Research Article Travelling Wave Solutions of Nonlinear ...such as bell-shaped solitary waves and periodic waves for the coupled nonlinear dynamical equations of DNA model are obtained

Research ArticleTravelling Wave Solutions of Nonlinear Dynamical Equations ina Double-Chain Model of DNA

Zheng-yong Ouyang1 and Shan Zheng2

1 Department of Mathematics, Foshan University, Foshan Guangdong 528000, China2Guangzhou Maritime College, Guangzhou Guangdong 510725, China

Correspondence should be addressed to Zheng-yong Ouyang; zyouyang [email protected]

Received 6 January 2014; Revised 24 February 2014; Accepted 10 March 2014; Published 3 April 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Z.-y. Ouyang and S. Zheng. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We consider the nonlinear dynamics in a double-chain model of DNA which consists of two long elastic homogeneous strandsconnectedwith each other by an elasticmembrane. By using themethodof dynamical systems, the bounded travelingwave solutionssuch as bell-shaped solitary waves and periodic waves for the coupled nonlinear dynamical equations of DNA model are obtainedand simulated numerically. For the same wave speed, bell-shaped solitary waves of different heights are found to coexist.

1. Introduction

In 1953, Waston and Crick discovered the structure ofdeoxyribonucleic acid (DNA) double helix [1], which openedthe area of molecular biology. The significance of the doublehelix model not only means the proven structure of DNA,but, more importantly, also helps to reveal the replicationmechanism of DNA. To further study the characteristicsof DNA, nonlinear science has been used to deal withDNA system, because its properties can be investigated bynonlinear models that combine the methods of physics withbiological tools [2]. The soliton theory especially whichpertains to nonlinear science has been widely applied in thestudy of DNA [3–6].

To study DNA structure from the view of nonlinearscience, it is necessary to look for the right nonlinearmathematical models. A number of researchers have tried toestablish mathematical models to describe DNA system. Atthe beginning, it is difficult to use a specific mathematicalmodel to simulate DNA system due to its complex structureand the presence of various movements [7]. So some simpli-fiedmodels were used to study only some internal movementin DNA [8–12]. Further, two movements which made themain contribution to the DNA denaturation process weretaken into account together in one model. For example, atwo-dimensional discrete model was used to describe the

two movements in DNA [13]. Another new double-chainmodel of DNA that consists of two long elastic homogeneousstrands was given to describe transverse movements alongthe hydrogen bond and longitudinal movements along thebackbone direction [14, 15].

In [14, 15], the nonlinear dynamical equations werederived and nonlinear dynamics of DNA were studied. Theexact solutions of the general nonlinear dynamic systemin the double-chain model of DNA were obtained anddiscussed numerically and analytically under some specialapproximate conditions. The nonlinear dynamical equationsfor the double-chain model were given as follows:

𝑢𝑡𝑡− 𝑐2

1𝑢𝑥𝑥

= 𝜆1𝑢 + 𝛾1𝑢V + 𝜇

1𝑢3+ 𝛽1𝑢V3,

V𝑡𝑡− 𝑐2

2V𝑥𝑥

= 𝜆2V + 𝛾2𝑢2+ 𝜇2𝑢2V + 𝛽

2V3 + 𝑐0,

(1)

where 𝑢 denotes the difference of the longitudinal displace-ments of the bottom and top strands, that is, the displace-ments of the bases from their equilibrium positions alongthe direction of the phosphodiester bridge that connects thetwo bases of the same strands; V represents the difference ofthe transverse displacements of the bottom and top strands,that is, the displacements of the bases from their equilibriumpositions along the direction of the hydrogen bond that

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 317543, 5 pageshttp://dx.doi.org/10.1155/2014/317543

2 Abstract and Applied Analysis

connects the two bases of the base pair. The constants 𝑐1, 𝑐2,

𝜆1, 𝜆2, 𝛾1, 𝛾2, 𝜇1, 𝜇2, 𝛽1, 𝛽2, and 𝑐

0in (1) are written as

𝑐1= ±√

𝑌

𝜌

, 𝑐2= ±√

𝐹

𝜌

, 𝜆1=

−2𝜇

𝜌𝜎ℎ

(ℎ − 𝑙0) ,

𝜆2=

−2𝜇

𝜌𝜎

, 𝛾1= 2𝛾2=

2√2𝜇𝑙0

𝜌𝜎ℎ2

, 𝜇1= 𝜇2=

−2𝜇𝑙0

𝜌𝜎ℎ3,

𝛽1= 𝛽2=

4𝜇𝑙0

𝜌𝜎ℎ3, 𝑐

0=

√2𝜇 (ℎ − 𝑙0)

