Traveling Wave Solutions for CH2 Equations

Mathematical Computation June 2015, Volume 4, Issue 2, PP.33-40

Traveling Wave Solutions for CH2 Equations Ye Zhang#, Shengqiang Tang

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin Guangxi 541004, China #

Email: 2554031440@qq.com

Abstract In this paper, we use a method in order to find exact explicit traveling solutions in the subspace of the phase space for CH2 equations. The key idea is removing a coupled relation for the given system so that the new systems can be solved. The existence of solitary wave solutions is obtained. It is shown that bifurcation theory of dynamical systems provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

Keywords: Solitary Wave Solution; Bifurcation Theory; Dynamical Systems; CH2 Equations

1 INTRODUCTION The study on the various physical structures of nonlinear dispersive equations has attracted much attention in connection with the important problems that arise in scientific applications. Mathematically, these physical structures have been studied by using various analytical methods, such as inverse scattering method [1], Darboux transformation method [2,3], Hirota bilinear method [4], Lie group method [5,6], sine-cosine method [7,8], tanh function method [9,10], Fan-expansion method [11,12] and so on. Practically, there is no unified technique that can be employed to handle all types of nonlinear differential equations.

In 2006, Chen et al. [13] presented the following CH2 equations,

0, qt + uqx + 2qu x + ρρ x =  ρt + (u ρ ) x = 0, 

(1.1)

where q =u − u xx + ω with ω being a constant. Chen et al. in [13] based on an inverse scattering transform method, a given function expansion which was applied to Eq. (1.1), a series of periodic solutions, solitary wave solutions and singular solutions are obtained by aid of symbolic computation.

Recently, Holm et al. [14] considered the solitary wave solutions of Eq. (1.1), by using an inverse scattering method.

In this paper, we will discuss the solitons problems on CH2 equations by bifurcation theory of dynamic systems: 1.

Change the partial differential equations into the ordinary differential equations by traveling wave transform; 2.

Draw the phase portraits and analyze them; 3. Solve the equations by the theory of dynamic systems, discuss the type of solitons and make known the conditions of the existence.

2 SOME PRELIMINARY RESULTS In this paper, we assume the limit relation lim ( ρ (ξ ) − ρ 0 ) = 0 , where ρ 0 ≠ 0 is a constant, and consider the ξ →±∞

dynamical bifurcation behavior for the traveling wave solutions of Eq. (1.1). = Let u u= (ξ ), q q= (ξ ), ρ ρ (ξ ), ξ =− x ct , v = v(ξ ) = u (ξ ) − c , where c is the wave speed. We have

u x === u ′, qx q′, qt −cq′, ρt = (u ρ )′, vx = v′, vt = −c ρ ′,(u ρ ) x = −cv′ .

(2.1)

Then Eq. (1.1) become to

0, −cq′ + uq′ + 2qu ′ + ρρ ′ =  −c ρ ′ + (u ρ )′ = 0.  - 33 www.ivypub.org/mc

(2.2)

Namely,

0, vq′ + 2qv′ + ρρ ′ =  0.  v ρ ′ + ρ v′ =

(2.3)

Then we have

ρ=

g1 , v

(2.4)

where g1 is a non-zero integral constant(Otherwise, we have the constant solution that u ≡ c, ρ ≡ ρ 0 ).

Thus, we obtain lim v(ξ )= ξ →±∞

g1

ρ0

, and substitute (2.4) into (2.3), we have:

(vq′ + qv′) + (v + c − v′′ + ω )v′ + ρρ ′ = 0.

(2.5)

Integrating (2.5) once, we have:

(v + c − v′′ + ω )v +

( g )2 v 2 (v′) 2 − + (ω + c)v + 1 2 = g2 , 2 2 2v

(2.6)

where g 2 is an integral constant. Eq. (2.6) is equivalent to the two-dimensional system as follows:

dv  = y,  dξ   2 2  dy = 1  − g + 3 v 2 + 2(ω + c)v + ( g1 ) − y  ,   d ξ v  2 2 2v 2 2 

(2.7)

which has the following first integral

y2 =

g ( g )2 g (v ) =v 2 + 2(ω + c)v − 2 g 2 + 3 − 12 , v v v

(2.8)

where g3 is an integral constant.

