Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model

Ghazala Akram; Saima Arshed; Maasoomah Sadaf; Manuel De la Sen; Muhammad Abbas; Yasser Salah Hamed

doi:10.1515/dema-2025-0142

Article Open Access

Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model

Ghazala Akram , Saima Arshed , Maasoomah Sadaf , Manuel De la Sen , Muhammad Abbas and Yasser Salah Hamed

Published/Copyright: July 15, 2025

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

The Peyrard-Bishop DNA model is investigated in this study. Two most reliable and efficient analytical techniques, the Jacobi elliptic function method, and the tanh - coth method, are being employed for finding new and novel soliton solutions of the suggested model. The Painlevé-test (P-test) is applied on the proposed DNA model to test the integrability. Few extracted results are graphically examined for studying the dynamical behavior.

Keywords: soliton solutions; fractional Peyrard-Bishop DNA model; β and M-fractional derivatives; Jacobi elliptic function; the tanh-coth expansion method; Painlevé-test

MSC 2010: 35C08; 68Q07; 26A33; 34A08; 35R11

1 Introduction

The topic of fractional calculus (FC) has acquired the attraction of researchers in the last few decades. FC has the ability to solve problems of complex nature occurring in different fields of science and engineering. The first international conference on FC, held in 1974, played a vital role in the development of this branch of mathematics. Many mathematicians, for instance, Lagrange, Abel, Euler, Liouville, Riemann, Caputo, Mainardi, and many others, played a prominent role in the development of FC. Riemann-Liouville, Caputo-Hadamard, Erdelyi-Kober, Katugampola, and Weyl [1–4] have introduced different fractional derivatives and integrals that have been very useful to tackle with different complicated problems.

But after some time, it has been realized by the researchers that Riemann-Liouville and Caputo fractional derivatives have some severe drawbacks. These shortcomings are stated as follows: Riemann-Liouville fractional derivative of an arbitrary constant is not zero. In Caputo definition, the function is assumed to be differentiable. The product law and quotient law for derivatives of functions are not satisfied by Riemann-Liouville and Caputo derivatives. Moreover, the chain rule and the index rule are also not satisfied by these definitions.

FC is used in control systems, signal processing, electrochemistry, economics, biomedical engineering, fluid mechanics, materials science, geophysics, telecommunications, and renewable energy systems.

The shortcomings of Riemann-Liouville and Caputo derivatives have been overcome by introducing β and M-truncated derivatives. The Hunter-Saxton equation has been solved using β -derivative [5]. Atangana introduced β -derivative in nonlocal version [6]. In 2017, Sousa and Capelas de Oliveira [7] introduced M-truncated derivative involving a Mittag-Leffler function with one parameter satisfying some properties of integer-order calculus. Both these definitions are proved to satisfy useful mathematical laws of differential calculus, which encourages further explorations using these newly defined concepts. Fractional differentiation can have a constructive role in modeling various oscillatory processes with damping [1]. The FC of special functions has significant importance and applications in various fields of science and engineering. The fractional integral and differential formulas of the extended hypergeometric-type functions by using the Marichev-Saigo-Maeda operators are extracted [2].

Different techniques have been developed to solve the nonlinear equations. Some numerical approaches are also used to solve fractional nonlinear equations [8]. Shifted Legendre polynomials have been used for constructing the numerical solution for a class of multiterm variable-order fractional differential equations [9].

Many physical phenomena in mechanics, viscoplasticity, optics, biology, and visco-fluid dynamics are modeled by fractional partial differential equations (FPDEs). Therefore, a number of versatile techniques have been introduced by different researchers to solve such FPDEs. Bibi et al. [10] extracted exact solutions for fractional Sharma-Tasso-Olever equation and ( 3 + 1 ) -dimensional Korteweg-de Vries-Zakharov Kuznetsov equation via the ( G ′ G 2 ) -expansion method. Mohyud-Din extracted solitons for space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation and fractional-coupled Burgers equation using the ( G ′ G 2 ) -expansion method [11]. Rizvi et al. [12] obtained the exact soliton solutions of ( 2 + 1 ) -dimensional fractional Schrödinger equation via the F-expansion and improved F-expansion method.

Numerous branches of fundamental science, including biology, optical fibers, and plasma, exhibit nonlinear phenomena. The truth is that nonlinear evolution equations represent these events. Consequently, a variety of effective techniques for obtaining exact solutions to nonlinear evolution equations had been discovered and improved. Raza et al. [13] determined the soliton solution and qualitative analysis of modified equal width Burger’s equation using the unified method. Sadaf et al. [14] investigated soliton solutions to the Fokas system using two variables G ′ G , 1 G -expansion method and generalized projective Riccati equation method. Kundu et al. [15] examined higher-dimensional nonlinear evolution equations by utilizing sine-Gordon expansion method.

