Auto-Bäcklund transformation and similarity reductions for coupled Burger's equation

Auto-Bäcklund transformation and similarity reductions for coupled

Burger’s equation

Ibrahim E. Inan

a

, Dog˘an Kaya

b

, Yavuz Ugurlu

c,* a

Firat University, Faculty of Education, 23119 ELAZIG, TURKEY b

Ardahan University, Faculty of Engineering, 75000 ARDAHAN, TURKEY c

Firat University, Department of Mathematics, 23119 ELAZIG, TURKEY

a r t i c l e

i n f o

Keywords:

Bäcklund transformation Similarity reduction Coupled Burger’s equation Solitary wave solution Traveling wave solution

a b s t r a c t

In this work, we applied Bäcklund transformation and similarity reduction for coupled Bur-ger’s equation. Clarkson and Kruskal developed a direct and simple method to obtain more similarity solutions of nonlinear partial differential equation. We received our inspiration from Fan’s article, as far as we think that our work is an extended one from the Fan’s has done in his paper. As a result of this study, we obtained solitary wave solutions and traveling wave solutions of coupled Burger’s equation.

Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction

The theory of nonlinear dispersive wave motion has recently undergone much study. We do not attempt to characterize the general form of nonlinear dispersive wave equations[1,2]. Nonlinear phenomena play a crucial role in applied mathe-matics and physics. Furthermore, when an original nonlinear equation is directly calculated, the solution will preserve the actual physical characters of solutions[3]. Explicit solutions to the nonlinear equations are of fundamental importance. There are many studies which obtain explicit solutions for nonlinear differential equations. Many explicit exact methods have been introduced in literature[4–13]. Some of them are generalized Miura transformation, Darboux transformation, Cole–Hopf transformation, Hirota’s dependent variable transformation, the inverse scattering transform and the Bäcklund transformation, tanh method, Sine–Cosine method, Painleve method, homogeneous balance method (HB), similarity reduc-tion method, improved tanh method and so on.

In this study, we will obtain wave solutions of coupled Burger’s equation by using Bäcklund transformation and similarity reduction. For this do, let’s consider coupled Burger’s equation as following:

ut uxxþ 2uuxþ ux

v

þ

v

xu ¼ 0;

v

t

v

xxþ 2

vv

xþ

v

xu þ ux

v

¼ 0:

ð1Þ According to the idea of improved HB[14], we seek for Bäcklund transformation of Eq.(1). When balancing u

_v

xwith uxx

and vuxwith

v

xxthen gives M1= 1 and M2= 1. Therefore, we may choose

u ¼ @ @xf ðwÞ þ u0¼ f 0_w xþ u0;

v

¼ @ @xgðwÞ þ

v

0¼ g 0_w xþ

v

0; ð2Þ

0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.119

*Corresponding author.

E-mail addresses:[email protected](I.E. Inan),[email protected](D. Kaya),[email protected],[email protected](Y. Ugurlu).

Contents lists available atScienceDirect

Applied Mathematics and Computation

where f = f(w), g = g(w), w = w(x, t), u0= u0(x, t) and

v

0=

v

0(x, t). Here f = f(w), g = g(w) and w = w(x, t) are undetermined

func-tions, u, u0,

v

and

v

0are four solutions of Eq.(1). Substituting(2)into Eq.(1), we obtain

2f0_f00_{þ g}0_f00_{þ f}0_g00_{ f}000 ð Þw3 xþ 3f00wxwxxþ f00wtwxþ 2f02wxwxxþ 2f00w2xu0  þ f0_g0_w xwxxþ f00w2x

v

0þ f0g0wxwxxþ g00w2xu0  þ f0_w xxxþ f0wxtþ 2f0wxðu0Þxþ 2f0wxxu0þ g0wxðu0Þx  þ f0_w xx

v

0þ g0wxxu0þ f0wxð

v

0Þx  ¼ 0; ð3Þ 2g0_g00_{þ g}0_f00_{þ f}0_g00_{ g}000 ð Þw3 xþ 3g 00_w xwxxþ g00wtwxþ 2g02wxwxxþ 2g00w2x

v

0þ f0g0wxwxxþ f00wx2

v

0þ f0g0wxwxx  þ g00_w2 xu0  þ g0_w xxxþ g0wxtþ 2g0wxð

v

0Þxþ 2g 0_w xx

v

0þ g0wxðu0Þxþ f 0_w xx

v

0þ g0wxxu0þ f0wxð

v

0Þx   ¼ 0: ð4Þ

Setting the coefficients of w3

x in(3) and (4)to zero respectively, we obtain a set of ordinary differential equations