Let us solve (2.7) with the following boundary condition

g1

lim v(ξ= ) A=(

ρ0

ξ →±∞

≠ 0) ,

(2.9)

where A is a constant, (2.7) can be cast into the following ordinary differential equation

(v′) 2 =

(v − A) 2 [v 2 + 2( A + ω + c)v − ( ρ 0 ) 2 ] , v2

(2.10)

and

H (v, y ) = (v − A) 2 [v 2 + 2( A + ω + c)v − ( ρ0 ) 2 ] − v 2 y 2 .

(2.11)

For ( A + ω + c) 2 + ( ρ 0 ) 2 > 0 , then (2.10) reduces to

v′ = ±

(v − A) (v − B1 )(v − B2 ) v

,

(2.12)

where

B1 = −( A + ω + c) + ( A + ω + c) 2 + ( ρ 0 ) 2 , B2 = −( A + ω + c ) − ( A + ω + c ) 2 + ( ρ 0 ) 2 .

Obviously, B1 > 0 > B2 .

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(2.13)

Definition 2.1. A function v(ξ ) is said to be a single peak soliton solution for the CH2 (2.3) if v(ξ ) satisfies the following conditions: (A1) v(ξ ) is continuous on R and has a unique peak point ξ 0 where v(ξ ) attains its global maximum or minimum value; (A2) v(ξ ) ∈ C 3 ( R − {ξ 0 }) satisfies (2.9) on R − {ξ 0 } ; (A3) lim v(ξ ) = A . ξ →±∞

Definition 2.2. A wave function v(ξ ) is called smooth soliton if v(ξ ) is smooth locally on either side of ξ 0 and − lim+ v′(ξ ) = lim- v′(ξ ) = 0. ξ →ξ0

ξ →ξ0

Without loss of generality, we assume ξ 0 = 0 .

Lemma 2.3. Suppose that one of the following two conditions holds

( ρ 0 ) 2 − 2 Aω − 3 A2 ,A>0, 2A ( ρ ) 2 − 2 Aω − 3 A2 (2) c > 0 ,A<0.

2A Then (2.3) has trivial solution v ≡ A .

(1) c <

Proof

( ρ 0 ) 2 − 2 Aω − 3 A2 , A > 0 , then we have A − B1 < 0 under the condition v(±∞)= A < B1 , it implies that 2A B2 < 0 < v < B1 and (v′) 2 ≤ 0 , thus v′ = 0 and v ≡ A .

(1) If c <

( ρ 0 ) 2 − 2 Aω − 3 A2 , A < 0 , then we have A − B2 > 0 under the condition v(±∞= ) A > B2 , it implies 2A that B2 < v < 0 < B1 and (v′) 2 ≤ 0 , thus v′ = 0 and v ≡ A .

(2) If c >

Theorem 2.4. Suppose that v(ξ ) is a single peak soliton for (2.3) at the peak point ξ 0 = 0 , then we have v(0) = B1 or v(0) = B2 .

Proof If v(0) ≠ 0 then v(ξ ) ≠ 0 for any ξ ∈ R since v(ξ ) ∈ C 3 ( R − {0}) . Differentiating both sides of (2.12) yields

v(ξ ) ∈ C 3 ( R − {0}) . For ( A + ω + c) 2 + ( ρ 0 ) 2 > 0 , if v(0) ≠ 0 , by (2.9) we know v′ exists. According to the definition of peak point, we obtain v(0) = B1 or v(0) = B2 from (2.12) (If v(0) = 0 , thus we can find a sufficient small neighborhood near zero which B2 < v < B1 , namely, according to (2.12), the derivative does not exist), since the fact v(0) = A contradicts that zero is the unique peak point.

3 EXACT EXPLICIT TRAVELING SOLUTIONS OF SYSTEM (2.3) In this section, we will present all possible soliton solutions for (2.3). By virtue of Theorem 2.4, all single peak soliton solutions for (2.3) must satisfy the following initial and boundary values problem

 (v − A) (v − B1 )(v − B2 ) , v ′ = ± v  v(0) ∈{B1 , B2 },   lim v(ξ ) = A.  ξ →±∞  - 35 www.ivypub.org/mc

(3.1)

In

the

following,

c<

( ρ 0 ) 2 − 2 Aω − 3 A2 ,A<0.

2A

Case (a). c >

let

us

separate

two

cases

to

discuss:

(a)

c>

( ρ 0 ) 2 − 2 Aω − 3 A2 ,A>0 , 2A

(b)

( ρ 0 ) 2 − 2 Aω − 3 A2 ,A>0.