Two exact methods are utilized in the presented work. These methods give nontrivial distinct solutions for the considered model. Jacobi elliptic function method (JEFM) utilizes Jacobi elliptic functions and is versatile for periodic and elliptic solutions, including solitons when m → 1 . tanh - coth method uses hyperbolic functions specifically suited for soliton solutions. This work is used to extract soliton solutions of the Peyrard-Bishop DNA model in its fractional form. The extraction of new solitary wave solutions is constructed via the JEFM and the tanh - coth expansion method. The suggested analytical techniques are also found to be very reliable in obtaining soliton solutions for nonlinear partial differential equations (NLPDEs). The integrability of the proposed model is tested using the Painlevé-test.

The proposed model for DNA denaturation was presented in 1989 [16]. DNA denaturation is the process of breaking down the DNA in such a way that the hydrogen bonds in the DNA break to unwind the double strands. The proposed model was a simple yet useful extension of Ising models, which were previously being used to investigate DNA denaturation. Recently, the proposed model has been examined by employing different analytical techniques [17–19], which results in different types of traveling wave solutions. The mathematical model of Peyrard-Bishop DNA is as follows:

(1) ϕ t t − ( δ 1 + 3 δ 2 ϕ x 2 ) ϕ x x − 2 λ ρ e − λ ϕ ( e − λ ϕ − 1 ) = 0 ,

where δ 1 = β 1 p d 2 , δ 2 = β 2 p d 4 , ρ = ν p , λ = 2 ε , and d is the inter-site nucleotide distance in the DNA strands [20,21].

Equation (1) effectively captures the essential physics of DNA denaturation by combining elements of wave propagation, nonlinear elasticity, and potential energy interactions. It models how nucleotides move and interact within the DNA strands, considering both linear and nonlinear effects, as well as the nature of the hydrogen bonds. This comprehensive approach justifies the equation’s form and its relevance to describing DNA denaturation processes.

The fractional Peyrard-Bishop DNA model is essential for studying DNA denaturation due to its ability to accurately represent complex, nonlocal, and memory-dependent biological processes. It addresses the limitations of traditional models, provides precise and versatile descriptions, and supports advanced analytical techniques, ultimately leading to a deeper understanding and better predictions of the DNA denaturation process.

In β -derivative, the proposed model has the following form:

(2) D t 2 γ ϕ − ( δ 1 + 3 δ 2 ( D x γ ϕ ) 2 ) D x 2 γ ϕ − 2 λ ρ e − λ ϕ ( e − λ ϕ − 1 ) = 0 ,

where D t γ and D x γ represent the β -derivatives.

The dynamical model in M-truncated derivative has the following form:

(3) D M , t 2 γ , β ϕ − ( δ 1 + 3 δ 2 ( D M , x γ , β ) 2 ) D M , x 2 γ , β ϕ − 2 λ ρ e − λ ϕ ( e − λ ϕ − 1 ) = 0 ,

where D M , t γ , β and D M , x γ , β are the M-truncated derivatives.

The article is divided into six sections. Preliminaries are included in Section 2. The mathematical analysis of the proposed model and the solution obtained by proposed methods are included in Section 3. The algorithmic steps of Painlevé-test and its application on the proposed model are carried out in Section 4. Section 5 presents the graphical illustrations of some constructed solutions. Section 6 includes concluding remarks.

2 Preliminaries

Although various attempts have been made over the years to define fractional differential operators, most of them fail to satisfy one or more important mathematical properties exhibited by the classical integer-order derivatives. Recently, the β -derivative and M-truncated derivative have been proposed, which satisfy many useful mathematical properties just like classical derivatives. Due to these advantageous properties, both these definitions are applicable in a wide range of mathematical and theoretical investigations. The definitions of fractional derivatives and integrals along with their fundamental properties are discussed in this section.

2.1 β -derivative and its properties

The conformable fractional derivative was recently proposed and is proved to satisfy quotient law, product law, and many other important mathematical properties. Motivated by the successful studies using conformable derivative, another kind of conformable derivative is defined, which is known as β -derivative.

Definition 1

The β -derivative can be defined as [5]:

D x γ 0 β g ( x ) = lim ε → 0 g x + ε x + 1 Γ ( γ ) 1 − γ − g ( x ) ε , 0 < γ < 1 .

