Travelling Wave Solutions of Nonlinear Dynamical Equations in a Double-Chain Model of DNA

and Applied Analysis 3


Introduction
In 1953, Waston and Crick discovered the structure of deoxyribonucleic acid (DNA) double helix [1], which opened the area of molecular biology.The significance of the double helix model not only means the proven structure of DNA, but, more importantly, also helps to reveal the replication mechanism of DNA.To further study the characteristics of DNA, nonlinear science has been used to deal with DNA system, because its properties can be investigated by nonlinear models that combine the methods of physics with biological tools [2].The soliton theory especially which pertains to nonlinear science has been widely applied in the study of DNA [3][4][5][6].
To study DNA structure from the view of nonlinear science, it is necessary to look for the right nonlinear mathematical models.A number of researchers have tried to establish mathematical models to describe DNA system.At the beginning, it is difficult to use a specific mathematical model to simulate DNA system due to its complex structure and the presence of various movements [7].So some simplified models were used to study only some internal movement in DNA [8][9][10][11][12].Further, two movements which made the main contribution to the DNA denaturation process were taken into account together in one model.For example, a two-dimensional discrete model was used to describe the two movements in DNA [13].Another new double-chain model of DNA that consists of two long elastic homogeneous strands was given to describe transverse movements along the hydrogen bond and longitudinal movements along the backbone direction [14,15].
In [14,15], the nonlinear dynamical equations were derived and nonlinear dynamics of DNA were studied.The exact solutions of the general nonlinear dynamic system in the double-chain model of DNA were obtained and discussed numerically and analytically under some special approximate conditions.The nonlinear dynamical equations for the double-chain model were given as follows: where  denotes the difference of the longitudinal displacements of the bottom and top strands, that is, the displacements of the bases from their equilibrium positions along the direction of the phosphodiester bridge that connects the two bases of the same strands; V represents the difference of the transverse displacements of the bottom and top strands, that is, the displacements of the bases from their equilibrium positions along the direction of the hydrogen bond that connects the two bases of the base pair.The constants  1 ,  2 ,  1 ,  2 ,  1 ,  2 ,  1 ,  2 ,  1 ,  2 , and  0 in (1) are written as where , , , , ,  0 , and ℎ are, respectively, the mass density, the area of transverse cross-section, the Young modulus, the tension density of the strand, the rigidity of the elastic membrane, the distance of the two strands, and the height of the membrane in the equilibrium position.
In fact, system ( 6) is a planar dynamical system whose phase orbits defined by the vector field of (6) determine traveling wave solutions, so we first need to investigate some different phase portraits of system (6) in different parameter regions.Then, by using the theory of dynamical system, we will identify some different wave profiles and compute exact representations for solitary wave solutions and periodic wave solutions.
This paper is organized as follows.In Section 2, we draw bifurcation phase portraits of system (6), where explicit parametric conditions will be derived.In Section 3, we give exact representations of solitary wave and periodic wave solutions of ( 6) in explicit form.A short conclusion will be given in Section 4.

Properties of Singular Points and Bifurcation Phase Portraits
Firstly, we compute the equilibrium points of system (6) under different parametric conditions.It is clear that neither the parameters  nor  takes value zero from the forms of parameters , , and ; then  equals zero if  = ± √ 2/2.So we give equilibrium points in different parametric cases as follows.
Secondly, we compute the intersection points of phase portraits of (6) on the -axis when the orbits pass through the equilibrium point (0, 0), namely, when the Hamiltonian constant  = ℎ(0, 0 /, and  2 = ((−2/3) + √ 2 )/.When  = 0, there are a double zero point 0 and a zero point  0 of () on the axis.When  ̸ = 0 and  2 > 0, there are a double zero point 0 and two zero points  1 and  2 of () on the -axis.In order to draw bifurcation phase portraits of system (6), we need to discuss the sign of parameters , , and .The intersection points of phase portraits of (6) on the -axis are given as follows.
Similarly, in Figure 1(c) the homoclinic orbit has the following expression: Thus the corresponding smooth solitary wave solution is obtained as (10) and its profile is shown in Figure 4(b).
In this case, on the  −  plane the homoclinic orbit of (6) passing through points ( 2 , 0) and (0, 0) has the following expression: Choosing (0) =  2 , substituting (12) into the first equation of ( 6), and integrating along the homoclinic orbit, we have Solving (13), it follows that where The profiles of solutions  1 and  2 both have figure caption Figure 4(a).
In this case, on the  −  plane the homoclinic orbit of (6) passing through points ( 1 , 0) and (0, 0) has the following expression: Choosing (0) =  1 , substituting (15) into the first equation of ( 6), and integrating along the homoclinic orbit, we have Solving ( 16), we get where . The profile of  3 is shown in Figure 4(d).
In this case, there exist two homoclinic orbits and their expressions are (12) and (15), respectively.Similarly we get the corresponding solitary wave solutions as  2 and  3 , which coexist for the same speed.The plane graphs of the coexisting solitary wave solutions are shown in Figure 4(c), when  < 0,  > 0, and  > 0, and Figure 4(d), when  < 0,  < 0, and  > 0.
In this case, on the − plane the orbit of (6) which passes through points ( 1 , 0) and ( 2 , 0) is a periodic orbit; it has the following expression: Choosing (0) =  1 , substituting (18) into the first equation of ( 6), and integrating along the periodic orbit, we have Solving (19) we obtain the following periodic wave solution: The profile of  4 is shown in Figure 4(e).
Similarly, corresponding to the periodic orbit of (6) which passes through points ( 1 , 0) and ( 2 , 0), there exists a periodic wave solution.Now, the expression of the periodic orbit is The profile of  5 is shown in Figure 4(f).

Conclusion
In this paper, we employ the method of dynamical systems to study the nonlinear dynamical equations of DNA model.The bifurcation phase portraits of the DNA under some parametric conditions are drawn, and explicit exact expressions of bell-shaped solitary waves and periodic waves are obtained via some special homoclinic and periodic orbits; their planar graphs are simulated.These solutions are new and different from those in [14][15][16]; some existing results are improved.Furthermore, in general, the larger the wave speed, the higher the wave crest.So for the same wave speed, it is interesting that the bell-shaped solitary waves of different heights are found to coexist.