Explicit exact solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order

Chaos, Solitons and Fractals 13 (2002) 311±319

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Explicit exact solitary-wave solutions for compound KdV-type and compound KdV±Burgers-type equations with nonlinear terms of any order Weiguo Zhang a,*, Qianshun Chang b, Baoguo Jiang c a

Department of Basic Sciences, University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China b Institute of Applied Mathematics, The Chinese Academy of Sciences, Beijing 100080, People's Republic of China c Department of Mathematics and Mechanics, Changsha Railway University, Changsha 410075, People's Republic of China Accepted 20 December 2000

Abstract In this paper, we consider compound KdV-type and KdV±Burgers-type equations with nonlinear terms of any order. The explicit exact solitary-wave solutions for the equations are obtained by means of proper transformation, which degrades the order of nonlinear terms, and an undetermined coecient method. A solitary-wave solution with negative velocity for the generalized KdV±Burgers equation ut ‡ up ux auxx ‡ uxxx ˆ 0 is found. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction In this paper we consider the compound KdV-type equation with nonlinear terms of any order ut ‡ aup ux ‡ bu2p ux ‡ duxxx ˆ 0;

a; b; d; p ˆ consts; p > 0

…1:1†

and the compound KdV±Burgers-type equation with nonlinear terms of any order ut ‡ aup ux ‡ bu2p ux ‡ ruxx ‡ duxxx ˆ 0;

a; b; r; p; d ˆ consts; r 6ˆ 0; p > 0:

…1:2†

These equations arise in a variety of physical contexts and have been studied by many authors [1±8]. The KdV-type equations have application in quantum ®eld theory, plasma physics and solid-state physics. For example, the kink soliton can be used to calculate energy and momentum ¯ow and topological charge in the quantum ®eld. In [5] and [6], Dey and Co€ey considered kink-pro®le solitary-wave solutions for Eq. (1.1) with p ˆ 1 and p ˆ 2. Eq. (1.2) with p ˆ 1 describes the propagation of undulant bores in shallow water and weakly nonlinear plasma waves with certain dissipative e€ects. The compound KdV±Burgers-type equation (1.2) with p > 1 is a model for long-wave propagation in nonlinear media with dispersion and dissipation. The kink solution of Eq. (1.2) with p ˆ 1 was obtained in [8]. The stability of traveling-wave solutions of Eq. (1.2) with b ˆ 0; r < 0 and p P 1 are studied by Pego et al. [4]. In [7], we obtained kinkand bell-pro®le solitary-wave solutions of Eq. (1.1) with p ˆ 1, and kink solutions of Eq. (1.2) with p ˆ 1. In this paper, the motivation is to use an approach to study both the compound KdV-type equation (1.1) and the compound KdV±Burgers-type equation (1.2), and ®nd many possible solitary-wave solutions of *

Corresponding author. E-mail address: [email protected] (Q. Chang).

0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 2 7 2 - 1

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W. Zhang et al. / Chaos, Solitons and Fractals 13 (2002) 311±319

these equations. Two kinds of solitary-wave solutions of (1.1) and kink solution of (1.2) are obtained by means of proper transformation which degrades the order of nonlinear terms and the undetermined coecient method. A solitary-wave solution with negative velocity for the generalized KdV±Burgers equation ut ‡ up ux auxx ‡ uxxx ˆ 0 is found. This paper is organized as follows. Some basic results are presented in Section 2. In Section 3, we ®nd the bell-pro®le solitary-wave solution of Eq. (1.1). The explicit kink-pro®le solitary-wave solutions of Eqs. (1.1) and (1.2) are obtained in Section 4. Finally, a solitary-wave solution with negative velocity for the generalized KdV±Burgers equation is found in Section 5. 2. Basic results Let u…x; t† ˆ u…x yields

vt†  u…n† be a traveling-wave solution for Eq. (1.2). Substituting u…n† in Eq. (1.2)

vu0 …n† ‡ aup …n†u0 …n† ‡ bu2p …n†u0 …n† ‡ ru00 …n† ‡ du000 …n† ˆ 0: Integrating the above equation once, we have vu…n† ‡

a b up‡1 …n† ‡ u2p‡1 …n† ‡ ru0 …n† ‡ du00 …n† ˆ k; p‡1 2p ‡ 1

…2:1†

where k is an integrating constant. Let C ˆ limn!1 u…n†. Assume that asymptotic values of the solitary-wave solution of (1.2) have u0 …n†;

u00 …n† ! 0;

as jnj ! 1;