2f0_f00_{þ g}0_f00_{þ f}0_g00_{ f}000_{¼ 0; 2g}0_g00_{þ g}0_f00_{þ f}0_g00_{ g}000_{¼ 0;}

which have solutions f ¼ g ¼ 1

2ln w; ð5Þ

there by from(5)it holds that f02_{¼ g}02_{¼ f}0_g0_¼1

2f

00_¼1

2g

00_{: ð6Þ}

By using(6), Eqs.(3) and (4)can be written as the sum of some terms with f0_{and f}00_{setting their coefficients to zero will}

lead to wxðwxxþ wtþ 3wxu0þ wx

v

0Þ ¼ 0; @ @xðwxxþ wtþ 3wxu0þ wx

v

0Þ ¼ 0; wxðwxxþ wtþ 3wx

v

0þ wxu0Þ ¼ 0; @ @xðwxxþ wtþ 3wx

v

0þ wxu0Þ ¼ 0: Above equations are satisfied provided that

 wxxþ wtþ 3wxu0þ wx

v

0¼ 0; ð7Þ

 wxxþ wtþ 3wx

v

0þ wxu0¼ 0: ð8Þ

From(2) and (5), we obtain Bäcklund transformation of Eq.(1)

u ¼ 1 2 @ @xln w þ u0;

v

¼  1 2 @ @xln w þ

v

0; ð9Þ

where w satisfying(7) and (8). We take initial solutions of Eq.(1)as u0=

v

0= 0, then(7), (8), and(9)respectively reduce to

 wxxþ wt¼ 0; ð10Þ u ¼ 1 2 @ @xln w;

v

¼  1 2 @ @xln w: ð11Þ

Specially, we take a solution of(10)

w ¼ 1 þ exp½cðx þ ctÞ;

then(11)will yield the solitary wave solutions of Eq.(1)

u ¼ c 4 1 þ tanh c 2ðx þ ctÞ h i n o ;

v

¼ c 4 1 þ tanh c 2ðx þ ctÞ h i n o :

According to the idea of improved HB[14], we seek for similarity reduction of Eq.(1). Similarly, when balancing u

v

xwith

uxxand vuxwith

v

xxthen gives M1= 1 and M2= 1. Therefore, we may choose

u ¼ @ @xf ðwÞ þ u0¼ f 0_w xþ u0;

v

¼ @ @xgðwÞ þ

v

0¼ g 0_w xþ

v

0; ð12Þ

where f = f(w), g = g(w), w = w(x,t), u0= u0(x,t) and

v

0=

v

0(x, t). Here f = f(w), g = g(w) and w = w(x, t) are undetermined

2f0_f00_{þ g}0_f00_{þ f}0_g00_{ f}000 ð Þw3xþ 3f00wxwxxþ f00wtwxþ 2f02wxwxxþ 2f00wx2u0þ f0g0wxwxxþ f00w2x

v

0þ f0g0wxwxx  þ g00_w2 xu0þ f0wxxxþ f0wxtþ 2f0wxðu0Þxþ 2f 0_w xxu0þ g0wxðu0Þxþ f 0_w xx

v

0þ g0wxxu0þ f0wxð

v

0Þx  

þ ðu0Þt ðu0Þxxþ 2u0ðu0Þxþ

v

0ðu0Þxþ u0ð

v

0Þx

  ¼ 0; ð13Þ 2g0_g00_{þ g}0_f00_{þ f}0_g00_{ g}000 ð Þw3 xþ 3g00wxwxxþ g00wtwxþ 2g02wxwxxþ 2g00w2x