The β -derivative has the following properties [5,6]:

The β -derivative is a linear operator:
D x γ 0 β ( e g ( x ) + s h ( x ) ) = e D x γ 0 B g ( x ) + s D x γ 0 B h ( x ) , ∀ r , s ∈ ℜ .
It obeys the product rule:
D x γ 0 β ( g ( x ) ⁎ h ( x ) ) = h ( x ) D x γ 0 β g ( x ) + g ( x ) D x γ 0 β h ( x ) .
It obeys the quotient rule:
D x γ 0 β g ( x ) h ( x ) = h ( x ) D x γ 0 β g ( x ) − g ( x ) D x γ 0 β h ( x ) h 2 ( x ) .
The β -derivative of a constant is zero:
D x γ 0 β c = 0 , for c any constant .

The β -fractional integral [5] is defined as

0 β I x γ f ( x ) = ∫ 0 x v + 1 Γ [ α ] γ − 1 f ( v ) d v .

2.2 M-truncated derivative

The M-truncated derivative, M-truncated integral, and their properties have been defined in this subsection. The motivation to choose the M-truncated derivative is driven by its ability to overcome the limitations of traditional fractional derivatives, its mathematical consistency, flexibility, and effectiveness in modeling and solving complex problems across different fields.

Definition 2

Suppose g : [ r , ∞ ) → ℜ and x > 0 . Also, 0 < γ < 1 , and τ > 0 . The M-truncated derivative of g of order γ is represented by D γ , τ j M and defined as [22]

D γ , τ j M g ( x ) = lim ε → 0 g ( x E τ j ( ε x − γ ) ) − g ( x ) ε ,

∀ t > 0 , where i E τ ( ⋅ ) , τ > 0 is a truncated Mittag-Leffler function of one parameter defined as

E τ j ( x ) = ∑ k = 0 j x k Γ ( τ k + 1 ) .

Definition 3

Suppose that g ( x ) is defined on ( r , x ] , r ≥ 0 ; x ≥ r , and 0 < γ < 1 , then the left M-truncated integral is defined as

I γ , τ M r g ( x ) = ∫ r x g ( t ) d ρ ( t , r ) = Γ ( τ + 1 ) ∫ r x f ( t ) ( t − r ) γ − 1 d t ,

where d ρ ( t , s ) = Γ ( τ + 1 ) ( t − s ) γ − 1 .

Likewise, the right M-truncated integral is defined as

I M s γ , τ g ( x ) = ∫ x s g ( t ) d ρ ( s , t ) = Γ ( τ + 1 ) ∫ x s g ( t ) ( s − t ) γ − 1 d t .

The M-truncated derivative has the following properties [23]. Suppose 0 < γ < 1 , τ > 0 , e , s ∈ ℜ , and g a n d h are γ -differentiable at x > 0 . The M-truncated derivative has the following properties [23]. Suppose 0 < γ < 1 , τ > 0 , e , s ∈ ℜ , and g a n d h are γ -differentiable at x > 0 .

M-truncated derivative is a linear operator:
D γ , τ j M ( e g ( x ) + s h ( x ) ) = e D γ , τ j M g ( x ) + s j M D γ , τ h ( x ) , ∀ e , s ∈ ℜ .
It satisfies the product rule:
D γ , τ j M ( g ( x ) * h ( x ) ) = g ( x ) j M D γ , τ h ( x ) + h ( x ) D γ , τ j M g ( x ) .
It satisfies the quotient rule:
D γ , τ j M g h ( x ) = h ( x ) j M D γ , τ g ( x ) − g ( x ) D γ , τ j M h ( x ) h ( x ) 2 .
The M-truncated derivative for a differentiable function g ( x ) is defined as
D γ , τ j M ( g ( x ) ) = x 1 − γ Γ [ τ + 1 ] d g ( x ) d x .

3 Mathematical analysis of the proposed model

In this section, the exact soliton solutions of equations (2)–(3) are determined using the Jacobi elliptic function method and the tanh - coth expansion method. The application of the proposed methods requires that equations (2)–(3) must be reduced to an ordinary differential equation using traveling wave transformation [24,25]. Hence, the transformation

(4) ϕ ( x , t ) = U ( ξ ) ,

has been employed, where ξ is the suitably defined traveling wave parameter.

The traveling wave parameter ξ is defined in the following two ways depending on the definition of fractional derivative.

For β -derivative, ξ is defined in the following form:

(5) ξ = 1 γ x + 1 Γ ( γ ) γ − μ γ t + 1 Γ ( γ ) γ .