𝜌𝜎ℎ

,

(2)

where 𝜌,𝜎,𝑌,𝐹, 𝜇, 𝑙0, and ℎ are, respectively, themass density,

the area of transverse cross-section, the Young modulus,the tension density of the strand, the rigidity of the elasticmembrane, the distance of the two strands, and the height ofthe membrane in the equilibrium position.

In [16], introducing the transformation V = 𝑎𝑢 + 𝑏 (𝑎 and𝑏 are constants) in (1) yields


1𝑢𝑥𝑥

= 𝑢3(𝜇1+ 𝛽1𝑎2) + 𝑢2(2𝛽1𝑎𝑏 + 𝑎𝛾

1)

+ 𝑢 (𝜆1+ 𝑏𝛾1+ 𝛽1𝑏2) ,


2𝑢𝑥𝑥

= 𝑢3(𝜇2+ 𝛽2𝑎2) + 𝑢2(

𝛾2

𝑎

+

𝜇2𝑏

𝑎

+ 3𝛽2𝑎𝑏)

+ 𝑢 (𝜆2+ 3𝛽2𝑏2) +

𝜆2𝑏

𝑎

+

𝛽2𝑏3

𝑎

+

𝑐0

𝑎

.

(3)

Comparing (3) and using the physical quantities (2), it followsthat 𝑏 = ℎ/√2 and 𝐹 = 𝑌; (3) can be written as


1𝑢𝑥𝑥

= 𝐴𝑢3+ 𝐵𝑢2+ 𝐶𝑢 + 𝐷, (4)

where 𝐴 = ((−2𝛼/ℎ3) + (4𝑎2𝛼/ℎ3)), 𝐵 = 6√2𝑎𝛼/ℎ2, 𝐶 =((−2𝛼/𝑙

0) + (6𝛼/ℎ)), and𝐷 = 0 with 𝛼 = 𝜇𝑙

0/𝜌𝜎.

Reference [16] studied the traveling wave solutions of(4) and obtained one particular family of solitary kink-type solutions for different values of Riccati parameter of(4). Unfortunately the results are not complete, because (4)admits tanh-type, periodic-type, and bell-type solutions; [16]only presented the tanh-type solutions.

In this paper, we use the bifurcationmethod of dynamicalsystems [17–20] to study bell-type and periodic-type solu-tions of (6), which can improve the results in [16]. We lookfor solutions of (4) in the form of 𝑢(𝑥, 𝑡) = 𝜑(𝜉) with 𝜉 =(𝑥/𝑐1) − √2𝑡. Substituting 𝑢(𝑥, 𝑡) = 𝜑((𝑥/𝑐

1) − √2𝑡) into (4),

we have

𝜑𝜉𝜉

= 𝐴𝜑3+ 𝐵𝜑2+ 𝐶𝜑, (5)

which is equivalent to the following two-dimensional system:𝑑𝜑

𝑑𝜉

= 𝑦,

𝑑𝑦

𝑑𝜉

= 𝐴𝜑3+ 𝐵𝜑2+ 𝐶𝜑. (6)

Obviously, system (6) has the first integral

𝐻(𝜑, 𝑦) =

1

4

𝐴𝜑4+

1

3

𝐵𝜑3+

1

2

𝐶𝜑2−

1

2

𝑦2= 𝑀, (7)

where𝑀 is the Hamiltonian constant.

In fact, system (6) is a planar dynamical system whosephase orbits defined by the vector field of (6) determinetraveling wave solutions, so we first need to investigate somedifferent phase portraits of system (6) in different parameterregions. Then, by using the theory of dynamical system, wewill identify some different wave profiles and compute exactrepresentations for solitary wave solutions and periodic wavesolutions.

This paper is organized as follows. In Section 2, we drawbifurcation phase portraits of system (6), where explicitparametric conditions will be derived. In Section 3, we giveexact representations of solitary wave and periodic wavesolutions of (6) in explicit form. A short conclusion will begiven in Section 4.

