Bifurcation Analysis and Exact Wave Solutions for the Double-Chain Model of DNA

This work aims to study analytically the nonlinear model for deoxyribonucleic acid (DNA). Based on the complete discrimination and direct method, some new wave solutions are introduced. These solutions are sorted into solitary, periodic, kink (antikink), and singular solutions. Moreover, a part of them is illustrated graphically. Based on Hamilton concepts, we study the bifurcation and phase portrait for the Hamilton system corresponding to the model under consideration.


Introduction
Deoxyribonucleic acid (DNA) molecule carries the information for living beings that is required to live and propagate themselves. e nonlinear model for deoxyribonucleic acid (DNA) is attractive for the study because its properties can be investigated precisely by experiments gathering both the physical methods and biological tools [1]. In 1953, Watson and Crick [2] initially discovered the double helix construction of DNA; in spite of that, it is not easy until now to nd a speci c mathematical model that involved all its characteristics. e reason is its complicated structure and the existence of several motions such as the torsional, transverse, and longitudinal motions [3]. However, the existence of several motions of the DNA on thoroughly distinct time scales becomes aidable to model a few of these that predominate in the given time scale range. e idea that was introduced by Davydov [4] in his pioneer works related to the theoretical studies of the nonlinear characteristics of DNA has been rstly utilized by Englander and coauthors in 1980 to investigate dynamics of DNA regarding the nitrogen rational motion [5]. is idea has been further developed in several subsequent works. For instance, Yomosa proposed a model of the dynamics plane of the base rotator [6], and this study was followed by Takeno and Homma who got better this model by considering the degree of freedom describing base rotations in the plane perpendicular to the helical axis around the structure of the backbone [7]. e denaturation process in which the base transverse motion along the hydrogen bond is regarded was investigated by Peyrard and Bishop [8]. Two kinds of internal motions, which have been proposed by Muto et al., contribute mainly to the denaturation process of DNA. ese motions are longitudinal motions over the backbone and transverse motions over the hydrogen bond [9]. is model has been developed and improved in several works in order to study distinct motions and construct solitary wave-type solutions [10][11][12][13][14]. ese waves acquire their signi cance from their ability to transmit energy without losing, i.e., the energy is conserved [8,15], and moreover, they explicate the long-range interaction of kink solitons in the double chain [16,17] and transcription regulation [18].
Taking into account some acceptable approximations from the point of biological science, the two equations describe a DNA model with double chains consisting of elastic two long homogeneous strands. ese strands characterize two polynucleotide chains of DNA molecules attached by an elastic membrane which represents the hydrogen bonds between the base pair of two chains. e dynamical nonlinear system characterizing the double-chain model of DNA takes the form [19,20] I tt − e 2 1 I xx � a 1 where I refers to the longitudinal displacement difference between the top and bottom wires, while J indicates the transverse displacements between the upper and lower strands, and e i , a i , b i , c i , d i , and m 0 , i � 1, 2, are constants given by where ρ, σ, R 1 , and R 2 refer to density of mass, area of cross section, Young's modulus, and density of the tension of each strand, μ indicates the stiffness of the elastic membrane, h is the distance between the two strands, and l 0 is the height of membrane in the equilibrium.
To transform the nonlinear system (1) into a single partial differential equation, we present where α and β are two arbitrary constants, and furthermore, we assume β � h/ � 2 √ and R 1 � R 2 . us, the linear system (1) is reduced to where f i are arbitrary constants introduced for suitability, and they are given by e nonlinear model (1) has been investigated in several works. Riccati parameterized factorization method has been applied in [20] to construct some solitary wave solutions for the DNA model (1). e Φ 6 − model expansion method has been utilized in [21] to construct some solutions which are assorted into solitary, kink, and singular waves. Some solitary wave solutions for the double-chain model of DNA have been introduced and discussed [19]. Some exact wave solutions of this model have been constructed by using Conte's Painlevé truncation expansion and Pickering's truncation expansion [22]. Some bounded wave solutions for this model such as bell-shaped solitary waves and periodic waves have been formulated based on the method of the dynamical systems [23]. e generalized exponential rational function method has been applied to introduce exact form solutions and solitonic structures for this model [24].
In this work, we are interested in constructing some traveling wave solutions for the nonlinear model (1) which is equivalent to building a wave solution for the reduced equation (1). We apply the complete discriminant system in addition to determining the intervals of permitted real propagations. e significance of finding these intervals enables us to construct only real wave solutions, and furthermore, for the same constraints on the system's parameters, there are several intervals of possible real wave propagations. Hence, the missing of such study in previous works leads to missing some wave solutions and the appearance of complex solutions. e bifurcation analysis is introduced which plays an important role in determining the types of the solutions before constructing them. We are also interested in studying the influence of the system's parameters on the solutions. is work is organized as follows: Section 2 involves the reduction of the DNA model to ordinary differential equations and using the complete discrimination method to construct some traveling wave solutions for equation (1). Section 3 contains the study of some dynamical properties of equation (1) by employing the complete discrimination and Hamiltonian concepts. Section 4 is a graphic representation of some of the obtained solutions. Furthermore, it examines the influence of the one-parameter changing on the solutions keeping the other parameters fixed. Section 4 is a collection and the summary of the obtained results.