v

0þ f0g0wxwxxþ f00w2x

v

0  þ f0_g0_w xwxxþ g00w2xu0  þ g0_w xxxþ g0wxtþ 2g0wxð

v

0Þxþ 2g0wxx

v

0þ g0wxðu0Þxþ f0wxx

v

0þ g0wxxu0þ f0wxð

v

0Þx   þ ð

v

0Þt ð

v

0Þxxþ 2

v

0ð

v

0Þxþ

v

0ðu0Þxþ u0ð

v

0Þx   ¼ 0: ð14Þ

From(13) and (14), we obtain wxwxx 3f00þ 2f02þ 2f0g0   þ f00 _w twxþ 2w2xu0þ w2x

v

0   þ g00_w2 xu0þ f0 wxxxþ wxtþ 2wxðu0Þx  þ 2wxxu0þ wxx

v

0þ wxð

v

0Þx  þ g0 _w xðu0Þxþ wxxu0  

þ ðu0Þtþ 2u0ðu0Þxþ

v

0ðu0Þxþ u0ð

v

0Þx ðu0Þxx

  ¼ 0; ð15Þ wxwxx 3g00þ 2g02þ 2f0g0   þ g00 _w twxþ 2w2x

v

0þ w2x

v

0   þ g00_w2 xu0þ g0 wxxxþ wxtþ 2wxð

v

0Þxþ 2wxx

v

0  þ wxxu0þ wxðu0Þx  þ f0 _w xð

v

0Þxþ wxxu0   þ ð

v

0Þtþ 2

v

0ð

v

0Þxþ

v

0ðu0Þxþ u0ð

v

0Þx ð

v

0Þxx   ¼ 0: ð16Þ

To take Eqs.(15) and (16)are ODEs of f and g only for w, the rations of the coefficients of different derivative and power of f and g must be functions of w. That is to say, the following constrained conditions are satisfied

wwwxx¼ w3x

C

1ðwÞ; wtwxþ 2w2xu0þ w2x

v

0¼ w3x

C

2ðwÞ; w2 xu0¼ w3x

C

3ðwÞ;  wxxxþ wxtþ 2wxðu0Þxþ 2wxxu0þ wxx

v

0þ wxð

v

0Þx¼ w3x

C

4ðwÞ; wxðu0Þxþ wxxu0¼ w3x

C

5ðwÞ;

ðu0Þtþ 2u0ðu0Þxþ

v

0ðu0Þxþ u0ð

v

0Þx ðu0Þxx¼ w 3 x

C

6ðwÞ; ð17Þ and wwwxx¼ w3x

C

7ðwÞ; wtwxþ 2w2x

v

0þ w2x

v

0¼ w3x

C

8ðwÞ; w2 xu0¼ w3x

C

9ðwÞ;  wxxxþ wxtþ 2wxð

v

0Þxþ 2wxx

v

0þ wxxu0þ wxðu0Þx¼ w 3 x

C

10ðwÞ; wxð

v

0Þxþ wxxu0¼ w3x

C

11ðwÞ; ð

v

0Þtþ 2

v

0ð

v

0Þxþ

v

0ðu0Þxþ u0ð

v

0Þx ð

v

0Þxx¼ w 3 x

C

12ðwÞ: ð18Þ

Here we take f(w) = g(w) and u0(x, t) =

v

0(x, t), if we satisfy according to f(w), u0(x, t) and Eq.(17), we can write following

equals instead of Eqs.(17) and (18)

wwwxx¼ w3x

C

13ðwÞ; wtwxþ 4w2xu0¼ w3x

C

14ðwÞ; wxt¼ w3x

C

ð1Þ 15ðwÞ wxxxþ 4wxðu0Þxþ 4wxxu0¼ w3x

C

ð2Þ 15ðwÞ )

C

15ðwÞ ¼

C

ð1Þ15ðwÞ þ

C

ð2Þ 15ðwÞ;

ðu0Þtþ 4u0ðu0Þx ðu0Þxx¼ w3x

C

16ðwÞ

ð19Þ

(Without loss of generality, we take C15ðwÞ ¼Cð1Þ15ðwÞ þC ð2Þ

15ðwÞ fordhdt¼ Ah

3_{Þ. WhereC}

i(i = 0, 1, . . ., 16) are some arbitrary

functions which are determined later. There are three freedoms in the determination of u0,w which we can exploit the

fol-lowing rules, without loss of generality:

(a) If u0has the form u0¼ u0ðx; tÞ þ@x@X, then we can assume thatX= 0 (make the transformation f(w) ? f(w) X);

(b) If w(x, t) is defined by an equation of the formX(w) = w0(x, t), we can also assume thatX= w (make the transformation

w ?X1(w)).

