Applied Mathematics and Computation 204 (2008) 957–962

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

The Cole–Hopf transformation and improved tanh–coth methodapplied to new integrable system (KdV6)

Cesar A. Gómez S a,*, Alvaro H. Salas b,c

a Department of Mathematics, Universidad Nacional de Colombia, Bogotá, Colombiab Department of Mathematics, Universidad Nacional de Colombia, Manizales, Colombiac Universidad de Caldas, Manizales, Colombia

a r t i c l e i n f o

Keywords:Improved tanh–coth methodKdV equationSixth-order KdV equationKdV6Coupled equationsCole–Hopf transformationMathematica

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.08.006

* Corresponding author.E-mail addresses: cagomezsi@unal.edu.co (C.A. G

a b s t r a c t

In this paper, we study a new coupled system (KdV6) from the point of view of constructingexact solutions for it by using the Cole–Hopf transformation and one improved tanh–cothmethod. Traveling wave solution which includes periodic and soliton solutions are for-mally derived.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

The sixth-order nonlinear wave equation (KdV6) have the most general form

Uxxxxxx þ aUxUxxxx þ bUxxUxxx þ cU2x Uxxþ

dUtt þ qUxxxt þxUxUxt þ rUtUxx ¼ 0;

(ð1:1Þ

where a, b, c, d, q, x, r are arbitrary parameters and U ¼ Uðx; tÞ is a differentiable function. Several important forms of (1.1)can be derived from the considerations given by the authors in [1–3]. However, the five authors of [4] have found four casesin which (1.1) passes the Painlevé test. One of those cases is the equation

ðo3x þ 8Uxox þ 4UxxÞðUt þ Uxxx þ 6U2

x Þ ¼ 0; ð1:2Þ

which is equivalent to the Korteweg-de Vries equation with a source satisfying a third-order ordinary differential equation:

vt þ vxxx þ 12vvx �Wx ¼ 0;Wxxx þ 8vWx þ 4Wvx ¼ 0;

�ð1:3Þ

where

v ¼ Ux;

W ¼ Ut þ Uxxx þ 6U2x :

�ð1:4Þ

. All rights reserved.

ómez S), asalash2002@yahoo.com (A.H. Salas).

958 C.A. Gómez S, A.H. Salas / Applied Mathematics and Computation 204 (2008) 957–962

Recently, Kupershmidt [5] has showed that the (1.3) is integrable in the usual sense, which is a truly remarkable fact,since W ¼ 0 leaves only the unperturbed KdV itself. It is easy to see that with the change of variable

vðx; tÞ ¼ 12 uðx;�tÞ;

Wðx; tÞ ¼ 12 wðx;�tÞ;

(ð1:5Þ

the system given by (1.3) reduces to new integrable system

ut � 6uux � uxxx þwx ¼ 0;wxxx þ 4uwx þ 2uxw ¼ 0:

�ð1:6Þ

The objective of this work is to present exact solutions to system (1.6), therefore solutions for the (KdV6) (1.2), and in thisway, to further complement other studies related to this new equation. For this, we will use two methods: In the first meth-od, we use a Cole–Hopf transformation and in the second method, we consider a improved tanh–coth method which uses theidea introduced by the author in [6] to solve nonlinear partial differential equations. This paper is organized as follows: InSection 2, we will review briefly the use the Cole–Hopf transformation, and how to use the improved tanh–coth method tosolve a coupled system. In Section 3 we give the mathematical framework to search exact solutions for the system (1.6).Finally, some conclusions are given.

2. The methods

2.1. The Cole–Hopf transformation

Consider a system of two coupled PDE’s in the variables x and t

Pðu;w;ux;wx;ut ;wt ;uxt ;wx;t;uxx;wxx; . . .Þ ¼ 0;Qðu;w;ux;wx;ut ;wt; uxt;wx;t ;uxx;wxx; . . .Þ ¼ 0:

�ð2:1Þ

We seek solutions to (2.1) in the form

uðx; tÞ ¼ aðlogð1þ enÞÞxx þ b;wðx; tÞ ¼ cðlogð1þ enÞÞxx þ d;

(ð2:2Þ

where n ¼ kxþxt þ d and k, x, d, a, b, c and d are arbitrary constants to be determined later. Substituting (2.2) into (2.1) andusing the notation T ¼ en, we obtain a system of two polynomial equations in the variable T. Equating the coefficients of theseveral powers of T to zero, we obtain an algebraic system in the unknowns k, x, d, a, b, c and d. Solving this system andreversing, we finally obtain solutions of (2.1).