2. Properties of Singular Points andBifurcation Phase Portraits

Firstly, we compute the equilibrium points of system (6)under different parametric conditions. It is clear that neitherthe parameters 𝐵 nor 𝐶 takes value zero from the forms ofparameters 𝐴, 𝐵, and 𝐶; then 𝐴 equals zero if 𝑎 = ±√2/2.So we give equilibrium points in different parametric cases asfollows.

(1) When 𝐴 = 0, there are two equilibrium points (0, 0)and (−𝐶/𝐵, 0).

(2) When 𝐴 ̸= 0 and 𝛿1> 0, there are three equilibrium

points (0, 0), (𝜑−, 0), and (𝜑

+, 0) of (6) on the 𝜑-axis,

where 𝛿1

= 𝐵2− 4𝐴𝐶, 𝜑

−= (−𝐵 − √𝛿

1)/2𝐴, and

𝜑+= (−𝐵 + √𝛿

1)/2𝐴.

Define 𝑓0(𝜑) = 𝐴𝜑

3+ 𝐵𝜑2+ 𝐶𝜑; then the type of

equilibrium point (𝜑, 0) can be decided by 𝜆±

= √𝑓

0(𝜑),

which are eigenvalues of linearized system of (6) at point(𝜑, 0).

Secondly, we compute the intersection points of phaseportraits of (6) on the 𝜑-axis when the orbits pass throughthe equilibrium point (0, 0), namely, when the Hamiltonianconstant𝑀 = ℎ(0, 0) = 0. Let 𝑓(𝜑) = (1/2)𝐴𝜑4 + (2/3)𝐵𝜑3 +𝐶𝜑2, 𝛿2= (4/9)𝐵

2− 2𝐴𝐶, 𝜑

0= −3𝐴/2𝐵, 𝜑

1= ((−2/3)𝐵 −

√𝛿2)/𝐴, and 𝜑

2= ((−2/3)𝐵+√𝛿

2)/𝐴. When𝐴 = 0, there are

a double zero point 0 and a zero point 𝜑0of 𝑓(𝜑) on the 𝜑-

axis. When 𝐴 ̸= 0 and 𝛿2> 0, there are a double zero point 0

and two zero points 𝜑1and 𝜑

2of 𝑓(𝜑) on the 𝜑-axis. In order

to draw bifurcation phase portraits of system (6), we need todiscuss the sign of parameters 𝐴, 𝐵, and 𝐶. The intersectionpoints of phase portraits of (6) on the 𝜑-axis are given asfollows.

(1) If 𝐴 = 0, there are two intersection points (0, 0) and(𝜑0, 0) of (6) on the 𝜑-axis (see Figure 1).

(2) If 𝐴 > 0, there are three intersection points (0, 0),(𝜑1, 0), and (𝜑

2, 0) of (6) on the 𝜑-axis and 𝜑

1< 𝜑2

(see Figure 2).(3) If 𝐴 < 0, there are three intersection points (0, 0),

(𝜑1, 0), and (𝜑

2, 0) of (6) on the 𝜑-axis and 𝜑

2< 𝜑1

(see Figure 3).

Abstract and Applied Analysis 3

y

0 𝜑𝜑0

(a) 𝐵 > 0, 𝐶 > 0

y

0 𝜑𝜑0

(b) 𝐵 > 0, 𝐶 < 0

y

0 𝜑𝜑0

(c) 𝐵 < 0, 𝐶 > 0

y

0 𝜑𝜑0

(d) 𝐵 < 0, 𝐶 < 0

Figure 1: The bifurcation phase portraits of system (6) with 𝐴 = 0.

y

0 𝜑𝜑1 𝜑2

(a) 𝐵 > 0, 𝐶 > 0

y

0 𝜑𝜑1 𝜑2

(b) 𝐵 > 0, 𝐶 < 0

y

0 𝜑𝜑1 𝜑2

(c) 𝐵 < 0, 𝐶 > 0

y

0 𝜑𝜑1 𝜑2

(d) 𝐵 < 0, 𝐶 < 0

Figure 2: The bifurcation phase portraits of system (6) with 𝐴 > 0.

y

0 𝜑𝜑1𝜑2

(a) 𝐵 > 0, 𝐶 > 0

y

0 𝜑𝜑1𝜑2

(b) 𝐵 > 0, 𝐶 < 0

y

0 𝜑𝜑1𝜑2

(c) 𝐵 < 0, 𝐶 > 0

y

0 𝜑𝜑1𝜑2

(d) 𝐵 < 0, 𝐶 < 0

Figure 3: The bifurcation phase portraits of system (6) with 𝐴 < 0.

