Forming localized waves of the nonlinearity of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model

1.   Introduction

Traveling wave and soliton solutions are one of the most interesting and fascinating areas of research in different fields of engineering and physical sciences. These models are basic ingredients of sciences in which play important roles in numerous areas such as biology, physics, chemistry, fluid mechanics and many engineering and science applications among others ^[1,2,3,4,5]. Furthermore, the approaches to solving these types of equations alongside nonlinear PDEs ranging from analytical to numerical methods are very important in many engineering and sciences applications. Some of these methods include finding the exact solutions by using the special techniques in which can be manifested to new works with vigorous references. Consequently, it is imperative to address the dynamics of these soliton pulses from a mathematical aspect. This will lead to a deeper understanding of the engineering perspective of these solutions ^{[6,7,8,9,10,11,12,13]}.

In this paper, we will study the different kinds of traveling wave solutions in mathematical model in DNA dynamics from a purely mathematical viewpoint. Therefore, the importance of this paper will be to extract the exact traveling wave solution for the nonlinear model. This model is described for first by Peyrard-Bishop, that takes into consideration the inclusion of nonlinear interaction between adjacent displacements along with the Hydrogen bonds ^[14]. There are several integration tools available to solve the model. Many such nonlinear equations as DNA dynamics have been examined with regards to soliton theory, where complete integrability was emphasized by various analytical techniques.

For investigating the appearance of solitonic structures of the oscillator-chain of Peyrard-Bishop model has been analyzed by ^[14,15]. The balance between weak nonlinearity and dispersion in DNA dynamic model with linear dispersion and nonlinear dispersion arise in works of Dusuel et al. ^[16] and Alvarez et al. ^[17]. Treatment of mathematical and physical modeling of equations of DNA dynamics show that those can be reduced to a significant nonlinear formations. The nonlinearity of the DNA dynamic model arises in localized waves in which have a few considerable features, as example in transporting energy without dissipation. A few methods in which physical properties of DNA dynamics have been investigated by the numerous authors ^{[18,19,20,21,22,23]}. There are techniques usually used in biological systems such as the discrete derivative operator (DDO) technique applying to long-range interactions systems ^[24,25], the semi-discrete approximation ^[26,27,28], the solitary perturbation technique ^[29,30], the modified extended tanh function method ^[31,32,33].

Author of ^[34] made use of the Hirota bilinear method of the bidirectional Sawada-Kotera equation to obtain new lump-type solutions and interaction phenomenon. In ^[35], author found the lump soliton and novel solitary wave solutions for the (3+1)-dimensional extended Jimbo-Miwa equations. Manafian and co-author found the interaction phenomenon to the (2+1)-dimensional Breaking Soliton equation ^[36]. Moreover, Ilhan et al. determined lump wave solutions and the interactions between lump solutions for a variable-coefficient Kadomtsev-Petviashvili equation ^[37]. Authors of ^[38] obtained the stationary solutions of various nonlinear Schrödinger equations. Younas and co-authors ^[38] studied the nonlinear chirp solitons for the model of Schröodinger-Hirota equation with concluding the bright, dark and singular solitons. Ali and co-workers ^[39] utilized the extended trial equation method and retrieved Jacobi elliptic, periodic, bright and singular solitons for paraxial nonlinear Schröodinger equation. In ^[40], the first and second-order rogue wave solutions were gained for the coupled Schrödinger equations. Arif et al. investigated the solitons and lump wave solutions to the graphene thermophoretic motion system ^[41]. Structures of this paper as follows, the nonlinear DNA dynamics model has been summarized in section 2. In sections 3–7, an overview of the integration schemes are given along with the analysis of the model including the improved $\tan(\phi/2)$-expansion method, $\exp(-\Omega(\eta))$-expansion method, improved $\exp(-\Omega(\eta))$-expansion method, generalized (G'/G)-expansion method, and exp-function method, respectively. The next section gives the discussions about the model. In the last section, the conclusions have been given.