2.2. The improved tanh–coth method

The traveling wave transformation

n ¼ xþ kt þ n0; ð2:3Þ

convert (2.1) to a system of ordinary differential equations in the unknowns uðnÞ, wðnÞ

P1ðu;w;u0;w0;u00;w00; . . .Þ ¼ 0;Q 1ðu;w;u0;w0; u00;w00; . . .Þ ¼ 0:

�ð2:4Þ

We seek solutions to (2.4) by using the expansions

uðnÞ ¼PMi¼0

ai/ðnÞi þP2M

i¼Mþ1ai/ðnÞM�i

;

wðnÞ ¼PMi¼0

bi/ðnÞi þP2M

i¼Mþ1bi/ðnÞM�i

;

8>>>><>>>>:

ð2:5Þ

where M is a positive integer that will be determined and / ¼ /ðnÞ satisfies the Riccati equation

/0ðnÞ ¼ aþ b/ðnÞ þ c/ðnÞ2; ð2:6Þ

where a, b, c are constants. The solutions of (2.6) are given by

1. If a ¼ 0,

/ðnÞ ¼ b�cþ be�bn

: ð2:7Þ

C.A. Gómez S, A.H. Salas / Applied Mathematics and Computation 204 (2008) 957–962 959

2. If b ¼ 0,

/ðnÞ ¼ffiffiffiffiacp

c tanð ffiffiffiffiffiffiacpnÞ; a > 0; c > 0

/ðnÞ ¼ffiffiffiffiacp

c tanhð ffiffiffiffiffiffiacpnÞ; a > 0; c < 0

/ðnÞ ¼ffiffiffiffiffiffi�acp

c tanhð� ffiffiffiffiffiffiffiffiffiffi�acpnÞ; a < 0; c > 0

/ðnÞ ¼ffiffiffiffiacp

c tanð� ffiffiffiffiffiffiacpnÞ; a < 0; c < 0:

8>>>>>><>>>>>>:

ð2:8Þ

3. If c ¼ 0,

/ðnÞ ¼ �aþ bebn

b: ð2:9Þ

4. If a ¼ b ¼ 0,

/ðnÞ ¼ � 1cn: ð2:10Þ

5. If b2 < 4ac and c–0,

/ðnÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qtan 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qn

� �� b

2c: ð2:11Þ

6. If b2 > 4ac and c–0,

/ðnÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qtanh 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qn

� �� b

2c: ð2:12Þ

Substituting (2.5) into (2.4) and using (2.6) results in an algebraic system of two equations in powers of /ðnÞ. Balancingthe linear terms of highest order in the resulting equations with the highest order nonlinear term to obtain M, will yields aset of algebraic equations for a; b; c; k; a0; . . . ; a2M; b0; . . . ; b2M , because all coefficients of /ðnÞi (i ¼ 1;2; . . .) have to vanish.Solving the algebraic system, and reversing, we obtain solutions to (2.1) in the original variables.

3. Using the Cole–Hopf transformation

In this section we obtain soliton solutions to (1.6) using the Cole–Hopf transformation. By using the first method de-scribed in the previous section and after simplifications we obtain the algebraic system

4bck3 þ 2adk3 þ ck5 ¼ 0;axk2 þ 6abk3 þ ak5 � ck3 ¼ 0;6abk3 þ ak5 þ axk2 � ck3 ¼ 0;4bck3 � 11ck5 þ 2adk3 þ 6ack5 ¼ 0;6ack5 � 11ck5 þ 4bck3 þ 2adk3 ¼ 0;6a2k5 þ axk2 � ck3 þ 6abk3 � 11ak5 ¼ 0:

8>>>>>>>><>>>>>>>>:

ð3:1Þ

Solving the previous algebraic system we get

a ¼ 2;

c ¼ 2ðxþ6bkþk3Þk ;

d ¼ � 12k ðk

5 þ 10bk3 þxk2 þ 24b2kþ 4bxÞ:

8><>: ð3:2Þ

According with (2.2) and using (3.2), after simplifications we obtain the following solution to (1.6):

u1ðx; tÞ ¼ bþ k2

1þcoshðkx�xtþdÞ ;

w1ðx; tÞ ¼ � k3þ6bkþx2k 4bþ k2 1� 2

1þcoshðkx�xtþdÞ

h in o;