Finally, from the above analysis about system (6), we givetwelve different phase portraits of (6) which are shown inFigures 1–3.There exist homoclinic orbits and periodic orbitsin some phase portraits of Figures 1–3, which correspond tobell-type and periodic-type waves and are of special interestto us.

3. Exact Explicit Representations ofSolitary Wave Solutions

(1) The case 𝐴 = 0, 𝐵 > 0, and 𝐶 > 0 (see Figure 1(a)).

Corresponding to the homoclinic orbit of (6) passingthrough points (𝜑

0, 0) and (0, 0) on the𝜑−𝑦 plane, there exists

a smooth solitary wave solution. Now, (7) becomes

𝑦2=

2

3

𝐵𝜑3+ 𝐶𝜑2

(𝜑0⩽ 𝜑 ⩽ 0) . (8)

Choosing 𝜑(0) = 𝜑0, substituting (8) into the first equation of

(6), and integrating along the homoclinic orbit, we have

∫

𝜑0

𝜑

1

𝑠√(2/3) 𝐵𝑠 + 𝐶

𝑑𝑠 =𝜉. (9)

Solving (9) yields

𝑢1= −

3𝐶

𝐵 (cosh (√𝐶 𝜉) + 1)

. (10)

(2) The case 𝐴 = 0, 𝐵 < 0, and 𝐶 > 0 (see Figure 1(c)).

Similarly, in Figure 1(c) the homoclinic orbit has thefollowing expression:

𝑦2=

2

3


(0 ⩽ 𝜑 ⩽ 𝜑0) . (11)

Thus the corresponding smooth solitary wave solution isobtained as (10) and its profile is shown in Figure 4(b).

(3) The case 𝐴 > 0, 𝐵 > 0, and 𝐶 > 0 (see Figure 2(a)).

In this case, on the 𝜑 − 𝑦 plane the homoclinic orbit of(6) passing through points (𝜑

2, 0) and (0, 0) has the following

expression:

𝑦2=

1

2

𝐴𝜑4+

2

3


(𝜑2⩽ 𝜑 ⩽ 0) . (12)

4 Abstract and Applied Analysis

0

𝜑

𝜉

(a)

0

𝜑

𝜉

(b)

u2

u3

0

𝜑

𝜉

(c)

u2

u3

0

𝜑

𝜉

(d)

0

𝜑

𝜉

(e)

0

𝜑

𝜉

(f)

Figure 4: Smooth solitary wave and periodic wave solutions.

Choosing 𝜑(0) = 𝜑2, substituting (12) into the first equation

of (6), and integrating along the homoclinic orbit, we have

∫

𝜑2

𝜑

1

𝑠√(1/2) 𝐴𝑠2+ (2/3) 𝐵𝑠 + 𝐶

𝑑𝑠 =𝜉. (13)

Solving (13), it follows that

𝑢2=

4𝐶𝛼1

𝛼2

1𝑒−√𝐶|𝜉|

+ ((4/9) 𝐵2− 2𝐴𝐶) 𝑒

√𝐶|𝜉|− (4/3) 𝐵𝛼

1

, (14)

where 𝛼1

= (2(−2𝐵2+ 9𝐴𝐶 + 𝐵√4𝐵

2− 18𝐴𝐶))/(−6𝐵 +

3√4𝐵2− 18𝐴𝐶).The profiles of solutions 𝑢

1and 𝑢2both have

figure caption Figure 4(a).

(4) The case 𝐴 > 0, 𝐵 < 0, and 𝐶 > 0 (see Figure 2(c)).

