The classification of single travelling wave solutions to the Camassa–Holm–Degasperis–Procesi equation for some values of the convective parameter

P

— journal of October 2011

physics pp. 759–764

The classification of single travelling wave solutions to the Camassa–Holm–Degasperis–Procesi equation for some values of the convective parameter

CHUN-YAN WANG^∗, JIANG GUAN and BAO-YAN WANG

Department of Mathematics, Northeast Petroleum University, Daqing 163318, China

∗Corresponding author. E-mail: chunyanmyra@163.com

MS received 2 December 2010; revised 4 March 2011; accepted 21 March 2011

Abstract. By the complete discrimination system for the polynomial, we give the classification of single travelling wave solutions to the Camassa–Holm–Degasperis–Procesi equation for some values of the convective parameter.

Keywords. Complete discrimination system for the polynomial; single travelling wave solution;

Camassa–Holm–Degasperis–Procesi equation.

PACS Nos 02.30.Jr; 05.45.Yv; 03.65.Ge

1. Introduction

Classifications of single travelling wave solutions to some nonlinear differential equations have been obtained extensively by the complete discrimination system for polynomial method proposed by Liu [1–7]. Furthermore, Wang and Li [8] used Liu’s method and factorization method proposed by Cornejo-Pérez and Rosu [9–11] to give single solitary and multi-solitary solutions to some nonlinear differential equations. Yang [12] studied the classification of envelope solutions to SD equation by Liu’s method. In the present paper, we consider the Camassa–Holm–Degasperis–Procesi (CH–DP) equation which reads as

ut−c0ux+(b+1)uux−α²(ux xt+uux x x+buxux x)+γux x x =0, (1) where b is a convective parameter and b=1, b=0. Using the dynamical system approach, the bifurcation of travelling wave solutions to the CH–DP equation has been studied (see, for example, ref. [13] and the references therein). By Liu’s method, we shall give the classification of single travelling wave solutions to the CH–DP equation for some values of the convective parameter.

2. Classification of travelling wave solutions

Taking the travelling wave transformation u=u(ξ)andξ =x−ct , the CH–DP equation is reduced to the following ODE:

−(c+c0)u+(b+1)uu−α²(uu+buu)+(α²c+γ )u=0. (2)

This equation can be turned into the following first-order differential equation by integrat- ing it once and using y=(u)²as the new dependent variable:

y+pb(u)y+qb(u)=0, (3)

where pb(u)= _u−⁽_c^b_{−γ /α}⁻¹⁾ 2 and qb(u)= ^2(c+c_α⁰⁾2^uu⁻⁽−c^b−γ /α^+1)u22−2C1, C1 is an integral constant. By the method of variation of constants, we can obtain the general solution of eq. (3) [6],

y=exp

−

pb(u)du

×

C₂−

qb(u)exp

pb(u)du

du

, (4)

where C2is an integral constant. The solutions of u can be given from±(ξ−ξ0)= _du

√y. When b = −1 and −2, we can easily give the classifications of single travelling wave solutions by using the discrimination system for two-order and three-order polynomial respectively. For saving space, we only consider the case of b = 3. Of course, we can give same treatment for other values of b.

Substituting the expressions of p3(u)and q3(u)into eq. (4), and performing the trans- formationv = u −c−γ /α², the general solution can be obtained from the following quadrature:

±1

α(ξ−ξ0)=

vdv

v⁴+a3v³+a2v²+a0

, (5)

where a3=2c−²₃c0+₃^8γ_α2, a2=C1−(c+c0) c+_α^γ2

, a0=C2α².

Case 1. a0 =0. Denote=a²₃−4a2. There exist two cases to be discussed:

Case 1.1. If=0, the corresponding solutions are u = ±exp

±1

α(x−ct−ξ0)

+c₀ 3 − γ

3α². (6)

Case 1.2. If >0 or <0, the corresponding solutions are u = ±1

2exp

±1

α(x−ct−ξ0)

+ c²₀

18−17γ² 18α⁴+cc₀

6 + c₀γ 18α²−cγ

6α²−C₁ 2

×exp

∓1

α(x−ct−ξ0)

+c₀ 3 − γ

3α². (7)