2.   Peyrard-Bishop DNA dynamic model equation

It is popular that DNA molecule is a double helix. This means that it consists of two complementary polymeric chains twisted around each other ^[42]. The B-form DNA in theWatsonCrick model is a double helix, which contains of two strands. The masses of nucleotides do not vary too much which means that one can assume a homogeneous crystal structure. The strands are coupled to each other through the hydrogen bonds, so that these bonds are weak while the harmonic longitudinal are strong the PB model neglects all the displacements beside the transversal ^[43]. The Hamiltonian model of Peyrard and Bishop ^[43], and the equations found in the literature, is modeled by the Morse potential as

in which $u_n$ and $v_n$ are the displacements of the nucleotides. Also, the Hamiltonian for the DNA chain was described by Zdravković ^[43]. Moreover, the improved version of the PB model, introduced by Dauxois ^[44]. The Hamiltonian for describing the strand aperture the hydrogen bonds can be stated as ^[45]

in which $k_1$ and $k_2$ denote the strength for the linear and nonlinear couplings respectively and $q_n = m \dot{u_n}$ is the momentum for the displacement $u_n$. Searching Starting with the hamiltonian (2.2) the equation of motion in the continuum limit can be stated by the following form

with $l_1 = \frac{k_1}{m}d^2, \ \ l_2 = \frac{k_2}{m}d^4, \ \ D = \frac{\delta}{m}, \ \ \alpha\equiv \sqrt{2}a$ and being $d$ the inter-site nucleotide distance in the DNA ladder (^[46,47,48]). In this paper, consider the Peyrard-Bishop DNA dynamic model equation as follows

where $l_1, l_2, \alpha$ and $\Omega = D$ are constants. By make the following transformations

then the Eq. (2.4), can be reduced to the ordinary differential equation as

By multiplying the Eq. (2.6) by $u'$ and integrating once with respect to $\xi$, we get

By starting hypothesis is taken to be

By appending (2.8) into Eq. (2.7), the nonlinear equation is achieved as follows

3.   Description of the ITEM

In this section, the improved $\tan(\phi/2)$-expansion method ^[8,9] has been summarized to obtain the solutions of nonlinear partial differential equations (NPDEs). Hence, consider the NPDEs in the following way:

where $\mathcal{N}$ is a polynomial of $u$ and its partial derivatives in which the relationship of higher order derivatives and nonlinear terms. To find the traveling wave solutions, we outline the following sequence of steps towards the extended tanh method:

Step 1. Firstly, by using traveling wave transformation

where $\beta$ is non-zero arbitrary constant, permits to reduce Eq. (6.1) to an ODE of $u = u(\xi)$ in the following form

Step 2. Assuming that the solution of Eq. (6.1) can be expressed by the following ansatz:

where $A_k (0\leq k \leq m)$ are the parameters to be determined and $A_m\neq0$ and $\phi = \phi(\xi)$ satisfies in the ordinary differential equation as follows:

The particular solutions of Eq. (3.5) will be read as:

Family 1: When $\Delta = a^2+b^2-c^2 < 0$ and $b-c\neq0$, then $\phi(\xi) = 2\tan^{-1}\left[\frac{a}{b-c}-\frac{\sqrt{-\Delta}}{b-c}\tan\left(\frac{\sqrt{-\Delta}}{2}\overline{\xi}\right)\right].$

Family 2: When $\Delta = a^2+b^2-c^2 > 0$ and $b-c\neq0$, then $\phi(\xi) = 2\tan^{-1}\left[\frac{a}{b-c}+\frac{\sqrt{\Delta}}{b-c}\tanh\left(\frac{\sqrt{\Delta}}{2}\overline{\xi}\right)\right].$

For see the rest seventeen families refer to Ref. ^[8,9]. Also, $\overline{\xi} = \xi+C, p, A_k, B_k (k = 1, 2, ..., m), a, b$ and $c$ are constants to be determined later.

Step 3. To determine the positive integer $m$, we usually balance linear terms of the highest order in the resulting equation with the highest order nonlinear terms appearing in equation (3.3).

Step 4. We collect all the terms with the same order of $\tan(\phi/2)^k, (k = 0, 1, 2, ...)$ together. Equate each coefficient of the polynomials to zero, yields the set of algebraic equations for $A_0, A_k(k = 1, 2, ..., m), a, b$ and $c$ with the aid of the Maple.