For M-truncated derivative, ξ is defined in the following form:

(6) ξ = Γ ( τ + 1 ) γ ( x γ − μ t γ ) ,

where μ represents the speed of traveling wave. Utilizing the transformation equation (4) together with equations (5)–(6), the obtained ordinary differential equation is

(7) μ 2 ( U ″ ) − δ 1 + 3 δ 2 ( U ′ 2 ) U ″ − 2 λ ρ e − λ U ( e − λ U − 1 ) = 0 .

Equation (7) can be simplified as follows: multiplying equation (7) by U ′ and integrating the obtained equation imply

(8) ( μ 2 − δ 1 ) 2 ( U ′ ) 2 − 3 4 δ 2 ( U ′ ) 4 + ρ e − λ U ( e − λ U − 2 ) + C = 0 ,

where C is the constant of integration.

Taking

(9) u ( ξ ) = e − λ U ( ξ )

and substituting equation (9) into equation (8), the aforementioned expression becomes

(10) ( μ 2 − δ 1 ) 2 λ 2 u 2 ( u ′ ) 2 − 3 4 λ 4 δ 2 ( u ′ ) 4 + ρ u 5 ( u − 2 ) + C u 4 = 0 .

Equation (10) is solved in the following subsections using proposed analytical methods. Both these techniques determine the exact closed form solutions in the form of a truncated power series where the number of terms is determined by the well-known homogeneous balance principal. According to this principal, the highest order of derivative and the highest degree of nonlinear term in equation (10) are balanced. Consequently, the balancing number N = 2 is obtained. In the next subsections, the proposed methods are applied on equation (10) to determine the exact solutions, which in turn yield the soliton solutions of the governing model given by equations (2)–(3).

3.1 JEFM

According to JEFM [24], the solution for balancing number N = 2 can be written, as

(11) u ( ξ ) = A 0 + A 1 θ ( ξ ) + A 2 θ 2 ( ξ ) ,

where A 0 , A 1 , and A 2 are to be determined. Inserting equation (11) into equation (10), the three different possibilities of θ ( ξ ) can be considered

Case 1

(12) θ ( ξ ) = sn ( p ξ , m ) ,

or Case 2

(13) θ ( ξ ) = cn ( p ξ , m ) ,

or Case 3

(14) θ ( ξ ) = dn ( p ξ , m ) ,

where sn , cn , and dn represent the Jacobi elliptic functions, and m represents the modulus of ellipticity.

An algebraic system of equations has been constructed by assembling the coefficients of different powers of θ ( ξ ) and equating to zero. Upon solving the system of equations for Case 1, the values of parameters are calculated as

(15) A 0 = − 2 3 δ 2 m 2 λ 2 ρ , A 1 = 0 , A 2 = 2 3 δ 2 m 2 λ 2 ρ , μ = − 6 δ 2 λ 2 ρ − 3 ( λ 2 δ 1 + 12 m 2 δ 2 ) 3 4 λ , C = − 4 ( 3 δ 2 m 2 λ 2 ρ + 3 m 4 δ 2 ) λ 4 .

The obtained result for equation (1) is

(16) ϕ ( x , t ) = − 1 λ ln 2 3 δ 2 m 2 ( − 1 + sn 2 ( p ξ , m ) ) λ 2 ρ .

For m → 1 , equation (16) becomes

(17) ϕ ( x , t ) = − 1 λ ln 2 3 δ 2 m 2 s e c h 2 ( p ξ ) λ 2 ρ , ρ > 0 , δ 2 > 0 ,

representing solitons for equation (1).

According to Case 2, the values of parameters are calculated as

(18) A 0 = 0 , A 1 = 0 , A 2 = 2 3 δ 2 m 2 λ 2 ρ , μ = 6 δ 2 λ 2 ρ + 3 ( λ 2 δ 1 + 12 m 2 δ 2 ) 3 4 λ , C = − 4 ( 3 δ 2 m 2 λ 2 ρ + 3 m 4 δ 2 ) λ 4 .

The result obtained for equation (1) is

(19) ϕ ( x , t ) = − 1 λ ln 2 3 δ 2 m 2 cn 2 ( p ξ , m ) λ 2 ρ .

For m → 1 , equation (19) becomes

(20) ϕ ( x , t ) = − 1 λ ln 2 3 δ 2 m 2 s e c h 2 ( p ξ ) λ 2 ρ ,

with constraint conditions, ρ > 0 and δ 2 > 0 .