Exact Wave Solutions
Applying the wave transformation I(x, t) � u(ζ), ζ � kx − ωt, to equation (4), we obtain where k is a constant that specifies the cosine of angle of propagation with ζ− axis and ω is an arbitrary constant that characterizes the speed of the wave and ω 2 − k 2 e 2 1 ≠ 0. For simplicity, we insert into equation (6), and we get where ′ indicates derivatives with respect to ζ and n i , i � 2, 3, 4, are constants introduced for suitability and they are given by Integrating both sides of equation (8) with respect to p, we obtain where c i are arbitrary parameters. Separating the variables, we obtain the differential form where in which To integrate both sides of equation (11), the range of the parameters is required to be determined. e cause for this is that different values of the parameters imply different solutions to the integral. Hence, the key steps are to find the range of these parameters and consequently integrate both sides of equation (11). ere are many tools utilized to find these ranges of parameters. In this work, we will apply a complete discrimination system for a polynomial. is method is a natural generalization of the discrimination Δ � b 2 − 4ac for the quadratic polynomial ax 2 + bx + c, but it becomes difficult to calculate it for the higher degree polynomials.
is problem had been solved with aid of computer algebra programs by Yang et al. by presenting an algorithm to compute the complete discrimination system for polynomial [54]. e complete discrimination system for the quartic polynomial F 4 (p) � p 4 + c 2 p 2 + c 1 p + c 0 is given in [55], and it admits the form We study eight cases that describe different types of the roots for polynomial (12). To avoid confounding, we collect the classification of all the different types of the roots of the polynomial F 4 (p) by utilizing the discriminant system in Table 1. Furthermore, we integrate only on certain intervals for p in which n 4 F 4 (p) is positive in order to get real solutions.

Journal of Mathematics
then the quartic polynomial (12) has two real roots: one is simple and the other is triple.
Hence, it can be expressed as where p 1 is assumed to be positive, i.e., p 1 > 0. We consider two subcases according to n 4 is positive or negative: and integrate both sides of equation (11), we obtain It follows us, the solution of equation (1) becomes Similarly, we can calculate the solution if (ii) If n 4 < 0, the possible interval for p to obtain real propagation is p ∈ ] − 3p 1 , p 1 [. us, we assume p(ζ 0 ) � − 3p 1 and integrate both sides of equation (11), and we get Both solutions (19) and (20) are singular solutions for equation (1).

Case 3.
e polynomial F 4 (p) has two double real zeros, namely, We consider the following two cases in which n 4 is either positive or negative.
(i) If n 4 > 0, the intervals for real propagation are p < − p 1 , − p 1 < p < p 1 , and p > p 1 . If we consider the case in which p < − p 1 and assume p(ζ 0 ) � − ∞, equation (11) gives It follows Using equations (7) and (3), we obtain a wave solution for equation (1) in the form When p > p 1 , equation (1) has the same solution shown in equation (23) if (1) has a solution in the form (ii) e case in which n 4 < 0 is excluded since n 4 F 4 (p) < 0 for all p ∈ R.

Case 4.
e polynomial F 4 (p) has four real zeros, namely, Table 1: Types of the roots of the polynomial F 4 (p).