Step 5. Solving the algebraic equations in Step 4, then substituting $A_0, A_1, ..., B_m, a, b, c$ in (3.4).

3.1.   Application of the ITEM for Eq. (2.4)

Consider the homogeneous balance principle between the highest order derivatives $(v')^4$ and nonlinear terms $v^6$, and get

Therefore, the equation (3.4) takes the form

Substitute equation (3.6) and its derivatives into equation (2.9). Algebraic equations set can be obtained after equating the coefficients of $\tan^p(\phi/2)$ for $p = 0, 1, ..., 12$, and setting equal to zero. After solving the nonlinear algebraic equations, the following values of $a, b, c, \beta, A_0, A_1, A_2$ can be obtained:

Set Ⅰ.

By utilizing of Family 1, the trigonometric function solution becomes

The existence of the solution for the constraint condition is as $\frac{A_2(\beta^2-2l_1)}{3l_2} < \left(\frac{{\frac {\sqrt [4]{108\Omega\, {A_{{2}}}^{2}{l_{{2}} }^{3}}\alpha}{3l_{{2}}}}-c}{\alpha}\right)^2$.

By utilizing of Family 2, the hyperbolic function solution becomes

The existence of the solution for the constraint condition is as $\frac{A_2(\beta^2-2l_1)}{3l_2} > \left(\frac{{\frac {\sqrt [4]{108\Omega\, {A_{{2}}}^{2}{l_{{2}} }^{3}}\alpha}{3l_{{2}}}}-c}{\alpha}\right)^2$.

4.   The $\exp(-\Omega(\eta))$-expansion method

This section elucidates a systematic explanation of the $\exp(-\Omega(\eta))$-expansion method to obtain the solutions of nonlinear partial differential equations (NPDEs). Hence, take the NPDEs in the following way:

where $\mathcal{N}$ is a polynomial of $u$ and its partial derivatives in which the relationship of higher order derivatives and nonlinear terms. To find the traveling wave solutions, we outline the following sequence of steps towards the extended tanh method:

Step 1. Firstly, by utilizing the traveling wave transformation

where $\beta$ is non-zero arbitrary constant, permits to reduce equation (4.1) to an ODE of $u = u(\xi)$ in the following form

Step 2. Assuming that the solution of equation (4.1) can be expressed by the following ansatz:

where $F(\eta) = \exp(-\Phi(\xi))$ and $A_j (0\leq j\leq m)$, are the parameters to be determined $A_m\neq0$, and, $\Phi = \Phi(\xi)$ satisfying the ODE given below

The particular solutions of equation (4.5) will be read as:

Solution-1: When $\mu\neq0$ and $\lambda^2-4\mu > 0,$ therefore we attain $\Phi(\eta) = \ln\left(-\frac{\sqrt{\lambda^2-4\mu}}{2\mu} \tanh\left(\frac{\sqrt{\lambda^2-4\mu}}{2}(\xi+E)\right)-\frac{\lambda}{2\mu}\right)$.

Solution-2: When $\mu\neq0$ and $\lambda^2-4\mu < 0,$ therefore we attain $\Phi(\eta) = \ln\left(\frac{\sqrt{-\lambda^2+4\mu}}{2\mu} \tan\left(\frac{\sqrt{-\lambda^2+4\mu}}{2}(\xi+E)\right)-\frac{\lambda}{2\mu}\right)$.

Solution-3: When $\mu = 0$, $\lambda\neq0,$ and $\lambda^2-4\mu > 0,$ therefore we attain $\Phi(\eta) = -\ln\left(\frac{\lambda}{\exp(\lambda(\xi+E))-1}\right)$.

Solution-4: When $\mu\neq0$, $\lambda\neq0,$ and $\lambda^2-4\mu = 0,$ therefore we attain $\Phi(\eta) = \ln\left(-\frac{2\lambda(\xi+E)+4}{\lambda^2(\xi+E)}\right)$.

Solution-5: When $\mu = 0$, $\lambda = 0,$ and $\lambda^2-4\mu = 0,$ therefore we attain $\Phi(\eta) = \ln\left(\xi+E\right)$, where $A_j (0\leq j\leq m)$, $E, \lambda$ and $\mu$ are also the constants to be explored later.