According to Case 3, the values of parameters are calculated as

(21) A 0 = − 2 3 δ 2 m 2 λ 2 ρ , A 1 = 0 , A 2 = 2 3 δ 2 m 2 λ 2 ρ , μ = 6 δ 2 λ 2 ρ − 3 ( − λ 2 δ 1 + 12 m 2 δ 2 ) 3 4 λ , C = 4 ( 3 δ 2 m 2 λ 2 ρ − 3 m 4 δ 2 ) λ 4 .

The result obtained for equation (1) is

(22) ϕ ( x , t ) = − 1 λ ln 2 3 δ 2 m 2 ( − 1 + dn 2 ( p ξ , m ) ) λ 2 ρ .

For m → 1 , equation (22) becomes

(23) ϕ ( x , t ) = − 1 λ ln 2 3 δ 2 m 2 tanh 2 ( p ξ ) λ 2 ρ .

The obtained solution represents the soliton and valid if ρ > 0 and δ 2 > 0 .

The traveling wave variable ξ has been defined in equations (5) and (6).

3.2 tanh - coth expansion method

According to tanh - coth expansion method [25], the predicted solution for balancing number N = 2 can be written as

(24) u ( ξ ) = A 0 + A 1 tanh m ξ + A 2 tanh 2 m ξ + B 1 tanh − 1 m ξ + B 2 tanh − 2 m ξ ,

where A 0 , A 1 , and A 2 are the arbitrary parameters to be determined. Then, equation (24) is inserted along with the derivatives into equation (10). An algebraic system of equations has been constructed by assembling the coefficients of different powers of tanh ξ and equating to zero. Upon solving the system of equations, the following sets of solutions have been reported.

Set 1

According to the suggested technique, the values of parameters are calculated as

(25) A 0 = 2 3 δ 2 m 2 λ 2 ρ , A 1 = 0 , A 2 = − 2 3 δ 2 m 2 λ 2 ρ , B 1 = 0 , B 2 = 0 , μ = λ 2 ( δ 1 ∓ 2 3 ρ δ 2 ) + 12 m 2 δ 2 λ , C = 4 ( 3 δ 2 m 2 λ 2 ρ − 3 m 4 δ 2 ) λ 4 .

The bright soliton solutions have been extracted for equation (1) as

(26) ϕ ( x , t ) = − 1 λ ln 2 3 δ 2 m 2 sech 2 ( m ξ ) λ 2 ρ , ρ > 0 , δ 2 > 0 .

Set 2

According to the suggested technique, the values of parameters are calculated as

(27) A 0 = − 4 3 δ 2 m 2 λ 2 ρ , A 1 = 0 , A 2 = 2 3 δ 2 m 2 λ 2 ρ , B 1 = 0 , B 2 = 2 3 δ 2 m 2 λ 2 ρ , μ = λ 2 ( δ 1 ∓ 2 3 ρ δ 2 ) + 48 m 2 δ 2 λ , C = 16 ( 3 δ 2 m 2 λ 2 ρ − 12 m 4 δ 2 ) λ 4 .

The obtained solution for equation (1) is

(28) ϕ ( x , t ) = − 1 λ ln 8 3 δ 2 m 2 csch 2 ( 2 m ξ ) λ 2 ρ ,

with constraint conditions ρ > 0 and δ 2 > 0 .

Set 3

According to the suggested technique, the values of parameters are calculated as

(29) A 0 = − 2 3 δ 2 m 2 λ 2 ρ , A 1 = 0 , A 2 = 0 , B 1 = 0 , B 2 = 2 3 δ 2 m 2 λ 2 ρ , μ = λ 2 ( δ 1 ∓ 2 3 ρ δ 2 ) + 12 m 2 δ 2 λ , C = 4 ( 3 δ 2 m 2 λ 2 ρ − 3 m 4 δ 2 ) λ 4 .

The obtained solution for equation (1) is

(30) ϕ ( x , t ) = − 1 λ ln 2 3 δ 2 m 2 csch 2 ( m ξ ) λ 2 ρ .

The obtained results hold for ρ > 0 and δ 2 > 0 .

4 Painlevé test

The Painlevé test is the most efficient approach for investigating the integrability of NLPDEs [26]. The integrability of NLPDEs is a key concept and has vital importance among researchers [27]. The algorithm for the Painlevé test is as follows:

Step 1 Consider the following expression and insert it into equation (1) for evaluating dominant behavior:

(31) ϕ ( x , t ) = α θ ε ( x , t ) .

The parameter ε indicates the dominant behavior that has been evaluated first, whereas α is an arbitrary parameter.

Assuming

(32) ϕ ( x , t ) = α 0 θ ε ( x , t ) ,

and inserting the value of ε into equation (32), the resulting equation is inserted into equation (1). Now, α 0 is evaluated by putting the coefficient the smallest power of θ ( x , t ) to zero.