…2:2†

and C satis®es the following algebraic equation vx ‡

a p‡1 b x ‡ x2p‡1 ˆ 0: p‡1 2p ‡ 1

…2:3†

Then, the integrating constant k is equal to zero and r u00 …n† ‡ u0 …n† d

v a b u…n† ‡ up‡1 …n† ‡ u2p‡1 …n† ˆ 0 d d…p ‡ 1† d…2p ‡ 1†

…2:4†

holds. Multiplying (2.4) by u0 …n† and integrating once yields Z 1 0 r n 0 v 2 a b …u …n††2 ‡ u …n† ‡ up‡2 …n† ‡ u2…p‡1† …n† ˆ k1 ; …u …n††2 dn 2 d 1 2d d…p ‡ 1†…p ‡ 2† d…2p ‡ 1†…2p ‡ 2† …2:5† where k1 is a integrating constant. As n ! ‡1 and n ! 1, we obtain from (2.5) that Z r ‡1 0 v 2 a b 2 C‡ ‡ C‡p‡2 ‡ C 2p‡2 ˆ k1 …u …n†† dn d 1 2d d…p ‡ 1†…p ‡ 2† d…2p ‡ 1†…2p ‡ 2† ‡

…2:6†

and v 2 a b C ‡ C p‡2 ‡ C 2p‡2 ˆ k1 ; 2d d…p ‡ 1†…p ‡ 2† d…2p ‡ 1†…2p ‡ 2† respectively. Then substituting (2.7) into (2.6) implies that Z ‡1 v 2 a …C …C p‡2 r …u0 …n††2 dn ˆ C‡2 † ‡ 2 …p ‡ 1†…p ‡ 2† 1

C‡p‡2 † ‡

…2:7†

b …C 2p‡2 …2p ‡ 1†…2p ‡ 2† ‡

C‡2p‡2 †: …2:8†

W. Zhang et al. / Chaos, Solitons and Fractals 13 (2002) 311±319

The limits of (2.4) as n !

313

1 and n ! ‡1 are the following formulas:

vC ‡

a b C p‡1 ‡ C 2p‡1 ˆ 0 p‡1 2p ‡ 1

…2:9†

vC‡ ‡

a b C p‡1 ‡ C 2p‡1 ˆ 0; p‡1 ‡ 2p ‡ 1 ‡

…2:10†

and

respectively. Combining (2.9) and (2.10), we have a …C p‡2 p‡1

C‡p‡2 † ˆ v…C 2

C‡2 †

b …C 2p‡2 2p ‡ 1

C‡2p‡2 †:

By substituting (2.11) into (2.8), the following useful formula is obtained,   Z ‡1 p b 2 v…C‡2 C 2 † ‡ …C‡2p‡2 C 2p‡2 † ; …u0 …n†† dn ˆ 2r…p ‡ 2† …p ‡ 1†…2p ‡ 1† 1

…2:11†

…2:12†

which should be satis®ed by the solitary-wave solutions of (1.2). In view of (2.12), we have the following results for the solitary-wave solution of Eq. (1.2) with the assumptions (2.2) and (2.3) of asymptotic values. (i) If u…n† is a solitary-wave solution of Eq. (1.2), then r and v…C‡2