Step 3. To determine the positive integer $m$, we usually balance the linear terms of the highest order in the resulting equation with the highest order nonlinear terms appearing in equation (4.3).

Step 4. We collect all the terms with the same order of $F(\xi)^k, (k = 0, 1, 2, ...)$ together. Equate each coefficient of the polynomials of $F(\xi)^k$ to zero, yields the set of algebraic equations for $A_0, A_k(k = 1, 2, ..., m), \lambda$ and $\mu$ with the aid of the Maple.

Step 5. Solving the algebraic equations in Step 4, then substituting $A_0, A_1, ..., A_m, \lambda, \mu$ in (4.4).

4.1.   Application of the EEM for Eq. (2.4)

Consider the homogeneous balance principle between the highest order derivatives $(v')^4$ and nonlinear terms $v^6$, we obtain $4m+4 = 6m$, then $m = 2$. The exact solution can be expressed in the following form

Substitute equation (4.6) and its derivatives into equation (2.9). The algebraic equations set can be obtained after equating the coefficients of $F(\xi)$ for p = 0, 1, ..., 20, and setting equal to zero. After solving the nonlinear algebraic equations, the following values of $\lambda, \mu, \beta, A_0, A_1, A_2$ can be obtained:

Set Ⅰ.

By utilizing of Family 1, the hyperbolic function solution becomes

The existence of the solution for the constraint condition is as $l_2(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12Rl_2)}-e\alpha^3\Omega) > 0$.

By utilizing of Family 2, the trigonometric function solution becomes

The existence of the solution for the constraint condition is as $l_2(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12Rl_2)}-e\alpha^3\Omega) < 0$.

By utilizing of Family 3, the hyperbolic function solution becomes

The existence of the solution for the constraint condition is as $l_2(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12Rl_2)}-e\alpha^3\Omega) > 0,$ and

5.   The improved $\exp(-\Omega(\eta))$-expansion method

In this section the improved $\exp(-\Omega(\eta))$-expansion method is utilized to obtain the solutions of nonlinear partial differential equations (NPDEs). Hence, consider the NPDEs in the following way:

where $\mathcal{N}$ is a polynomial of $u$ and its partial derivatives in which the relationship of higher order derivatives and nonlinear terms. To find the traveling wave solutions, we outline the following sequence of steps towards the extended tanh method:

Step 1. Firstly, by using the traveling wave transformation

where $\beta$ is non-zero arbitrary constant, permits to reduce equation (4.1) to an ODE of $u = u(\xi)$ in the following form

Step 2. Assuming that the solution of equation (5.1) can be expressed by the following ansatz:

where $F(\eta) = \exp(-\Phi(\xi))$ and $A_j (0\leq j\leq m), B_j (1\leq j\leq m)$, are the parameters to be determined $A_m\neq0$, and, $\Phi = \Phi(\xi)$ satisfying the ODE given below

The particular solutions of equation (5.5) will be read like before section.

Step 3. To determine the positive integer $m$, we usually balance the linear terms of the highest order in the resulting equation with the highest order nonlinear terms appearing in equation (4.3).

Step 4. We collect all the terms with the same order of $F(\xi)^k, (k = 0, 1, 2, ...)$ together. Equate each coefficient of the polynomials of $F(\xi)^k$ to zero, yields the set of algebraic equations for $A_0, A_k, B_k(k = 1, 2, ..., m), \lambda$ and $\mu$ with the aid of the Maple.

Step 5. Solving the algebraic equations in Step 4, then substituting $A_0, A_1, B_1, ..., A_m, B_m, \lambda, \mu$ in (5.4).

5.1.   Application of the IEEM for Eq. (2.4)

The exact solution will be the same as the previous section as

Substitute equation (5.6) and its derivatives into equation (2.9). The algebraic equations set can be obtained after equating the coefficients of $F(\xi)$ for $p = 0, 1, ..., 20$, and setting equal to zero. After solving the nonlinear algebraic equations, the following values of $\lambda, \mu, \beta, A_0, A_1, B_1, A_2, B_2$ can be obtained:

Set Ⅰ.

By utilizing of Family 1, the hyperbolic function solution becomes

The existence of the solution for the constraint condition is as $l_2(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12Rl_2)}-e\alpha^3\Omega) > 0$.