Step 2 In order to evaluate the resonances, suppose that

(33) ϕ ( x , t ) = α 0 θ ε ( x , t ) + α r θ ε + r ( x , t ) .

Put equation (33) into equation (1), then the sum of coefficients of α r , including the lowest power of θ ( x , t ) , is assembled. The resonances are determined from the system of equations obtained by the proposed technique.

Step 3 The expression of ϕ ( x , t ) has the following form:

(34) ϕ ( x , t ) = θ ε ( x , t ) ∑ i = 0 r m α i θ i ( x , t ) ,

where r m represents the maximum value of resonances and α i represents the constant of integration. Equation (34) is substituted into equation (1), and α i is evaluated after equalizing all coefficients of different powers of θ ( x , t ) equals zero. The Painlevé test is said to hold if, the integrating constants α r at resonances r j , j = 1 , 2 , 3 , … determined to be arbitrary.

4.1 Application of Painlevé test on proposed model

In this section, Painlevé test has been successfully applied on the governing model. The dominant behavior of the proposed model has been obtained by plugging equation (31) into equation (1), which yields ε = − 2 . Now, ε is inserted into equation (32)

(35) ϕ ( x , t ) = α 0 θ − 2 ( x , t ) .

Substituting equation (35) into equation (1) and following the proposed algorithm as discussed in Section 4, we obtain

(36) α 0 = 2 3 δ 2 ( θ x ( x , t ) ) 2 λ 2 ρ .

Substituting ε into α 0 , equation (33) becomes

(37) ϕ ( x , t ) = 2 3 δ 2 ( θ x ( x , t ) ) 2 λ 2 ρ θ − 2 ( x , t ) + α s θ s − 2 ( x , t ) .

Substituting equation (37) into equation (1), then taking the coefficient of least power of θ ( x , t ) to zero, we obtain

(38) 1152 3 + 864 3 r − 288 r 2 = 0 .

Solving the aforementioned equation for r gives

(39) r 1 = − 1 , r 2 = 4 .

To determine the compatibility condition, insert ε and maximum value of resonances s = r m = 4 into equation (34)

(40) ϕ ( x , t ) = θ − 2 ( x , t ) ∑ r = 0 4 α r θ r ( x , t ) .

Equation (40) is substituted into equation (1). The value of constants of integration α r , r = 1 , 2, 3, 4 is being evaluated by equalizing the coefficients of least powers of θ ( x , t ) to zero.

For r = 1 ,

(41) α 1 = − 2 λ 2 3 δ 2 ρ θ x x ( x , t ) .

Similarly, at r = 2 ,

(42) α 2 = 3 288 ( θ x ( x , t ) ) 2 λ 2 ρ δ 2 ( 32 3 ( θ x ( x , t ) ) 2 λ 2 ρ δ 2 + 16 ( θ x ( x , t ) ) 2 λ 2 ρ δ 1 − 16 ( θ t ( x , t ) ) 2 λ 2 ρ + 192 θ x ( x , t ) θ x x x ( x , t ) ρ δ 2 − 219 ( θ x x ( x , t ) ) 2 ρ δ 2 ) .

Likewise, at r = 3 ,

(43) α 3 = − 3 1,152 δ 2 5 ⁄ 2 ( θ x ( x , t ) ) 4 λ 2 ρ 3 ( 880 δ 2 5 ⁄ 2 ( θ x ( x , t ) ) 2 θ x x ( x , t ) 3 λ 2 ρ 3 − 96 θ t t ( x , t ) δ 2 2 ( θ x ( x , t ) ) 2 λ 2 ρ 5 ⁄ 2 − 464 ( θ t ( x , t ) ) 2 θ 2 2 θ x x ( x , t ) λ 2 ρ 5 ⁄ 2 + 560 ( θ x ( x , t ) ) 2 θ x x ( x , t ) ρ 5 ⁄ 2 λ 2 δ 2 2 p 1 + 8448 δ 2 3 θ x ( x , t ) θ x x ( x , t ) θ x x x ( x , t ) ρ 5 ⁄ 2 − 10833 δ 2 3 ( θ x x ( x , t ) ) 3 ρ 5 ⁄ 2 ) .

Likewise, at r = 4 ,

(44) α 4 = α 4 .

After satisfying the compatibility conditions, it is proved that the proposed model is integrable.

5 Graphical illustration

This section contains the 3D and 2D plots of some of the obtained solutions for studying fractional impact. The numerical simulations for three of the obtained exact soliton solutions of the considered DNA model are presented.