No.
Conditions on the discriminant system Types of the roots for F 4 (p) Two real roots: one is triple and the other is simple 3 One double root and two complex conjugate roots 6 Two real roots and two complex conjugate roots 7 Four real roots: one double and others simple . Now, we consider the two cases n 4 > 0 and n 4 > 0, individually: (i) When n 4 > 0, the possible intervals of p for real propagation are p < − ( It implies to where complete elliptic integral of the first type [56]. Using equations (7) and (3), we obtain a new traveling wave solution for equation (1) in the form If we select p 1 < p < p 2 , postulate p(ζ 0 ) � p 1 , and follow the same procedures, we will obtain a new traveling wave solution for equation (1) in the form where ζ 0 < ζ < ζ 1 . Also, if we elect p > p 3 and suppose p(ζ 0 ) � p 3 , we will get a new wave solution for equation (1) in the form Journal of Mathematics us, if we chose p 1 < p < p and assume It gives where +p 3 )), and ζ 2 � (1/Ω 2 )K(k 2 ). Utilizing equations (7) and (3), we obtain a new solution for equation (1): Similarly, if we select p 2 < p < p 3 and assume p(ζ 0 ) � p 2 , we present a new wave solution for equation (1) in the form 2.5. Case 5. e polynomial F 4 (p) has one double real root and two conjugate complex roots if D 4 � 0 and D 2 D 3 < 0. erefore, it takes the form , where * refers to the complex conjugate and p 1 � − Rep 2 . We consider the case in which n 4 > 0, and sequentially, the allowed intervals for real propagation are p < p 1 or p > p 1 . Choosing p > p 1 , assuming p(ζ 0 ) � ∞, and integrating both sides of equation (11), we obtain where ϵ � ζ 0 − s(inh − 1 (2p 1 /ρ))/ ���������� � n 4 (4p 2 1 + ρ 2 ) is a new constant which is introduced for suitability and ρ � Imp 2 . Employing equations (7) and (3), we obtain a solution for equation (1) in the form 6 Journal of Mathematics It can be noted that the case in which n 4 is negative does not work because F 4 (p) ≥ 0 for all p ∈ R.

Case 6.
e polynomial F 4 (P) has two real roots and two complex conjugate roots if D 4 < 0 and D 2 D 3 > 0. Hence, it can be written in the form , where p 1 < p 2 and Rep 3 � (1/2)(p 1 + p 2 ). We consider the following: (i) If n 4 > 0, then the permitted intervals for real propagation are p > p 2 and p < p 1 . Selecting p < p 2 and p(ζ 0 ) � p 2 and integrating both sides of equation (11), we obtain where . Taking into account equations (7) and (3), we obtain a novel wave solution for equation (1) in the form (ii) If n 4 < 0, then the allowed intervals of possible propagation are p 1 < p < p 2 . Assuming p(ζ 0 ) � p 1 and integrating both sides of equation (11), we obtain where Utilizing equations (7) and (3), we construct a novel wave solution for equation (1) in the form where It can be noted that the case in which n 4 < 0 does not work because F 4 (p) > 0 for all p ∈ R.

Case 8.
e polynomial F 4 (p) has four real roots in which one of them is double and the others are simple if D 2 > 0, D 3 > 0, and D 4 � 0. Hence, it takes the form F 4 (p) � (p − p 1 ) 2 (p − p 2 )(p − p 3 ), where p 1 < p 2 < p 3 and p 3 � − (2p 1 + p 2 ). We consider the two cases in which n 4 is either positive or negative: (i) If n 4 > 0, then the possible intervals for real propagation are p < p 1 , p > p 3 , and p 1 < p < p 2 . With similar computations as in previous cases, we present the solution of equation (1) directly. If we choose p > − (2p 1 + p 2 ) and assume p(ζ 0 ) � − (2p 1 + p 2 ), we have a new wave solution for equation (1) in the form In similar calculations, we can calculate the wave solution for p < p 1 and p 1 < p < p 2 .
(ii) If n 4 < 0, the allowed interval for real propagation is p ∈ ]p 2 , p 3 [, and postulating p(ζ 0 ) � p 2 , we obtain a new wave solution for equation (1) in the form