By utilizing of Family 2, the trigonometric function solution becomes

The existence of the solution for the constraint condition is as $l_2(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12Rl_2)}-e\alpha^3\Omega) < 0$.

By utilizing of Family 3, the kink-soliton solution becomes

The existence of the solution for the constraint condition is as $l_2(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12Rl_2)}-e\alpha^3\Omega) > 0$ and

Set Ⅱ.

By utilizing of Family 1, the hyperbolic function solution becomes

The existence of the solution for the constraint condition is as $l_2\mu(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12\mu^2Rl_2)}-e\alpha^3\Omega) > 0$.

By utilizing of Family 2, the trigonometric function solution becomes

The existence of the solution for the constraint condition is as $l_2\mu(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12\mu^2Rl_2)}-e\alpha^3\Omega) < 0$.

By utilizing of Family 3, the exponential function solution becomes

The existence of the solution for the constraint condition is as $l_2\mu(\sqrt {\alpha^4(\Omega^2\alpha^2e^2-12\mu Rl_2)}-e\alpha^3\Omega) > 0$.

6.   Description of the GGM

As the fourth method, the generalized (G'/G)-expansion method has been summarized to obtain the solutions of NPDEs. Hence, consider the NPDEs of in the following way:

where $\mathcal{N}$ is a polynomial of $u$ and its partial derivatives in which the relationship of higher order derivatives and nonlinear terms. To find the traveling wave solutions, we outline the following sequence of steps towards the GGM:

Step 1. Firstly, by using traveling wave transformation

where $\beta$ is non-zero arbitrary constant, permits to reduce equation (6.1) to an ODE of $u = u(\xi)$ in the following form

Step 2. Assuming that the solution of equation (6.1) can be expressed by the following ansatz:

where, $A_k (0\leq k \leq m)$ are constants to be determined, such that $A_m\neq0$, and $\Phi(\xi) = G'(\xi)/G(\xi)$ satisfies the following ODE:

The particular solutions of equation (6.5) will be read as:

Family 1: When $k_2\neq0$, $f = k_1-k_3$ and $s = k_2^2+4k_4(k_1-k_3) > 0$, then $\Phi(\xi) = \frac{k_2}{2f}+\frac{\sqrt{s}}{2f}\frac{C_1\sinh\left(\frac{\sqrt{s}}{2k_1}\xi\right)+C_2\cosh\left(\frac{\sqrt{s}}{2k_1}\xi\right)} {C_1\cosh\left(\frac{\sqrt{s}}{2k_1}\xi\right)+C_2\sinh\left(\frac{\sqrt{s}}{2k_1}\xi\right)}.$

Family 2: When $k_2\neq0$, $f = k_1-k_3$ and $s = k_2^2+4k_4(k_1-k_3) < 0$, then $\Phi(\xi) = \frac{k_2}{2f}+\frac{\sqrt{-s}}{2f}\frac{-C_1\sin\left(\frac{\sqrt{-s}}{2k_1}\xi\right)+C_2\cos\left(\frac{\sqrt{-s}}{2k_1}\xi\right)} {C_1\cos\left(\frac{\sqrt{-s}}{2k_1}\xi\right)+C_2\sin\left(\frac{\sqrt{-s}}{2k_1}\xi\right)}.$

Family 3: When $k_2\neq0$, $f = k_1-k_3$ and $s = k_2^2+4k_4(k_1-k_3) = 0$, then $\Phi(\xi) = \frac{k_2}{2f}+\frac{C_2}{C_1+C_2\xi}.$

Family 4: When $k_2 = 0$, $f = k_1-k_3$ and $g = fk_4 > 0$, then $\Phi(\xi) = \frac{\sqrt{g}}{f}\frac{C_1\sinh\left(\frac{\sqrt{g}}{k_1}\xi\right)+C_2\cosh\left(\frac{\sqrt{g}}{k_1}\xi\right)} {C_1\cosh\left(\frac{\sqrt{g}}{k_1}\xi\right)+C_2\sinh\left(\frac{\sqrt{g}}{k_1}\xi\right)}.$