C2 † ‡

b …C 2p‡2 …p ‡ 1†…2p ‡ 1† ‡

C 2p‡2 †

have the same sign. R ‡1 (ii) u0 …n† is square integrable in the interval … 1; ‡1†, i.e., there exists the integral 1 ‰u0 …n†Š2 dn. (iii) When r 6ˆ 0, there exists only solitary-wave solution satisfying jC‡ j 6ˆ jC j for R ‡1the kink-pro®le 2 Eq. (1.2). If jC‡ j ˆ jC j, then 1 ‰u0 …n†Š dn ˆ 0 holds, and this leads to u…n† being a constant, which is not a solitary-wave solution. Therefore, there is neither bell-pro®le solitary-wave solution with the same asymptotic value nor kink-pro®le solitary-wave satisfying C‡ ˆ C for Eq. (1.2). (iv) If r ˆ 0, Eq. (1.2) has not only the bell-pro®le solitary-wave solution with the same asymptotic values, but also the kink-pro®le solitary-wave which satis®es v…C‡2

C2 † ‡

b …C 2p‡2 …p ‡ 1†…2p ‡ 1† ‡

C 2p‡2 † ˆ 0:

…2:13†

The above results are important in constructing the solitary-wave solutions of Eqs. (1.1) and (1.2). 3. The bell-pro®le solitary-wave solutions to the compound KdV-type equation By using the same deduction as of formula (2.4), we know that the following equation u00 …n†

v a b u…n† ‡ up‡1 …n† ‡ u2p‡1 …n† ˆ 0 d d…p ‡ 1† d…2p ‡ 1†

…3:1†

holds for the traveling-wave solutions of Eq. (1.1) with asymptotic conditions (2.2) and (2.3). Let u…n† ˆ w1=p …n†:

…3:2†

Then it follows from (3.1) that 1 1 p w…n†w00 …n† ‡ 2 w02 …n† p p

v 2 a b w …n† ‡ w3 …n† ‡ w4 …n† ˆ 0: d d…p ‡ 1† d…2p ‡ 1†

…3:3†

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W. Zhang et al. / Chaos, Solitons and Fractals 13 (2002) 311±319

Now, we assume that the solution of Eq. (3.3) has the following form: w…n† ˆ

A ea…n‡n0 † …1 ‡ ea…n‡n0 † †2 ‡ B ea…n‡n0 †

ˆ

A sech2 …a=2†…n ‡ n0 † ; 4 ‡ B sech2 …a=2†…n ‡ n0 †

…3:4†

where A; B and a are constants to be determined, and n0 is an arbitrary phase shift. Substituting (3.4) into (3.3), we obtain a2 p2

v ˆ 0; d

…3:5†

1 2 2v a …2 ‡ B† ‡ …2 ‡ B† p d

a A ˆ 0; d…p ‡ 1†

2…2p ‡ 1† 2 v 2 a ‡ ‰2 ‡ …2 ‡ B† Š p2 d

…3:6†

a A…2 ‡ B† d…p ‡ 1†

b A2 ˆ 0: d…2p ‡ 1†

It follows from (3.5) that r v : a ˆ p d

…3:7†

…3:8†

Substituting (3.8) into (3.6) leads to 2‡Bˆ

aA : …p ‡ 1†…p ‡ 2†v

…3:9†

Substituting (3.8), (3.9) into (3.7), we get A2 ˆ

2

2

4v2 …p ‡ 1† …p ‡ 2† …2p ‡ 1† a2 …2p ‡ 1† ‡ bv…p ‡ 1†…p ‡ 2†

2

:

Thus, we have two groups of solutions r v a1:2 ˆ p ; d A1:2 ˆ 2jvj…p ‡ 1†…p ‡ 2†

B1:2 ˆ

2ajvj 2 v

…3:10†

s 2p ‡ 1 a2 …2p ‡ 1† ‡ bv…p ‡ 1†…p ‡ 2†

s 2p ‡ 1 a2 …2p ‡ 1† ‡ bv…p ‡ 1†…p ‡ 2†

2

2

;

:

Therefore, there are two solutions of the form (3.4) in Eq. (3.3): q 2 p pv 2p‡1 …n ‡ n0 † jvj…p ‡ 1†…p ‡ 2† a2 …2p‡1†‡bv…p‡1†…p‡2† 2 sech 2 d  q ; w1 …n† ˆ p  2p‡1 sech2 p2 dv…n ‡ n0 † 2‡ 1 ‡ ajvj v a2 …2p‡1†‡bv…p‡1†…p‡2†2 q 2 p pv 2p‡1 …n ‡ n0 † jvj…p ‡ 1†…p ‡ 2† a2 …2p‡1†‡bv…p‡1†…p‡2† 2 sech 2 d  q : w2 …n† ˆ p  2p‡1 sech2 p2 dv…n ‡ n0 † 2‡ 1 ajvj v a2 …2p‡1†‡bv…p‡1†…p‡2†2

…3:11†

…3:12†

…3:13†

…3:14†

W. Zhang et al. / Chaos, Solitons and Fractals 13 (2002) 311±319

315

It is easy to verify that if bv > 0 or bv P 0 and ad > 0; then w1 …n† > 0; 8n 2 R; if bv > 0 or bv P 0 and ad < 0; then w2 …n† < 0; 8n 2 R: From the above discussion, the following theorem is obtained. Theorem 1. Suppose vd > 0. (1) If bv > 0 or bv P 0 and ad > 0, then Eq. (1.1) has the bell-profile solitary-wave solution u1 …x; t† ˆ u…x

vt† ˆ ‰w1 …x

vt†Š1=p ;

…3:15†

where w1 …n† is determined by (3.13). 1=p (2) If p makes …q† meaningful for any negative number q (for instance p is an odd number), and if the condition bv > 0 or the condition bv P 0; ad < 0 holds, then Eq. (1.1) has also the bell-profile solitary-wave solution u2 …x; t† ˆ u…x

vt† ˆ ‰w2 …x

vt†Š1=p ;

…3:16†

where w2 …n† is given by (3.14). It is not dicult to verify that the asymptotic assumptions (2.2) and (2.3) hold for the solitary-wave solutions (3.15) and (3.16). The general bell-pro®le solitary-wave solutions of Eq. (1.1) are obtained in Theorem 1, which includes results of [1,4,7] as special cases. Taking p ˆ 1 and a ˆ 0 in (3.15) and (3.16), we obtain the solutions r r 6v v u…x; t† ˆ  sech …x b d

 vt ‡ n0 † ;

which are bell-pro®le solitary-wave solutions of the mKdV equation ut ‡ bu2 ux ‡ duxxx ˆ 0: Therefore, the bell solitary-wave pro®le solutions of the famous mKdV equation are included in (3.15) and (3.16). 4. The kink-pro®le solitary-wave solution Let u…n† ˆ w1=p …n†:

…4:1†

Then we have from (2.4) that 1 1 p r w…n†w00 …n† ‡ 2 w02 …n† ‡ w…n†w0 …n† p p dp

v 2 a b w …n† ‡ w3 …n† ‡ w4 …n† ˆ 0: d d…p ‡ 1† d…2p ‡ 1†

…4:2†

Now, we assume that the solution of (4.2) has the following form: w…n† ˆ

i A ea…n‡n0 † Ah a 1 ‡ tanh …n ‡ n ˆ † : 0 2 1 ‡ ea…n‡n0 † 2

…4:3†

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W. Zhang et al. / Chaos, Solitons and Fractals 13 (2002) 311±319

Thus, there are the following relations between parameters A; a and v: 1 1 2 p v ˆ 0; a2 ‡ rpa d d 1 2 ap pv ‡ A ˆ 0; a2 ‡ ra d d d…p ‡ 1† b a A2 ‡ A v ˆ 0: 2p ‡ 1 p‡1

…4:4†

By solving system (4.4) of algebraic equations, two sets of solutions are obtained: s# " …p ‡ 1†…2p ‡ 1† pa p2 b A1 ˆ r ; pb…p ‡ 2† p‡1 d…p ‡ 1†…2p ‡ 1† s p2 b A1 ; a1 ˆ d…p ‡ 1†…2p ‡ 1† v1 ˆ and

b a A2 ‡ A1 ; 2p ‡ 1 1 p ‡ 1

s# " …p ‡ 1†…2p ‡ 1† pa p2 b A2 ˆ ‡r ; pb…p ‡ 2† p‡1 d…p ‡ 1†…2p ‡ 1† s p2 b A2 ; a2 ˆ d…p ‡ 1†…2p ‡ 1† v2 ˆ