8<: ð3:3Þ

where b–0, k–0, d, and x are arbitrary constants, and they may be real or complex. In the particular case that we change k, d,x by k

ffiffiffiffiffiffiffi�1p

, dffiffiffiffiffiffiffi�1p

, xffiffiffiffiffiffiffi�1p

, respectively we obtain the periodic solutions

u2ðx; tÞ ¼ b� k2

1þcosðkx�xtþdÞ ;

w2ðx; tÞ ¼ k3�6bk�x2k 4b� k2 1� 2

1þcosðkx�xtþdÞ

h in o:

8><>: ð3:4Þ

Remark 1. Note that, by (1.4) and (1.5), (1.2) is equivalent to

960 C.A. Gómez S, A.H. Salas / Applied Mathematics and Computation 204 (2008) 957–962

wxxxðx;�tÞ þ 4uðx;�tÞwxðx;�tÞ þ 2wðx;�tÞuxðx;�tÞ ¼ 0:

The solutions obtained in this section satisfies this equation.

Remark 2. It is easy to see that (1.2) convert to (KdV6) equation

Uxxxxxx þ Uxxxt þ 40UxxUxxx þ 20UxUxxxx þ 8UxUxt þ 120U2x Uxx þ 4UxxUt ¼ 0:

From the second equation in (1.4) we obtain Ut ¼W � Uxxx � 6U2x . By (1.5) we have

Ux ¼ 12 uðx;�tÞ;

Ut ¼ 12 wðx;�tÞ � 1

2 uxxðx;�tÞ � 32 u2ðx;�tÞ:

(

Taking the solutions obtained here, theses last expressions satisfies the (KdV6) equation (1.2).

4. Using the improved tanh–coth method

With the traveling wave transformation

uðx; tÞ ¼ uðnÞ;wðx; tÞ ¼ wðnÞ;n ¼ xþ kt þ n0;

8><>: ð4:1Þ

the system (1.6) reduces to

ku0ðnÞ � 6uðnÞu0ðnÞ � u000ðnÞ þw0ðnÞ ¼ 0;w000ðnÞ þ 4uðnÞw0ðnÞ þ 2u0ðnÞwðnÞ ¼ 0:

�ð4:2Þ

Substituting (2.5) into (4.2), and balancing the order of u000 with the order of uu0 and the order of w000 with the order of uw0

in the first and second equations, respectively in (4.2) we have

M þ 3 ¼ 2M þ 1;

so that M ¼ 2. Therefore we seek solutions to (4.2) as follows:

uðnÞ ¼ a0 þ a1/ðnÞ þ a2/ðnÞ2 þ a3/ðnÞ�1 þ a4/ðnÞ�2;

wðnÞ ¼ b0 þ b1/ðnÞ þ b2/ðnÞ2 þ b3/ðnÞ�1 þ b4/ðnÞ�2:

(ð4:3Þ

Inserting (4.3) into (4.2) and making use of (2.6), we obtain a system of algebraic equations in the unknowns a0, a1, a2, a3,a4, b0, b1, b2, b3, b4, k, a, b and c. Solving the previous system with aid of Mathematica, using (2.7)–(2.10) and after simpli-fications, we obtain the following set of solutions to (1.6):

First set: a1 ¼ 0, a2 ¼ 0, a3 ¼ �2ab, a4 ¼ �2a2, b0 ¼ � 12 ðb

2 þ 8acþ 4a0Þðb2 þ 8ac� kþ 6a0Þ, b1 ¼ 0, b2 ¼ 0, b3 ¼ �2abðb2

þ8ac� kþ 6a0Þ, b4 ¼ �2a2ðb2 þ 8ac� kþ 6a0Þ,

u1ðx; tÞ ¼ a0 þ4acðb2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qtan 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qðxþ kt þ n0Þ

� �b� 2acÞ

b�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qtan 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qðxþ kt þ n0Þ

� �� �2 ;

w1ðx; tÞ ¼12ðb2 þ 8ac� kþ 6a0Þ �b2 � 4a0 þ

8acb

b�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qtan 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qðxþ kt þ n0Þ