Dynamic Properties
e aim of this section is to investigate some dynamic properties for equation (4) by investigating the bifurcation and phase portrait for the traveling wave system corresponding to equation (8) which takes the form System (43) is a Hamiltonian system with one degree of freedom related to Hamilton function: where h is an arbitrary constant and is the potential function. It is well known that the equilibrium points for the Hamilton system (43) are also critical points for the potential function (45), i.e., they are the roots of dV dp us, we use the discrimination of (46) to determine the number of equilibrium points. e discrimination of (46) is Now, let us determine the number of equilibrium points for the Hamilton system (43) and study the properties of its phase space. us, we need to define energy curve corresponding to 8 Journal of Mathematics It is well known that any orbit for the Hamilton system (43) is an energy curve on a certain level of the energy. Case 1.
e dynamical system (43) has a unique equilibrium point if dV/dp � 0 has a unique real root. is happens in two cases which are studied individually: (i) If Δ � 0 and c 2 � 0, then dV/dp � 0 has one triple real root, i.e., dV/dp � − 2n 4 p 3 . is shows (0, 0) is a unique equilibrium point for system (43) which is saddle if n 4 > 0, and it is center if n 4 < 0. e phase space for this case is outlined by Figures 1(a) and 1(b). e value of the energy at the equilibrium point e following proposition describes Figure 1.
(ii) If Δ < 0, then dV/dp � 0 has one real zero and two complex conjugate roots, i.e., dV dp It is clear that the point (a, 0) is a unique equilibrium point for the Hamilton system (43) and it is saddle if n 4 > 0 and center if n 4 < 0. e phase space is clarified by Figures 2(a) and 2(b).

Proposition 1.
e Hamiltonian system (43) has a unique equilibrium point (0, 0) if c 1 � c 2 � 0. If n 4 > 0, it is a saddle point and all the orbits are unbounded, see Figure 1(a). If n 4 < 0, system (43) has a family of bounded periodic orbits C h : h > h 1 about the center point (0, 0) as outlined by Figure 1(b). A similar conclusion can be presented to describe Figure 2.
Case 2. If Δ � 0 an d c 2 < 0, then dV/dp � 0 has two real roots: one is simple and the other is double. us, we write dV/dp � − 2n 4 (p − a) 2 (p + 2a) and so (a, 0) an d (− 2a, 0) are two equilibrium points for system (43). It is clear that (a, 0) is a cusp while (− 2a, 0) is a center if n 4 < 0 and saddle if n 4 > 0. e phase portrait for this case is outlined in Figure 3. e following proposition gives a short description for the phase portrait for this case.

Proposition 2.
e Hamiltonian system (43) has two equilibrium points E 1 � (a, 0) and E 2 (− 2a, 0). If n 4 > 0, then E 1 is cusp point while E 2 is saddle, and furthermore, all the phase space orbits are unbounded as outlined in Figure 3(a). While if n 4 < 0, E1 is a cusp and E 2 is a center. e Hamilton system (43) has two bounded families of periodic orbits which are illustrated in green and blue and separated by the phase curve C h : h � V(a, 0) in red.
Case 3. If Δ > 0 and c 2 < 0, then dV/dp has three real roots, i.e., it can be written as dV/dp � − 2n 4 (p − a)(p − b) (p + a + b), where we assumed b > a > 0. Consequently, the dynamical system (43) has three equilibrium points (a, 0), (b, 0), and (− a − b, 0). If n 4 > 0, then (a, 0) is center  and (b, 0) and (− a − b, 0) are saddle points. While if n 4 < 0, (a, 0) is saddle point and the other two equilibrium points are center. e phase portrait for this case is described in Figures 4(a) and 4(b). We describe the phase portrait for the Hamiltonian system in the following proposition. Proposition 3. If Δ > 0 and c 2 < 0, then the Hamilton system (43) has three equilibrium points (a, 0), (b, 0), and (− a − b, 0). If n 4 > 0, there are two families of orbits C h : h ∈ ]V(a), V(b)[ in green in which one of them is bounded and surrounded by the homoclinic orbit C h : h � V(2a) while the other family is unbounded. Moreover, all the other orbits are unbounded, see Figure 4(a) for more clarification. If n 4 < 0, system (43) has three equilibrium points in which one is a saddle and the others are centers. It has three bounded families of periodic orbits. Two of them in blue and green are periodic orbits around the two center points (2a, 0) and (− a − b, 0), and they are separated by the homoclinic orbit in red C h : h � V(a) . e third one is a family of superperiodic orbits in brown C h : h > V(a) around the two centers points and lies outside the homoclinic orbit in red. For more details about superperiodic orbits, see, for example, [57]. e investigation of the type of the phase space orbits is helpful in determining the types of the solutions. For instance, the existence of periodic orbits, homoclinic orbits, and heteroclinic orbits for the traveling wave system (43) indicates the existence of periodic wave solutions, solitary, and kink solutions for equation (1). Furthermore, this analysis can be employed to construct the traveling wave solution by introducing the constraints on the coefficients of function (12). Consequently, we can prove the following theorem.