The graphical representations of equation (17) have been carried out in Figures 1, 2, and 3 by assuming the values of arbitrary parameters as λ = 1.8 , ρ = 1 , δ 1 = 0.5 , δ 2 = 0.7 , and τ = 0.70 and fractional parameter γ = 0.45 , γ = 0.75 , and γ = 0.95 , respectively. The evolution of soliton expressed by equation (17) is shown through Figures 1(a) to 3(a) using β -derivative. The evolution of the soliton is also shown using M-derivative through Figures 1 (b) to 3 (b). It is evident that the soliton takes a stable shape much faster with increasing value of fractional order γ for β -derivative as compared to M-derivative. For a closer observation, the comparison of the 2D plots of the soliton profiles presented in part (a) and (b) of Figures 1–3 is shown in part (c), keeping x constant. A similar 2 D comparison is also observed in part (d) of Figures 1–3, keeping t constant.

$Figure 1 (a) 3D graph w.r.t β \beta -derivative, (b) 3D graph w.r.t M-derivative, (c) 2D line graph with the variation in t t , and (d) 2D line graph with the variation in x x of equation (17), where the fractional operator γ = 0.45 \gamma =0.45 .$

Figure 1

(a) 3D graph w.r.t β -derivative, (b) 3D graph w.r.t M-derivative, (c) 2D line graph with the variation in t , and (d) 2D line graph with the variation in x of equation (17), where the fractional operator γ = 0.45 .

$Figure 2 (a) 3D plot of β \beta -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t t , and (d) 2D line graph with the variation in x x of equation (17), where the fractional operator γ = 0.75 \gamma =0.75 .$

Figure 2

(a) 3D plot of β -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t , and (d) 2D line graph with the variation in x of equation (17), where the fractional operator γ = 0.75 .

$Figure 3 (a) 3D plot of β \beta -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t t , and (d) 2D line graph with the variation in x x of equation (17), where the fractional order γ = 0.95 \gamma =0.95 .$

Figure 3

Similarly, the graphical representations of equation (23) are shown in Figures 4, 5 and 6 by assuming the values of arbitrary parameters as λ = 1.8 , ρ = 1 , δ 1 = 0.5 , δ 2 = 0.7 , and τ = 0.70 and fractional parameter γ = 0.45 , γ = 0.75 , and γ = 0.95 , respectively.

$Figure 4 (a) 3D plot of β \beta -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t t , and (d) 2D line graph with the variation in x x of equation (23), where the fractional operator γ = 0.45 \gamma =0.45 .$

Figure 4

(a) 3D plot of β -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t , and (d) 2D line graph with the variation in x of equation (23), where the fractional operator γ = 0.45 .

$Figure 5 (a) 3D plot of β \beta -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t t , and (d) 2D line graph with the variation in x x of equation (23), where the fractional operator γ = 0.75 \gamma =0.75 .$

Figure 5

$Figure 6 (a) 3D plot of β \beta -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t t , and (d) 2D line graph with the variation in x x of equation (23), where the fractional operator γ = 0.95 \gamma =0.95 .$

Figure 6

Moreover, the graphical representations of equation (30) are shown in Figures 7 and 8 λ = 1.8 , ρ = 1 , δ 1 = 0.5 , δ 2 = 0.7 , and τ = 0.70 and fractional parameter γ = 0.45 , and γ = 0.75 , respectively.

$Figure 7 (a) 3D plot of β \beta -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t t , and (d) 2D line graph with the variation in x x of equation (30), where the fractional operator γ = 0.45 \gamma =0.45 .$

Figure 7

(a) 3D plot of β -derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t , and (d) 2D line graph with the variation in x of equation (30), where the fractional operator γ = 0.45 .

$Figure 8 (a) 3D plot of γ-derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t, and (d) 2D line graph with the variation in x of equation (30), where the fractional operator γ = 0.75 \gamma =0.75 .$

Figure 8

(a) 3D plot of γ-derivative, (b) 3D plot of M-derivative, (c) 2D line graph with the variation in t, and (d) 2D line graph with the variation in x of equation (30), where the fractional operator γ = 0.75 .

The graphical observations indicate that the shape of soliton goes through a continuous change for increasing values of fractional parameter γ . The type of soliton is similar for the two types of fractional derivatives although there is a noticeable difference in the shape of solitons at different values of γ .