Family 5: When $k_2 = 0$, $f = k_1-k_3$ and $g = fk_4 < 0$, then $\Phi(\xi) = \frac{\sqrt{-g}}{f}\frac{-C_1\sin\left(\frac{\sqrt{-g}}{k_1}\xi\right)+C_2\cos\left(\frac{\sqrt{-g}}{k_1}\xi\right)} {C_1\cos\left(\frac{\sqrt{-g}}{k_1}\xi\right)+C_2\sin\left(\frac{\sqrt{-g}}{k_1}\xi\right)}.$

Family 6: When $k_4 = 0$ and $f = k_1-k_3,$ then $\Phi(\xi) = \frac{C_1k_2^2\exp\left(\frac{-k_2}{k_1}\xi\right)}{fk_1+C_1k_1k_2\exp\left(\frac{-k_2}{k_1}\xi\right)}.$

Family 7: When $k_2\neq0$ and $f = k_1-k_3 = 0,$ then $\Phi(\xi) = -\frac{k_4}{k_2}+C_1\exp\left(\frac{k_2}{k_1}\xi\right),$

Family 8: When $k_1 = k_3$, $k_2 = 0$ and $f = k_1-k_3 = 0,$ then $\Phi(\xi) = C_1+\frac{k_4}{k_1}\xi,$

Family 9: When $k_3 = 2k_1$, $k_2 = 0$ and $k_4 = 0,$ then $\Phi(\xi) = -\frac{1}{C_1+\left(\frac{k_3}{k_1}-1\right)\xi},$ where $d_0, d_j, e_j(j = 1, ..., m), k_1, k_2, k_3$ and $k_4$ are constants to be determined later.

Step 3. To determine the positive integer $m$, we usually balance linear terms of the highest order in the resulting equation with the highest order nonlinear terms appearing in equation (3.3).

Step 4. We collect all the terms with the same order of $\Phi(\xi)^k, (k = 0, 1, 2, ...)$ together. Equate each coefficient of the polynomials of i to zero, yields the set of algebraic equations for $A_0, A_k(k = 1, 2, ..., m), k_1, k_2, k_3,$ and $k_4$ with the aid of the Maple.

Step 5. Solving the algebraic equations in Step 4, then substituting $A_0, A_1, ..., B_m, k_1, k_2, k_3, k_4$ in (6.4).

6.1.   Application of GGM

By processing the generalized $G'/G$-expansion method and considering the homogeneous balance principle, we get the exact solution in the following form

Solving the nonlinear algebraic equations, we have the following sets of coefficients for the solutions of (6.6) as given below:

Subset Ⅰ.

Based on the Family 1, the exact soliton solution can be written as

in which $\xi = x-{\frac { \sqrt{A_2(2\, {\alpha}^{2}A_{{2}}{k_{{1}}}^{2}l_{{1}}-12\, {k_{{1}}}^{2}l_{{2}} \epsilon_{{1}} \left(\epsilon_{{1}}-2 \right) \left(A_{{0}}-1 \right) -3\, l_{{2}} \left(4\, A_{{0}}{k_{{1}}}^{2}-A_{{2}}{\epsilon_{ {3}}}^{2}-4\, {k_{{1}}}^{2} \right))}}{k_1\alpha A_{{2}}}} t$. The existence of the solution for the constraint condition is as $|\varepsilon_3| > \frac{\alpha^2|k_1|}{|\varepsilon_1-1|}\sqrt{\frac{A_0A_2\Omega}{3l_2}}$.

Based on the Family 2, the exact periodic solution can be written as

in which $\xi = x-{\frac { \sqrt{A_2(2\, {\alpha}^{2}A_{{2}}{k_{{1}}}^{2}l_{{1}}-12\, {k_{{1}}}^{2}l_{{2}} \epsilon_{{1}} \left(\epsilon_{{1}}-2 \right) \left(A_{{0}}-1 \right) -3\, l_{{2}} \left(4\, A_{{0}}{k_{{1}}}^{2}-A_{{2}}{\epsilon_{ {3}}}^{2}-4\, {k_{{1}}}^{2} \right))}}{k_1\alpha A_{{2}}}} t$. The existence of the solution for the constraint condition is as $|\varepsilon_3| < \frac{\alpha^2|k_1|}{|\varepsilon_1-1|}\sqrt{\frac{A_0A_2\Omega}{3l_2}}$.