2A

From (2.12) and with the first integral, we have

± ∫ f1 (v)dv = ξ + K1 , where f1 (v) =

v (v − A) (v − B1 )(v − B2 )

(3.2)

and .. is the integration constant.

In this case, according to theorem 2.4, we know that v(0) = B1 or v(0) = B2 . Besides, v ≥ B1 or v ≤ B2 .

For c >

( ρ0 ) 2 − 2 Aω − 3 A2 , A > 0 , we see that A > B1 (> 0 > B2 ) .

2A

In this case, we will only discuss v ≥ B1 (as v(ξ ) is continuous on R ), then we obtain the following implicit representations of breaking wave solutions: v

ξ= ξ1 (v) = ±∫

v f (v)dv B1 1

 A  v − φ (v) − A + φ ( A) ln = ± − ln v − φ (v) + ( A + ω + c)   φ ( A) v − φ (v) − A − φ ( A) B

1

 A  v − φ (v) − A + φ ( A) B − A + φ ( A) =± ln − ln 1  φ ( A)  v − φ (v) − A − φ ( A) B1 − A − φ ( A)  

where φ ( x)=

( x − B1 )( x − B2 )=

 v − φ (v ) + A + ω + c  − ln A + B1 + ω + c 

 ,  

(3.3)

x 2 + 2( A + ω + c) x − ( ρ0 ) 2 .

Employing the above formula to draw the graph of ξ1 (v) , we can conclude that [ B1 , A) of ξ1 (v) has the same meaning function v = v(ξ ) (which means function v = v(ξ ) and ξ = ξ1 (v) have the same functional curve and we can denote v(ξ ) = ξ1−1 (v) ) which satisfies (3.1).

Then we discuss this case:   A  v − φ (v) − A + φ ( A) B − A + φ ( A)  v − φ (v ) + A + ω + c ± − ln 1 ln ξ =  − ln  φ ( A)  v − φ (v) − A − φ ( A)  B1 − A − φ ( A)  A + B1 + ω + c    2 2 ( ρ ) − 2 Aω − 3 A  c> 0 , A > 0, v ∈ [ B1 , A),  2A

let = ξ11 (v)

 ,  

(3.4)

B − A + φ ( A) A v − φ (v) − A + φ ( A) v − φ (v ) + A + ω + c , ξ12 (v) = −ξ11 (v) . (ln − ln 1 ) − ln φ ( A) v − φ (v) − A − φ ( A) B1 − A − φ ( A) A + B1 + ω + c

(1) c >

( ρ0 ) 2 − 2 Aω − 3 A2 ( ρ0 ) 2 − 2 Aω − 3 A2 = ,A>0, , A > 0 , (2) c 2A 2A

(3) c <

( ρ0 ) 2 − 2 Aω − 3 A2 ( ρ ) 2 − 2 Aω − 3 A2 ,A<0, , A > 0 , (4) c > 0 2A 2A

( ρ ) 2 − 2 Aω − 3 A2 ( ρ0 ) 2 − 2 Aω − 3 A2 ,A<0. , A < 0 , (6) c < 0 2A 2A By the standard phase portrait analysis (see Fig. 1-(1), (2.13) and (3.2), we know:

(5) c

1. ξ11′ (v) = f1 (v) and ξ12′ (v) = − f1 (v) . 2. ξ11 (v) ≤ 0 if and only if v = B1 that ξ11 (v) = 0 . 3. ξ12 (v) ≥ 0 if and only if - 36 www.ivypub.org/mc

v = B1 that ξ12 (v) = 0 . 4. For any v ∈ ( B1 , A) , ξ11′ (v) < 0 < ξ12′ (v) . 5. lim+ ξ11′ (v) = −∞ while lim+ ξ12′ (v) = +∞ . v → B1

v → B1

From the definition of inverse function, we denote that v11 (ξ ) = ξ11−1 (v) and v12 (ξ ) = ξ12 −1 (v) , then we have: A. Function v11 (ξ ) and v12 (ξ ) satisfy (3.1). B. The domain of v11 (ξ ) is (−∞, 0] while v12 (ξ ) is [0, +∞) . C.

