0 ( ) 0 ( , n j i , ,2,1 "

， , j i ≠ , constant A and the function ( t x x x n, , , , 2 1 " ) ϕ ϕ = are to be determined later. Substituting (2) into the left hand side of system (1), noticing that

t u x A t u i t i i ∂ ∂ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂

∂ ∂ ) 0 ( ϕ ϕ ,

j i j x j i x i j i j x u u u x A x A x u u j j ∂ ∂ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂

∂ ∂ ) 0 ( ) 0 ( ) 0 ( 2 2 2 1 ϕ ϕ ϕ ϕ ,

) 0 ( 1 2 i n j x i i u x u j Δ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − Δ ∂ ∂

Δ ∑

ϕ ϕ ϕ ϕ , and using (3), we have

2

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + Δ − ∂ ∂ + ∂ ∂

Δ − ∂ ∂ + ∂ ∂ ∑ ∑ ∑

=

n j x n j j j t i i n j j i j i j A x u x A u x u u t u 1 2 1 ) 0 ( 1 2 1 ϕ ϕ μ ϕ ϕ μ ϕ ϕ μ , n i , ,2,1 "

. (4) In order to find A , setting the coefficient of ∑

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ n j x j 1 2 ϕ ϕ to zero，yields

0 2 1

• μ A , which implies that μ 2

= A . (5) Using (5) the expression (2) becomes

) 0 ( 2 i x i u u i + −

ϕ ϕ μ , (6) n i , ,2,1 "

and the expression(4) becomes 0 2

1 ) 0 ( 1

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ Δ − ∂ ∂ + ∂ ∂

Δ − ∂ ∂ + ∂ ∂ ∑

This content is AI-processed based on ArXiv data.