Based on the Family 3, the exact singular solution can be written as

in which $\xi = x-{\frac { \sqrt{A_2(2\, {\alpha}^{2}A_{{2}}{k_{{1}}}^{2}l_{{1}}-12\, {k_{{1}}}^{2}l_{{2}} \epsilon_{{1}} \left(\epsilon_{{1}}-2 \right) \left(A_{{0}}-1 \right) -3\, l_{{2}} \left(4\, A_{{0}}{k_{{1}}}^{2}-A_{{2}}{\epsilon_{ {3}}}^{2}-4\, {k_{{1}}}^{2} \right))}}{k_1\alpha A_{{2}}}} t$ and $A_2 = {\frac {24\Omega\, {A_{{0}}}^{2}l_{{2}} \left(\epsilon_{{2}}-k_{{1}} \right) ^{2} \left(A_{{0}}-1 \right) }{4\, {\Omega}^{2}{\alpha}^{4}{A _{{0}}}^{4}{k_{{1}}}^{4}+3\, \Omega\, {A_{{0}}}^{2}l_{{2}}-6\, \Omega\, A_ {{0}}l_{{2}}+3\, Rl_{{2}}}}$.

Based on the Family 6, the exact kink solution can be written as

in which $\xi = x-{\frac { \sqrt{A_2(2\, {\alpha}^{2}A_{{2}}{k_{{1}}}^{2}l_{{1}}+12\, {k_{{1}}}^{2}l_{{2}} \epsilon_{{1}} \left(\epsilon_{{1}}-2 \right) -3\, l_{{2}} \left(-A_{{2}}{\epsilon_{ {3}}}^{2}-4\, {k_{{1}}}^{2} \right))}}{k_1\alpha A_{{2}}}} t$.

7.   Basic idea of the Exp–function method

We first consider the nonlinear equation of form

and introduce a transformation as

where $\beta$ is constant to be determined later. Therefore the Eq. (7.1) is reduced to an ODE as follows

The EFM is based on the assumption that the travelling wave solutions can be expressed in the form

where $\mathrm{c}$, $\mathrm{d}$, $\mathrm{p}$ and $\mathrm{q}$ are positive integers which could be freely chosen, $\mathrm{a_n}$'s and $\mathrm{b_m}$'s are unknown constants to be determined.

7.1.   Application of EFM for Eq. (2.4)

We apply the Exp-function method to Eq. (2.9). In order to determine values of $\mathrm{c}$ and $\mathrm{p}$, we balance the terms $(v')^4$ and $v^6$ in Eq. (2.9) along with Eq. (7.4), then we get

respectively. Balancing highest order of the Exp–function in (7.5) and get $4c+4p = 6c+2p$, which leads to the result $c = p$. Similarly, to find values of $d$ and $q$, for the terms $(v')^4$ and $v^6$ in Eq. (2.9) by simple calculation, we attain

respectively. Balancing lowest order of the Exp–function in (7.6), we achieve $d = q$.

Case Ⅰ: $p = c = 1$ and $q = d = 1$.

For simplicity, we set $a_{-1} = 0, b_1 = 1$, $p = c = 1$ and $d = q = 1$. Then Eq. (7.4) reduces to

Substituting (7.7) into Eq. (2.9), we get an equation in the following form

where $C_n(-4\leq n\leq 4)$ are polynomial expressions in terms of $a_1, a_0, a_{-1}, b_0, b_{-1}$ and $\beta$. Thus, solving the resulting system $C_n = 0(-4\leq n\leq 4)$ simultaneously, we acquire the following set as

(Ⅰ) The first set is:

then the solution equation (2.4) will be as

If we choose $b_0 = 2$ and $b_0 = -2$, then the solution equation (7.11), respectively, give:

(Ⅱ) The second set is:

then the solution equation (2.4) will be as

Case Ⅱ: $\mathrm{p = c = 2}$ and $\mathrm{q = d = 2}$.