B1 ≤ v11,12 (ξ ) < A if and only if ξ = 0 that v11,12 (ξ ) = B1 . D. v11′ (ξ ) < 0 for any ξ < 0 and lim v11′ (ξ ) = 0 . E. ξ →0−

v12′ (ξ ) > 0 for any ξ > 0 and lim+ v12′ (ξ ) = 0 . ξ →0

(1)

(2)

(4)

(3)

(5) FIG. 1. THE PHASE PORTRAITS OF EQ. (2.5) ON THE H (v, v

(6) '

)

PLANE

Notice that the new function v11 (ξ ), ξ ∈ (−∞, 0), = v v= B= 0, 1 (ξ )  1,ξ v12 (ξ ), ξ ∈ (0, +∞).

(3.5)

Obviously, we have: a. Function v1 (ξ ) is continuous on R and has a unique peak point ξ0 = 0 , where v1 (ξ ) attains its global minimum value. b. v1 (ξ ) ∈ C 3 ( R − {0}) satisfies (1) c >

( ρ ) 2 − 2 Aω − 3 A2 ( ρ0 ) 2 − 2 Aω − 3 A2 ,A<0. , A > 0 , (2) c < 0 2A 2A

(2.11) on R − {0} . c. lim v1 (ξ ) = A . d. lim v1′ (ξ ) = 0. − lim v1′ (ξ ) = ξ →±∞

ξ →0−

ξ →0+

According to definition 2.1 and 2.2, we say the wave function v = v1 (ξ ) smooth soliton (see Fig 2-(1)).

Case (b). c <

( ρ0 ) 2 − 2 Aω − 3 A2 ,A<0.

2A

From (2.12) and with the first integral, we have

± ∫ f 2 (v)dv = ξ + K2 , - 37 www.ivypub.org/mc

(3.6)

where f 2 (v) =

v (v − A) (v − B1 )(v − B2 )

and K 2 is the integration constant.

In this case, according to theorem 2.4, we know that v(0) = B1 or v(0) = B2 . Besides, v ≥ B1 or v ≤ B2 .

For c <

( ρ0 ) 2 − 2 Aω − 3 A2 , A < 0 , we see that A < B2 (< 0 < B1 ) .

2A

In this case, we will only discuss v ≤ B2 (as v(ξ ) is continuous on R ), then we obtain the following implicit representations of breaking wave solutions: B2

ξ= ξ 2 (v ) = ±∫

B2 v

 A  v − φ (v) − A + φ ( A) f 2 (v)dv = ± ln − ln v − φ (v) + ( A + ω + c)   φ ( A) v − φ (v) − A − φ ( A) v

 A  v − φ (v) − A − φ ( A) B − A − φ ( A) =± ln − ln 2  φ ( A)  v − φ (v) − A + φ ( A) B2 − A + φ ( A)  

 v − φ (v ) + A + ω + c  + ln A + B2 + ω + c 

 ,  

(3.7)

Employing the above formula to draw the graph of ξ 2 (v) , we can conclude that ( A, B2 ] of ξ 2 (v) has the same meaning function v = v(ξ ) which satisfies (3.1).

Then we discuss this case:   A  v − φ (v) − A − φ ( A) B2 − A − φ ( A)  v − φ (v ) + A + ω + c ± ln − ln ξ =   + ln A + B + ω + c  φ ( A)  v − φ (v) − A + φ ( A)  B A ( A ) − + φ 2 2     ( ρ0 )2 − 2 Aω − 3 A2  , A < 0, v ∈ ( A, B2 ], c<  2A

let = ξ 21 (v)

 ,  

(3.8)

B − A − φ ( A) A v − φ (v) − A − φ ( A) v − φ (v ) + A + ω + c , ξ 22 (v) = −ξ 21 (v) . (ln − ln 2 ) + ln φ ( A) v − φ (v) − A + φ ( A) B2 − A + φ ( A) A + B2 + ω + c

(1)

(2)

FIG. 2. THE PROFILES OF WAVES WHICH SATISFY (3.1)

By the standard phase portrait analysis (see Fig. 1-(6)), (2.13) and (3.6), we know: 1. ξ 21′ (v) = f 2 (v) and ξ 22′ (v) = − f 2 (v) . 2. ξ 21 (v) ≤ 0 if and only if v = B2 that ξ 21 (v) = 0 . 3. ξ 22 (v) ≥ 0 if and only if v = B2 that ξ 22 (v) = 0 . 4. For any v ∈ ( A, B2 ) , ξ11′ (v) > 0 > ξ12′ (v) . 5. lim+ ξ 21′ (v) = −∞ while lim+ ξ 22′ (v) = +∞ . v → B2

v → B2

From the definition of inverse function, we denote that v21 (ξ ) = ξ 21−1 (v) and v22 (ξ ) = ξ 22 −1 (v) , then we have: A. Function v21 (ξ ) and v22 (ξ ) satisfy (3.1). B. The domain of v21 (ξ ) is (−∞, 0] while v22 (ξ ) is [0, +∞) . C.