Case 2. a₀ =0. By the transformationw=v+^a₄³, eq. (5) becomes

±1

α(ξ−ξ0)=

(w−^a₄³)dw

w⁴+pw²+qw+r, (8)

where p= −³₈a₃²+a2, q =^a₈³³−^a2₂^a3, r =a0+^a₁₆^2a32−^3a₂₅₆⁴³.Denote F(w)=w⁴+pw²+ qw+r and write its complete discrimination system as follows:

D₁=4, D₂= −p, D₃=8r p−2 p³−9q², D4=4 p⁴r−p³q²+36 pr q²−32r²p²−27

4 q⁴+64r³, E2=9q²−32 pr. (9)

Classification of single travelling wave solutions

Case 2.1. D₄=0,D₃=0,D₂<0. Then F(w)=(w²+l²)², and l>0. Hence D₂= −p= −2l² =3

8a₃²−a₂<0, q =a₃³ 8 −a2a3

2 =0.

So a3=0 or a3= ±4li .

Case 2.1.1. If a3=0, eq. (8) becomes

±1

α(ξ−ξ0)=

wdw

(w²+l²)². (10)

The corresponding solutions are u = ±

exp

±2

α(x−ct−ξ0)

−l²+c 2+ γ

3α² +c0

6. (11)

Case 2.1.2. If a3= ±4li , eq. (8) becomes

±1

α(ξ−ξ0)=

wdw (w²+l²)² ±i

dw

w²+l². (12)

The corresponding solutions are

±1

α(ξ−ξ0)=1 2ln

u−c

2 − γ 3α² −c0

6 2

+l²

±i arctanu−₂^c−_3α^γ2 −^c₆⁰

l . (13)

Case 2.2. D4 = 0,D3 = 0,D2 = 0. Then F(w) = w⁴. This case can be included in Case 1.

Case 2.3. D4 =0,D3 = 0,D2 >0,E2 > 0. Then F(w)= (w−w1)²(w+w1)², and w1>0. It follows that

D2= −p=2w²₁= 3

8a²₃−a2, q= a₃³ 8 −a2a3

2 =0.

Therefore, we have a3=0 or a3= ±4w1.

Case 2.3.1. If a3= ±4w1, then a0=0. It can be included in Case 1.

Case 2.3.2. If a3=0, eq. (8) becomes

±1

α(ξ−ξ0)=

dw w+w1

+w1

dw

(w−w1)(w+w1). (14) The corresponding solutions are

u = ±

w²₁±exp ±2

α(x−ct−ξ0)

+c+ γ

α². (15)

Case 2.4. D₄ =0,D₃ >0,D₂ >0. Then F(w)=(w−w1)²(w−w2)(w−w3), and 2w1+w2+w3=0, w2> w3. Since

p=w2w3−3w²₁= −3

8a₃²+a2, q = −2w1w2w3+2w₁³=a³₃ 8 −a₂a₃

2 we have the following two kinds of solutions:

Case 2.4.1. If a3=4w1, then a0=0. It can be included in Case 1.

Case 2.4.2. If a₃=4w1, eq. (8) becomes

±1

α(ξ−ξ0)=

√ dw

(w−w2)(w−w3) +

w1−a₃ 4

dw (w−w1)√

(w−w2)(w−w3). (16) Whenw1> w2andw > w2, or whenw1< w3andw < w3, we have

±1

α(x−ct−ξ0)

=ln|u+A+w1+

(u+A−w2)(u+A−w3)|

+

w1−c 2+c0

6 − 2γ 3α²

1

√(w1−w2)(w1−w3)

×ln [√

(u+A−w2)(w1−w3)−√

(w1−w2)(u+A−w3)]²

|u+A−w1|

. (17)

Whenw1> w2andw < w3, or whenw1< w3andw < w2, we have

±1

α(x−ct−ξ0)

=ln|u+A+w1+

(u+A−w2)(u+A−w3)|

+

w1−c 2+c0

6 − 2γ 3α²

1

√(w1−w2)(w1−w3)

×ln [√

(u+A−w2)(w3−w1)−√

(w2−w1)(u+A−w3)]²

|u+A−w1|

. (18)

Whenw2> w1> w3, we have

±1

α(x−ct−ξ0)

=ln|u+A+w1+

(u+A−w2)(u+A−w3)|

+

w1−c 2+c0

6 − 2γ 3α²

1

√(w2−w1)(w1−w3)

× arcsin(u+A−w2)(w1−w3)+(u+A−w3)(w1−w2)

|(u+A−w1)(w2−w3)| , (19) where A=₂^c+^c₆⁰ +_3α^γ2.