In this case, on the 𝜑 − 𝑦 plane the homoclinic orbit of(6) passing through points (𝜑

1, 0) and (0, 0) has the following

expression:

𝑦2=

1

2

𝐴𝜑4+

2

3


(0 ⩽ 𝜑 ⩽ 𝜑1) . (15)


of (6), and integrating along the homoclinic orbit, we have

∫

𝜑1

𝜑

1

𝑠√(1/2) 𝐴𝑠2+ (2/3) 𝐵𝑠 + 𝐶

𝑑𝑠 =𝜉. (16)

Solving (16), we get

𝑢3=

4𝐶𝛼2

𝛼2

2𝑒√𝐶|𝜉|

+ ((4/9) 𝐵2− 2𝐴𝐶) 𝑒

−√𝐶|𝜉|− (4/3) 𝐵𝛼

2

, (17)

where 𝛼2

= (2(2𝐵2

− 9𝐴𝐶 + 𝐵√4𝐵2− 18𝐴𝐶))/(6𝐵 +

3√4𝐵2− 18𝐴𝐶). The profile of 𝑢

3is shown in Figure 4(d).

(5) The case 𝐴 < 0, 𝐵 > 0, and 𝐶 > 0 and 𝐴 < 0, 𝐵 < 0,and 𝐶 > 0 (see Figures 3(a) and 3(c)).

In this case, there exist two homoclinic orbits and theirexpressions are (12) and (15), respectively. Similarly we getthe corresponding solitary wave solutions as 𝑢

2and 𝑢3, which

coexist for the same speed.The plane graphs of the coexistingsolitary wave solutions are shown in Figure 4(c), when𝐴 < 0,𝐵 > 0, and 𝐶 > 0, and Figure 4(d), when 𝐴 < 0, 𝐵 < 0, and𝐶 > 0.

(6) The case 𝐴 < 0, 𝐵 > 0, and 𝐶 < 0 (see Figure 3(b)).

In this case, on the𝜑−𝑦plane the orbit of (6)which passesthrough points (𝜑

1, 0) and (𝜑

2, 0) is a periodic orbit; it has the

following expression:

𝑦2=

1

2

𝐴𝜑4+

2

3

𝐵𝜑3+ 𝐶𝜑2, (0 < 𝜑

2⩽ 𝜑 ⩽ 𝜑

1) . (18)


of (6), and integrating along the periodic orbit, we have

∫

𝜑1

𝜑

1

𝑠√(1/2) 𝐴𝑠2+ (2/3) 𝐵𝑠 + 𝐶

𝑑𝑠 =𝜉. (19)

Solving (19) we obtain the following periodic wave solution:

𝑢4=

6𝐶

√4𝐵2− 18𝐴𝐶 cos (√−𝐶𝜉) − 2𝐵

. (20)

The profile of 𝑢4is shown in Figure 4(e).

Abstract and Applied Analysis 5

(7) The case 𝐴 < 0, 𝐵 < 0, and 𝐶 < 0 (see Figure 3(d)).

Similarly, corresponding to the periodic orbit of (6)which passes through points (𝜑

1, 0) and (𝜑

2, 0), there exists

a periodic wave solution. Now, the expression of the periodicorbit is

𝑦2=

1

2

𝐴𝜑4+

2

3

𝐵𝜑3+ 𝐶𝜑2, (𝜑

2⩽ 𝜑 ⩽ 𝜑

1< 0) . (21)


of (6), and integrating along the periodic orbit, we have

∫

𝜑

𝜑2

1

𝑠√(1/2) 𝐴𝑠2+ (2/3) 𝐵𝑠 + 𝐶

𝑑𝑠 =𝜉. (22)

Solving (22), it follows that

𝑢5=

6𝐶

−√4𝐵2− 18𝐴𝐶 cos (√−𝐶𝜉) − 2𝐵

. (23)

The profile of 𝑢5is shown in Figure 4(f).

4. Conclusion

In this paper, we employ the method of dynamical systemsto study the nonlinear dynamical equations of DNA model.The bifurcation phase portraits of the DNA under some para-metric conditions are drawn, and explicit exact expressions ofbell-shaped solitary waves and periodic waves are obtainedvia some special homoclinic and periodic orbits; their planargraphs are simulated. These solutions are new and differentfrom those in [14–16]; some existing results are improved.Furthermore, in general, the larger the wave speed, the higherthe wave crest. So for the same wave speed, it is interestingthat the bell-shaped solitary waves of different heights arefound to coexist.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (no. 11226303) and GuangdongProvince (no. 2013KJCX0189). The authors thank the editorsfor their hard working and also gratefully acknowledgehelpful comments and suggestions by reviewers.

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Research Article Travelling Wave Solutions of Nonlinear ...such as bell-shaped solitary waves and periodic waves for the coupled nonlinear dynamical equations of DNA model are obtained