A < v21,22 (ξ ) ≤ B2 if and only if ξ = 0 that v21,22 (ξ ) = B2 . D. v21′ (ξ ) > 0 for any ξ < 0 and lim v21′ (ξ ) = 0 . E. ξ →0−

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v22′ (ξ ) < 0 for any ξ > 0 and lim+ v22′ (ξ ) = 0 . ξ →0

Notice that the new function v21 (ξ ), ξ ∈ (−∞, 0), = v v= B= 0, 2 (ξ )  2 ,ξ v22 (ξ ), ξ ∈ (0, +∞).

(3.9)

Obviously, we have: a. Function v2 (ξ ) is continuous on R and has a unique peak point ξ0 = 0 , where v2 (ξ ) attains its global maximum value. b. v2 (ξ ) ∈ C 3 ( R − {0}) satisfies (2.11)

on R − {0} . c.

lim v2 (ξ ) = A . d.

ξ →±∞

lim v2′ (ξ ) = 0. − lim v2′ (ξ ) =

ξ →0−

ξ →0+

According to definition 2.1 and 2.2, we say the wave function v = v2 (ξ ) smooth soliton (see Fig 2-(2)).

4 CONCLUSION In this paper, we have considered traveling wave solutions for CH2 equations of Eq.(1.1) in its subspace of parameter space by using the method of dynamical systems. We obtain parametric representations for the solitary wave solutions of Eq. (1.1) in different parameter regions of the parameter space.

ACKNOWLEDGEMENTS This work is supported by Innovation Project of Guangxi Graduate Education (Nos. YCSZ2014143). The authors thank the referees and the editors for their valuable comments and suggestions on the improvement of this paper.

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G. W. Bluman, S. Kumei. Symmetries and Differential Equations, Springer-Verlag,Berlin; 1989

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L. Tian, J. Yin. New compacton solutions and solitary wave solutions of fully nonlinear generalized Camassa-Holm equations,

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Chaos Soliton. Fract.2004; 20: 289-299 Simul. 2007, 12: 314-327 [9]

A. M. Wazwaz. A reliable treatment of the physical structure for the nonlinear equation K(m,n), Appl. Math. Comput. 2005; 163: 1081-1095

[10] R. Senthamarai, L. Rajendran. Traveling wave solution of non-linear coupled reaction diffusion equation arising in mathematical chemistry, J. Math. Chem. 2009, 46: 550-561 [11] E. G. Fan. Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos.

Soliton. Fract. 2003, 16: 819-839 [12] D. Feng. G. Luo. The improved Fan sub-equation method and its application to the SK equation, Appl. Math. Comput .2009, 215: 1949-1967 [13] M. Chen. S.Q. Liu. Zhang Y. A two-component generalization of the Camassa-Holm equation and its solutions. Lett. Math. Phys.

75 (2006) 1-15; nlin.SI/0501028 [14] D. D. Holm. R. I. Ivanov Two-component CH system:Inverse Scattering, Peakons and Geometry. Inverse Problems 27 (2011)120; 045013

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AUTHORS Ye Zhang was born in Jiangsu in 1987,

Shengqiang Tang was born in 1951, and

and he is a Master of Science in Guilin

he is a supervisor of postgraduate of

University of Electronic Technology. His

Science in Guilin University of Electronic

major is mathematics in School of

Technology.

Mathematics and Computing Science.

qualitative

also taught English in a technical school for a few months. He

theory

working in

the

fields

are

differential

equations and symbol computation for

Mr. .Zhang worked as a businessman in a food factory from 2009 to 2012, and he

His

computers in School of Mathematics and Computing Science.

published his first article <Traveling Wave Solutions for a

Prof. Tang has lots of qualifications in his research area. He has

Coupled KdV Equations> in periodicalâ&#x20AC;?British Journal of

published hundreds of articles in all kinds of periodicals. Also,

Mathematics & Computer Scienceâ&#x20AC;? in 2014.

he has won several prizes in teaching achievements and

Mr. .Zhang also won the first prize in the competition of Mathematics Model Building of Guangxi Province in 2013.

excellent teachers. More, he has done quite a mount of researching projects.

- 40 www.ivypub.org/mc