By using (a) and (b), we can obtain the general solutions of Eq.(19)

C

13¼

C

14¼ 0;

C

ð1Þ15 ¼ A;

C

ð2Þ 15 ¼ A;

C

15¼ 0;

C

16¼  1 4ðA 2_{w þ BÞ; u} 0¼

v

0¼  1 4h x dh dtþ dr dt   ; ð20Þ and h = h(t),

r

=

r

(t) satisfy

dh dt¼ Ah

3

;

r

00 2Ah2

r

0_{¼ h}4

ðA2

r

þ BÞ; w ¼ xh þ

r:

ð21Þ

Therefore Eq.(15)is written as following

f000_{þ 4f}0_f00_1 4ðA 2 w þ BÞ ¼ 0: ð22Þ where if f0_{(w) = P(w), we have} P00þ 4PP01 4ðA 2 w þ BÞ ¼ 0: ð23Þ (i) In Eq.(23), if A = B = 0, P00_{þ 4PP}0_{¼ 0; ð24Þ}

and integrating(24)yields, we yield following equation

P0þ 2P2þ c2¼ 0; ð25Þ

where c2is a constant. If c2= 0, we have for solution ofEq. (25)

P ¼  1 2w:

From(21), if A = B = 0,

r

00_{= 0 and}

_r

0_{= c}

1,

r

= c1t + c0, u0¼ c₄1, w = x + c1t + c0, h = h0, where c0, c1, h0are arbitrary constants

(without loss of generality, we take h0= 1). Then(24)gives the traveling wave solutions of Eq.(1)

u ¼  1 2ðx þ c1t þ c0Þ c1 4;

v

¼  1 2ðx þ c1t þ c0Þ c1 4:

If c2> 0, we set c2¼1₂, we have for Eq.(25)

P ¼1 2tan w;

which lead to another traveling wave solution of Eq.(1)

u ¼1 2tanðx þ c1t þ c0Þ  c1 4;

v

¼1 2tanðx þ c1t þ c0Þ  c1 4:

If c2< 0, we set c2¼ 12, Eq.(25)have solution

P ¼ 1 2coth w;

which lead to another traveling wave solution of Eq.(1)

u ¼ 1 2cothðx þ c1t þ c0Þ  c1 4;

v

¼ 1 2cothðx þ c1t þ c0Þ  c1 4: (ii) In Eq.(23), if A = 0, B – 0, P00þ 4PP01 4B ¼ 0: In Eq.(21), if A = 0, B – 0,

r

00_{¼ B;}

_r

0_{¼ Bt þ c} 1;

r

¼12Bt 2 þ c1þ c0; u0¼ 14Bt  1 4c1; w ¼ x þ12Bt 2 þ c1t þ c0; h¼ h0where

c0, c1, h0are arbitrary constants (without loss of generality, we take h0= 1). Then(24)gives the traveling wave solutions

u ¼ PðwÞ 1 4Bt  1 4c1;

v

¼ PðwÞ 1 4Bt  1 4c1: and P(w) satisfy P00þ 4PP01 4B ¼ 0: (iii) A – 0, we set, in Eq.(23)

P00_{þ 4PP}0_ 1

16w ¼ 0: From Eq.(21), if A ¼ 1

2;B ¼ 0 (without loss of generality)

r

¼ 1 2Bt 2 þ c1þ c0; w ¼ x þ t 1 2þ c₁t12þ c_0; u₀¼1 8 x tþ 1 t   ; h¼ h0

where c0, c1, h0are arbitrary constants (without loss of generality, we take h0= 1). Then(24)gives the traveling wave

solu-tions of Eq.(1) u ¼ PðwÞ1 t12 þ1 8 x tþ 1 t   ;

v

¼ PðwÞ1 t12 þ1 8 x tþ 1 t   : and P(w) satisfy P00þ 4PP0 1 16w ¼ 0: 2. Conclusions

In this paper, we present Bäcklund transformation and similarity reduction. An implementation of the methods is given by applying it to coupled Burger’s equation. We also obtain solitary and traveling wave solutions of coupled Burger’s equa-tion at same time. It can be shown that the obtained soluequa-tions satisfy coupled Burger’s equaequa-tion by using Mathematica. The method can be used to many other nonlinear equations or coupled ones.

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