Since the values of $\mathrm{c}$ and $\mathrm{d}$ can be freely chosen, we set $\mathrm{p = c = 2}$ and $\mathrm{d = q = 2}$ and then the trial function (7.4) becomes

There are some free parameters in (7.17), we set $b_2 = 1$ and $a_1 = a_{-1} = a_{-2} = b_1 = b_{-1} = 0$ for simplicity, the trial function, (7.17) is simplified as follows

Substituting (7.18) into Eq. (2.9), we get an equation in the following form

where $C_n(-8\leq n\leq 16)$ are polynomial expressions in terms of $a_2, a_0, a_{-1}, b_0, b_{-2}$ and $\beta$. Thus, solving the resulting system $C_n = 0(-8\leq n\leq 16)$ simultaneously, we obtain the following set of algebraic equations

(Ⅰ) The first set is:

then the solution equation (2.4) will be as

If we choose $b_0 = 2$ and $b_0 = -2$, then the solution equation (7.22), respectively, give:

(Ⅱ) The second set is:

then the solution equation (2.4) will be as

8.   Discussion and remark

This paper finds many novel hyperbolic, trigonometric, kink, and kink-singular soliton solutions to governing model. With the help of some calculations, surfaces of results reported have been observed in Figures 1–5. These figures are depended on the family conditions which are of important physically. It has been investigated that all figures plotted have symbolized the nonlinear DNA dynamics. These mathematical properties come from trigonometric and hyperbolic function properties. In this sense, from the mathematical and physical points of views, these results take play an important role in explaining waves propagation in nonlinear dispersion. Hence, we consider surfaces plotted in this paper have proved such physical meaning of the solutions. In Figure 1 and Figure 2, we have depicted the 3D, 2D, contour, and density schematic representation of the analytical and numerical solutions at few space positions for three different waves at $x = -1$, $x = 0$, and $x = 1$ by taking $l_1 = 1, l_2 = -1, \alpha = 3, \Omega = 1, R = -1, \mu = 1.5$ for (5.8) and $l_1 = 1, l_2 = 2, \alpha = 3, \Omega = 10, R = 5, \mu = 1.5$ for (5.9). We observe that the breathe soliton wave move in direction $(x, t)$ and increases with move of negative $(x, t)$ to positive $(x, t)$. Also, the periodic wave solution for (6.9) by taking $A_0 = 1, A_2 = 2, k_1 = 2, l_1 = 3, l_2 = 2, \alpha = 2, \Omega = 5, C_1 = 2, C_2 = 3, R = 5$ is presented in Figure 3. Moreover, the rational kink wave solution for the DNA dynamics (6.10) by taking $A_0 = 1, A_2 = 1, k_1 = 2, l_1 = 3, l_2 = 2, \alpha = 2, \Omega = 5, C_1 = 2, C_2 = 3, R = 5,$ is offered in Figure 4. Likewise, the DNA dynamics for (6.11) by taking $A_0 = 0, A_2 = 1, k_1 = 2, l_1 = 3, l_2 = 2, \alpha = 2, \Omega = 5, R = 5,$ along with 3D plot, density plot, contour plot, and 2D plot with at spaces at $x = -1$, $x = 0$, and $x = 1$ are plotted in Figure 5.

9.   Conclusion

The article obtains, the traveling wave solutions of different kinds, which are solitary, topological, singular, periodic and rational solutions to the model for DNA dynamics. The integration mechanisms that are adopted, are improved $\tan(\phi/2)$-expansion scheme, $\exp(-\Omega(\eta))$-expansion scheme, improved $\exp(-\Omega(\eta))$-expansion scheme, generalized (G'/G)-expansion scheme, and exp-function scheme. It is quite visible that these integration schemes has its limitations. Thus, this paper are provides a lot of encouragement for future research in DNA dynamics. Afterwards extra solution methods will be applied to obtain lump and singular soliton solutions to the nonlinear model. In addition to, this model will be considered with other forms of nonlinear media. The constructed results may be helpful in explaining the physical meaning of the studied models and other related nonlinear phenomena models. Results are beneficial to the study of the nonlinear DNA dynamics. All calculations in this paper have been made quickly with the aid of the Maple.

Acknowledgments

The authors would like to thank the given comments and valuable recommendations by respected Editor and the reviewers provided to improve the paper.

Conflict of interest

The authors have declared no conflict of interest.