Graphic Interpretations
is section aims to illustrate some of the obtained solutions graphically. Moreover, we study the influence of the physical parameters on the obtained solutions by considering two Journal of Mathematics types of solutions: one of them is kink solution and the other is periodic. Figure 5 and 6 illustrates the kink solution (24) for di erent aspects. Figures 5(a) and 5(b) clarify the 3D and contour representation for the kink solution (24) when k 1, ω 2, σ 0.001, ρ 0.1, μ 0.0001, h 0.002, R 1 0.1, and l 0 0.002. Now, we illustrate graphically the in uence of some parameters on the kink solution while the other parameters are xed. Figure 6(a) illustrates the e ect of the change of the distance between the two strands. It is remarkable the amplitude of solution (24) is decreased when the distance between the two strands is increased. Figure 6(b) clari es the amplitude of the solution is increased when the sti ness of the elastic membrane is increased. Figure 6(c) clari es the amplitude of the solution is decreased when the area of the cross section of each strand is increased. Figure 6(d) outlines the amplitude of the kink solution (24) is increased when the height of the membrane in the equilibrium is increased (see Figure 7). Now, we are going to clarify solution (28) graphically and study the in uence of parameter changes on solution (28). Solution (28) is periodic as outlined in Figure 8(a), and its contour is illustrated in Figure 8(b). If the distance between the two strands is increased and the other parameters are xed, the amplitude of the solution is unchanged but the width of the solution is increased as outlined in Figure 8(a). e amplitude of solution (28) is not a ected by the changes in the sti ness of the elastic membrane, while the width of the solution is decreased when the sti ness of the elastic membrane is increased as clari ed by Figure 8(d). If the area of the cross section of each strand is increased, then the      amplitude keeps xed while the width is increased, see Figure 8(d). Figure 9(b) outlines the amplitude of the periodic solution (28) is kept unchanged while its width is decreased when the height of the membrane in the equilibrium is increased. Figure 9(a) illustrates the in uence of the superperiodic wave solution (39) due the changes in the sti ness of the elastic membrane. If the sti ness of the elastic membrane increases, the amplitude is kept unchanged while the width decreases. Figure 9(b) clari es the in uence of distance between the two strands on the superwave solution (39). If distance between the two strands is increased, then the amplitude and the width of superwave solution (39) are increased.
It is worth mentioning that we can make the same study for the remaining obtained solutions. However, we only give the 3D graphic and the 2D counter for some solutions. e 3D graphic representation and the singularity plane are outlined in Figure 10(a), while Figure 10(b) illustrates the 2D contour of solution (15). Solution (37) is illustrated in Figure 11.

Conclusion
is work is aimed to study analytically the double-chain model for deoxyribonucleic acid (DNA). A certain wave transformation has been applied to equation (1) to transform it into an ordinary di erential equation. e integration of this equation reacquired some studies on the parameters. is study has been performed by applying the complete discrimination of the polynomial F 4 (p). Moreover, we have determined the possible interval of real propagations. Such study is more signi cant because the missing of such study implies to loss some solutions and also, give rise to complex solutions which are undesirable in real problems. For instance, there are several solutions corresponding to the same conditions on the discriminant system as outlined in Case 4. We have introduced new waves' solutions for equation (1). Let us compare the results obtained in the present article with the well-known results obtained by other authors using di erent methods as follows: our results in a new doublechain model of DNA are new and di erent from those obtained in references [19][20][21][22][23][24]. We have studied the inuence of some parameters such as the distance between the two strands, the sti ness of the elastic membrane, the area of the cross section of each strand, and the height of the membrane in the equilibrium. We have considered two types of solutions: one is kink (24) and the other is periodic (28). We have shown graphically the amplitude of the kink solution is decreased when the distance between the two strands or the area of the cross section of each strand is increased, while it is increased when the sti ness of the elastic membrane or the height of the membrane in the equilibrium is increased. For more clari cation, see Figure 6. e amplitude of the periodic solution remains approximately unchanged when these physical parameters are    changed, but the width has been affected. e width is increased due to the increase of the distance between the two strands or the area of the cross section of each strand, while it is decreased as a result of increasing the stiffness of the elastic membrane or the height of the membrane in the equilibrium. For more illustrations, see Figure 10. From another point of view, this ODE has been expressed as a one-dimensional Hamiltonian system that describes the physical motion of a particle with one degree of freedom under the action of potential function V(p) given by (45). Based on the Hamiltonian concepts, we have studied some qualitative analyses such as phase portrait and bifurcation. e description of phase space has been presented through Propositions 1, 2, and 3. Moreover, these propositions contain the conditions for the existence of periodic and solitary wave solutions.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.