Classification of single travelling wave solutions

Case 2.5. D₄=0,D₃=0,D₂>0,E₂ =0. Then F(w)=(w−w1)³(w+3w1). Since

p= −6w₁²= −3

8a²₃+a2, q =8w³₁= a₃³ 8 −a2a3

2 , r = −3w₁⁴=a0+a2a²₃ 16 −3a₃⁴

256 we have the following two kinds of solutions:

Case 2.5.1. If a3=4w1, then a0=0. It can be included in Case 1.

Case 2.5.2. If a₃=4w1, eq. (8) becomes

±1

α(ξ−ξ0)=

√ dw

(w−w1)(w+3w1) +

w1−a3

4

dw (w−w1)√

(w−w1)(w+3w1)). (20) The corresponding solutions are

±1

α(x−ct−ξ0) =ln|u+A+w1+

(u+A−w1)(u+A+3w1)|

−(w1−^c₂+^c₆⁰ −_3α^2γ2) 2w1

u+A+3w1

u+A−w1

, (21)

where A=₂^c+^c₆⁰ +_3α^γ2.

Case 2.6. D₂D₃<0,D₄=0. Then F(w)=(w−w1)²[(w+w1)²+s²]. Since

p = −2w²₁+s²= −3

8a₃²+a2, q = −2w1s²= a₃³ 8 −a2a3

2 , r = w⁴₁+w₁²s² =a0+a₂a²₃

16 −3a⁴₃ 256 we have the following two kinds of solutions:

Case 2.6.1. If a3=4w1, then a0=0. It can be included in Case 1.

Case 2.6.2. If a₃=4w1, eq. (8) becomes

±1

α(ξ−ξ0)=

dw (w+w1)²+s² +

w1−a₃ 4

dw (w−w1)

(w+w1)²+s². (22)

The corresponding solutions are

±1

α(x−ct−ξ0)

=ln|u+A+w1+

(w+w1)²+s²| + (w1−₂^c+^c₆⁰ −_3α^2γ2)

4w²₁+s² ln

σ (u+A)+δ−

(w+w1)²+s² u+A−w1

, (23)

where A=₂^c+^c₆⁰ +_3α^γ2, δ= w²₁+s² /

4w₁²+s², σ =3w1/

4w₁²+s².

Case 2.7. D₂ > 0,D₃ > 0,D₄ > 0 or D₂D₃ > 0,D₄ < 0 or D₂D₃ < 0,D₄ > 0, the travelling wave solutions to CH–DP equation can be represented in terms of elliptical functions. For simplicity, they are omitted.

Remark. We must point out that the classification can be realized under the concrete parameters. For simplicity, we also omit them.

3. Conclusion

The complete discrimination system for polynomial method is a powerful method for exact solutions to nonlinear differential equations. By the method, we obtain the classification of single travelling wave solutions to CH–DP equation for some values of the convective parameter. These solutions show rich and varied constructions of the solutions to CH–DP equation.

Acknowledgements

The authors are grateful to the referee for the helpful suggestions. The project is supported by Scientific Research Fund of Education Department of Heilongjiang Province of China under Grant No. 11551020.

References

[1] C S Liu, Commun. Theor. Phys. 48, 601 (2007) [2] C S Liu, Commun. Theor. Phys. 45, 991 (2006) [3] C S Liu, Chin. Phys. 14, 1710 (2005)

[4] C S Liu, Chin. Phys. 16, 1832 (2007)

[5] C S Liu, Commun. Theor. Phys. 49, 153 (2008) [6] C S Liu, Commun. Theor. Phys. 49, 291 (2008) [7] C S Liu, Comput. Phys. Commun. 181, 317 (2010)

[8] D S Wang and H B Li, J. Math. Anal. Appl. 343, 273 (2008) [9] O Cornejo-Pérez and H C Rosu, Prog. Theor. Phys. 114, 533 (2005) [10] H C Rosu and O Cornejo-Pérez, Phys. Rev. E71, 046607 (2005)

[11] O Cornejo-Pérez, J Negro, L M Nieto and H C Rosu, Found. Phys. 36, 1587 (2006) [12] S Yang, Mod. Phys. Lett. B24, 363 (2010)

[13] J B Li and H H Dai, On the study of singular nonlinear travelling wave equations: Dynamical system approach (Science Press, Beijing, 2007)