� �0BB@

þ8ac � 2ac

b�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qtan 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac� b2

qðxþ kt þ n0Þ

� �� �2 � 1

0BBB@

1CCCA1CCCA;

u2ðx; tÞ ¼ a0 þ4ac b2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qtanh 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qðxþ kt þ n0Þ

� �b� 2ac

� �

bþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qtanh 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qðxþ kt þ n0Þ

� �� �2 ;

C.A. Gómez S, A.H. Salas / Applied Mathematics and Computation 204 (2008) 957–962 961

w2ðx; tÞ ¼12ðb2 þ 8ac� kþ 6a0Þ �b2 � 4a0 þ

8acb

bþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qtanh 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qðxþ kt þ n0Þ

� �0BB@

� 8ac2ac

bþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qtanh 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qðxþ kt þ n0Þ

� �� �2 þ 1

0BBB@

1CCCA1CCCA:

Second set: a1 ¼ 0, a2 ¼ �2c2, a3 ¼ 0, a4 ¼ �2a2, b ¼ 0, b0 ¼ �2ð2acþ a0Þð8ac� kþ 6a0Þ, b1 ¼ 0,b2 ¼ 2c2ð�8acþ k� 6a0Þ, b3 ¼ 0, b4 ¼ 2a2ð�8acþ k� 6a0Þ,

u3ðx; tÞ ¼ a0 � 2acðcot2ð ffiffiffiffiffiffiacp ðxþ ktÞÞ þ tan2ð ffiffiffiffiffiffiac

p ðxþ kt þ n0ÞÞÞ;

w3ðx; tÞ ¼ �2ð4accsc2ð2 ffiffiffiffiffiffiacp ðxþ kt þ n0ÞÞ þ a0Þð8ac� kþ 6a0Þ;

u4ðx; tÞ ¼ a0 þ 2acðcoth2ð ffiffiffiffiffiffiffiffiffiffi�acp ðxþ kt þ n0ÞÞ þ tanh2ð ffiffiffiffiffiffiffiffiffiffi�ac

p ðxþ kt þ n0ÞÞÞ;

w4ðx; tÞ ¼ 2ð4accsch2ð2 ffiffiffiffiffiffiffiffiffiffi�acp ðxþ kt þ n0ÞÞ � a0Þð8ac� kþ 6a0Þ:

Third set: a1 ¼ �2bc, a2 ¼ �2c2, a ¼ 0, a3 ¼ 0, a4 ¼ 0, k ¼ b2 þ 6a0 þ r22c2, b0 ¼ ðb

2þ4a0Þb24c2 , b1 ¼ br2

c , b3 ¼ 0, b4 ¼ 0,

u5ðx; tÞ ¼b2

coshðbðtb2 þ xþ 6ta0 þ tb22c2 þ n0ÞÞ þ 1

þ a0;

w5ðx; tÞ ¼b2ðb2tanh2 1

2 b tb2 þ xþ 6ta0 þ tb22c2 þ n0

� �� �þ 4a0Þ

4c2 :

Fourth Set: a1 ¼ �2bc, a2 ¼ �2c2, a3 ¼ 0, a4 ¼ 0, b0 ¼ � 12 ðb

2 þ 8acþ 4a0Þðb2 þ 8ac� kþ 6a0Þ,b1 ¼ �2bcðb2 þ 8ac� kþ 6a0Þ, b2 ¼ �2c2ðb2 þ 8ac� kþ 6a0Þ, b3 ¼ 0, b4 ¼ 0,

u6ðx; tÞ ¼ 2acþ a0 þb2 � 4ac

cosffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4ac� b2

qðxþ kt þ n0ÞÞ þ 1

;

w6ðx; tÞ ¼12

b2 þ 8ac� kþ 6a0�

b2 � 4ac tan2 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�b2 � 4ac

qðxþ kt þ n0Þ

� �� 8ac� 4a0

� �;

u7ðx; tÞ ¼ 2acþ a0 þb2 � 4ac

coshðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qðxþ kt þ n0ÞÞ þ 1

;

w7ðx; tÞ ¼12

b2 þ 8ac� kþ 6a0�

4ac� b2tanh2 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 4ac

qðxþ kt þ n0Þ

� �� 8ac� 4a0

� �:

Fifth set: a1 ¼ �2bc, a2 ¼ �2c2, a3 ¼ 0, a4 ¼ 0, k ¼ b2 þ 6a0 þ b22c2, b0 ¼ ðb

2þ4a0Þb24c2 , b1 ¼ bb2

c , b3 ¼ 0, b4 ¼ 0,

u8ðx; tÞ ¼ a0 �2b2c expðbxþ ðb3 þ 6a0bþ b2b

2c2Þt þ bn0Þðexpðbxþ ðb3 þ 6a0bþ b2b

2c2Þt þ bn0Þc� 1Þ2;

w8ðx; tÞ ¼b2b2

4c2

4a0 þ ðexpðbxþ ðb3 þ 6a0bþ b2b2c2Þt þ bn0Þcþ 1Þ2

ðexpðbxþ ðb3 þ 6a0bþ b2b2c2Þt þ bn0Þc� 1Þ2

:

Sixth set: a1 ¼ 0, a2 ¼ 0, a3 ¼ �2ab, a4 ¼ �2a2, b0 ¼ � 12 ðb

2 þ 4a0Þðb2 � kþ 6a0Þ, b1 ¼ 0, b2 ¼ 0, b3 ¼ �2abðb2 � kþ 6a0Þ,b4 ¼ �2a2ðb2 � kþ 6a0Þ,

u9ðx; tÞ ¼ a0 �2ebðxþktþn0Þab3

ða� ebðxþktþn0ÞbÞ2;

w9ðx; tÞ ¼ �ðb2 � kþ 6a0Þð4a0ða� ebðxþktþn0ÞbÞ2 þ b2ðaþ ebðxþktþn0ÞbÞ2Þ

2ða� ebðxþktþn0ÞbÞ2:

962 C.A. Gómez S, A.H. Salas / Applied Mathematics and Computation 204 (2008) 957–962

Seventh set: Rational solutions.a1 ¼ 0, a2 ¼ 0, a3 ¼ �2ab, a4 ¼ �2a2, b0 ¼ � 1

2 ð3b2 þ 4a0Þð3b2 � kþ 6a0Þ, b1 ¼ 0, b2 ¼ 0, b3 ¼ 2abð�3b2 þ k� 6a0Þ,b4 ¼ 2a2ð�3b2 þ k� 6a0Þ,

u10ðx; tÞ ¼12

1� 4

ðbðxþ 4aþ kt þ n0Þ þ 2Þ2

!b2 þ a0;

w10ðx;tÞ¼�3b2�kþ6a0

ðbðxþ4aþktþn0Þþ2Þ2ððbðxþ4aþktþn0Þðbðxþ4aþktþn0Þþ4Þþ6Þb2þ2ðbðxþ4aþktþn0Þþ2Þ2a0Þ:

Eighth set: Rational solutions.

a1 ¼ �b3

2a; a2 ¼ �

b4

8a2 ; a3 ¼ 0; a4 ¼ 0;

b0 ¼ �12ð3b2 þ 4a0Þð3b2 � kþ 6a0Þ; b1 ¼

b3ð�3b2 þ k� 6a0Þ2a

; b2 ¼b4ð�3b2 þ k� 6a0Þ

8a2 ; b3 ¼ 0; b4 ¼ 0;

u11ðx; tÞ ¼b2

2þ a0 �

2

ðxþ 4aþ kt þ n0Þ2;

w11ðx; tÞ ¼ �ð3b2 � kþ 6a0Þðb2ðxþ 4aþ kt þ n0Þ2 þ 2a0ðxþ 4aþ kt þ n0Þ2 þ 2Þ

ðxþ 4aþ kt þ n0Þ2:

Remark 3. In all cases, D ¼ b2 � 4ac.

5. Conclusions

In this paper, we have derived a lot of different solutions to new integrable system given by (1.6) from which solutions toKdV6 Eq. (1.2) can be derived in a more general form than the soliton solution of the potential Korteweg-de Vries equation.Two methods which can be considered as new have been used here. In one of the methods we have used a Cole–Hopf trans-formation and we have derived soliton solutions for the system (1.6). In the second method, we have considered a generalRiccati equation with constant coefficients and we have used the idea tanh–coth method to obtain periodic and soliton solu-tions for (1.6). This last method is more general that the method that appear in [7]. The methods used here, can be consideredas a powerful technique to analyze several forms of coupled systems.

Acknowledgements

We want to express our gratitude to professor A. Sinitsyn and we wish to thank the referee for providing valuable com-ments and suggestions.

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