…4:5†

…4:6†

b a A22 ‡ A2 : 2p ‡ 1 p‡1

Substituting (4.5) and (4.6) into (4.3), we have the following theorem. 1=p

1=p

Theorem 2. Suppose that b 6ˆ 0; db < 0 and p makes A1 or A2 meaningful. Then Eq. (1.2) has the kinkprofile solitary-wave solution s  ( "   #)1=p A1 1 p2 b b a 2 A ‡ A1 t ‡ n 0 A1 x 1 ‡ tanh …4:7† u1 …x; t† ˆ 2 d…p ‡ 1†…2p ‡ 1† 2p ‡ 1 1 p ‡ 1 2 or

( u2 …x; t† ˆ

" A2 1 2

1 tanh 2

s  p2 b A2 x d…p ‡ 1†…2p ‡ 1†



 #)1=p b a 2 A ‡ A2 t ‡ n 0 ; 2p ‡ 1 2 p ‡ 1

…4:8†

where A1 and A2 are given by (4.5) and (4.6), respectively. Direct calculation yields the solutions (4.7) and (4.8) satisfying (2.2) and (2.3). The results given in [8] are a special case of Theorem 2 …p ˆ 1†. Let r ˆ 0 in Theorem 2. We can obtain the following theorem. Theorem 3. Suppose that db < 0, and p makes … ab†1=p meaningful. Then the compound KdV-type equation (1.1) has a pair of kink-profile solitary-wave solutions  i1=p a…2p ‡ 1† h p p 1  tanh dv…x vt ‡ n0 † u…x; t† ˆ ; …4:9† 2b…p ‡ 2† 2d where v ˆ

2

a2 …2p ‡ 1†=b…p ‡ 1†…p ‡ 2† .

W. Zhang et al. / Chaos, Solitons and Fractals 13 (2002) 311±319

317

It is easy to prove that the asymptotic values of the above solution satisfy both algebraic equations (2.3) and (2.13). The results given in [5] are a corollary of Theorem 3 in the cases p ˆ 1 and 2, d > 0. From Theorems 1±3 we have obtained the bell-pro®le and kink-pro®le solitary-wave solutions of the compound KdV-type equation (1.1), and the kink-pro®le solitary-wave solutions of the compound KdV±Burgers-type equation (1.2).

5. The kink-pro®le solitary-wave solution to the generalized KdV±Burgers equation In this section, we consider the generalized KdV±Burgers equations ut ‡ kum ux ‡ ruxx ‡ duxxx ˆ 0 …d; k 6ˆ 0; m > 0†

…5:1†

ut ‡ u p u x

…5:2†

and auxx ‡ uxxx ˆ 0

…a > 0; p P 1†;

and seek their kink-pro®le solitary-wave solution of the form (4.1) and (4.3). Notice that Eq. (5.1) is a special form of Eq. (1.2) in the case a ˆ 0; b ˆ k; p ˆ m=2. Let p ˆ m=2. Then the formulas (4.1) and (4.3) can be rewritten as " u…n† ˆ

#1=m

A2 …1 ‡ e

…5:3†

a…n‡n0 † †2

and u…n† ˆ

  A a 1 ‡ tanh …n ‡ n0 † 2 2

1=m 1 a sech2 …n ‡ n0 † ; 2 2

…5:4†

respectively. Taking a ˆ 0; b ˆ k and p ˆ m=2 in the system of algebraic equations (4.4), we have following system: mr m2 a v ˆ 0; 2d 4d r m v ˆ 0; a2 ‡ a d d

a2 ‡

k A2 m‡1

…5:5†

v ˆ 0:

By solving the above system of algebraic equations, we obtain a set of solutions: aˆ vˆ A2 ˆ

mr ; d…m ‡ 4† 2r2 …m ‡ 2† d…m ‡ 4†

2

;

…5:6†

2r2 …m ‡ 1†…m ‡ 2† kd…m ‡ 4†

2

:

Substituting (5.6) into (5.3) or (5.4), we get the following result. Theorem 4. If m makes … kd† profile solitary-wave solution

1=m

meaningful, then the generalized KdV±Burgers equation (5.1) has kink-

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W. Zhang et al. / Chaos, Solitons and Fractals 13 (2002) 311±319

2

31=m 2

2

2r …m ‡ 1†…m ‡ 2†=kd…m ‡ 4† 6 7 u…x; t† ˆ 4    2 5 2r2 …m‡2† mr 1 ‡ exp d…m‡4† x ‡ d…m‡4†2 t ‡ n0

;

which can be rewritten as " ( r2 …m ‡ 1†…m ‡ 2† 1 u…x; t† ˆ kd…m ‡ 4†2

mr 2r2 …m ‡ 2† x‡ tanh t ‡ n0 2d…m ‡ 4† d…m ‡ 4†2 !#)1=m 1 mr 2r2 …m ‡ 2† 2 sech x‡ t ‡ n0 : 2 2d…m ‡ 4† d…m ‡ 4†2

…5:7†

!

…5:8†

Theorem 4 includes the results given in [7,8]. Finally, setting k ˆ 1; m ˆ p; r ˆ a; d ˆ 1 and v ˆ c in (5.7) and (5.8), we obtain a kink-pro®le solitary-wave solution of the generalized KdV±Burgers equation (5.2) 2 31=p 2

2a2 …p ‡ 1†…p ‡ 2†=…p ‡ 4† 6 7 u…x; t† ˆ 4    2 5 2a2 …p‡2† pa 1 ‡ exp x ‡ …p‡4†2 t ‡ n0 p‡4 ( " ! a2 …p ‡ 1†…p ‡ 2† pa 2a2 …p ‡ 2† x‡ ˆ 1 ‡ tanh t ‡ n0 2 2 2…p ‡ 4† …p ‡ 4† …p ‡ 4† !#)1=p 1 pa 2a2 …p ‡ 2† 2 sech x‡ t ‡ n0 : 2 2 2…p ‡ 4† …p ‡ 4†

…5:9†

Notice that, in the kink-pro®le solitary-wave solution (5.9) of Eq. (5.2), the wave velocity c ˆ 2 2a2 …p ‡ 2†=…p ‡ 4† is negative. However, the solution (5.9) still has the following properties (which has been mentioned in Theorem 1.1 in [4]): (i) the equation cu…n† ‡

1 up‡1 …n† ‡ o2n u…n† ˆ aon u; p‡1

n 2 R;

holds; (ii) the asymptotic values  2 1=p 2 u…n† ! uL ˆ ‰ 2a …p ‡ 1†…p ‡ 2†=…p ‡ 4† Š ; as n ! ‡1; 0; as n ! 1; 1=p

where uL ˆ ‰…p ‡ 1†cŠ ; (iii) the solution u…n† is a monotonic function. Acknowledgements This work was supported by the National Natural Science Foundation of China and the Natural Science Foundation of Hunan Province. References [1] Wadati M. Wave propagation in nonlinear lattice, I. J Phys Soc Jpn 1975;38:673±80. [2] Bona JL, Schonbek ME. Travelling wave solutions to the Korteweg±de Vries±Burgers equation. Proc R Soc Edinburgh Sect A 1985;101:207±26.

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Pego RL, Weinstein MI. Eigenvalue, and solitary wave instabilities. Phil Trans R Soc London Ser A 1992;340:47±94. Pego RL, Smereka P, Weinstein MI. Oscillatory instability of traveling waves for a KdV±Burgers equation. Phys D 1993;67:45±65. Dey B. Domain wall solution of KdV like equations with higher order nonlinearity. J Phys A 1986;19:L9±L12. Co€ey MW. On series expansions giving closed-form of Korteweg±de Vries-like equations. SIAM J Appl Math 1990;50:1580±92. Zhang WG. Exact solutions of the Burgers±combined KdV mixed equation. Acta Math Sci 1996;16:241±8 (in Chinese). Wang M. Exact solutions for a compound KdV±Burgers equation. Phys Lett A 1996;213:279±87.