6 Conclusion

This study addresses the fractional Peyrard-Bishop DNA model. Two fractional derivatives are employed in this article for studying the fractional behavior of the model. These fractional derivatives are very useful in studying memory and heredity properties of many complicated problems found in science and engineering. The proposed fractional model is solved using JEFM and tanh - coth expansion method for the first time in this article. By the application of β -derivative and M-truncated derivative, it has been noted that for assigning various values of fractional parameter γ , the β -derivative approaches the classical derivative (for γ = 1 ) more faster than M-truncated derivative. Some constructed solutions are plotted graphically by assigning particular values of arbitrary parameters. The proposed model equation (1) via Painlevé test. The nonlinear terms in the proposed model play a significant role in the construction of solutions using JEFM and tanh - coth expansion method. The balancing principle applied on equation (10) provides N = 2 . The value of N depends on nonlinear terms of the governing equation, which is essential for writing predicted solutions via proposed methods. On comparing our results with the results of [18], [19], and [28], it has been observed that Jacobi elliptic function solutions, dark solitons, bright solitons, and singular soliton solutions have been reported in this article with fractional effects.

Acknowledgment

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through Project Number (TU-DSPP-2024-47).

Funding information: M. De la Sen is grateful to the Basque Government for its support through Grants IT1555-22 and KK-2022/00090 and to MCIN/AEI 269.10.13039/501100011033 for Grant PID2021-1235430B-C21/C22. This research was funded by Taif University, Saudi Arabia, Project No. (TU-DSPP-2024-47).
Author contributions: GA: formal analysis, visualization, writing – review and editing. SA: writing – original draft, methodology. MS: supervision, methodology, writing – original draft. MDS: formal analysis, visualization, writing – review and editing. MA: visualization, software, writing – original. YSH visualization, methodology, writing – original draft. All authors have read and agreed to publish the manuscript.
Conflict of interest: The authors declare that they have no conflicts of interest to report regarding the present study.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: All data generated or analyzed during this study are included in this published article.

References

[1] S. Rekhviashvili, A. Pskhu, P. Agarwal, and S. Jain, Application of the fractional oscillator model to describe damped vibrations, Turkish J. Phys. 43 (2019), no. 3, 236–242. 10.3906/fiz-1811-16Search in Google Scholar

[2] S. Jain, C. Cattani, and P. Agarwal, Fractional hypergeometric functions, Symmetry 14 (2022), no. 4, 714. 10.3390/sym14040714Search in Google Scholar

[3] U. N. Katugampola, A new fractional derivative with classical properties, arXiv preaprint arXiv:1410.6535, 2014. Search in Google Scholar

[4] R. F. Camargo and E. de Oliveira, Fractional Calculus (in portuguese), Editora Livraria da Física, São Paulo, 2015. Search in Google Scholar

[5] A. Atangana, D. Baleanu, and A. Alsaedi, Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal, Open Phys. 14 (2016), no. 1, 145–149. 10.1515/phys-2016-0010Search in Google Scholar

[6] V. F. Morales-Delgado, J. F. Gómez-Aguilar, and M. A. Taneco-Hernandez, Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense, AEU - Int. J. Electron. Commun. 85 (2018), 108–117. 10.1016/j.aeue.2017.12.031Search in Google Scholar

[7] E. Capelas de Oliveira and C Sousa, On the local m-derivative, Prog. Fract. Differ. Appl. 4 (2018), 479–492. 10.18576/pfda/040207Search in Google Scholar

[8] A. A. El-Sayed, D. Baleanu, and P. Agarwal, A novel Jacobi operational matrix for numerical solution of multi-term variable-order fractional differential equations, J. Taibah Univ. Sci. 14 (2020), no. 1, 963–974. 10.1080/16583655.2020.1792681Search in Google Scholar

[9] A. A. El-Sayed and P. Agarwal, Numerical solution of multiterm variable-order fractional differential equations via shifted legendre polynomials, Math. Methods Appl. Sci. 42 (2019), no. 11, 3978–3991. 10.1002/mma.5627Search in Google Scholar

[10] S. Bibi, S.T. Mohyud-Din, R. Ullah, N. Ahmed, and U. Khan, Exact solutions for STO and (3+1)-dimensional KdV-ZK equations using G′G2-expansion method, Results Phys. 7 (2017), 4434–4439. 10.1016/j.rinp.2017.11.009Search in Google Scholar

[11] S.T. Mohyud-Din and S. Bibi, Exact solutions for nonlinear fractional differential equations using (G′G2)-expansion method, Alex. Eng. J. 57 (2018), no. 2, 1003–1008. 10.1016/j.aej.2017.01.035Search in Google Scholar

[12] S. T. R. Rizvi, K. Ali, S. Bashir, M. Younis, R. Ashraf, and M. O. Ahmad, Exact soliton of (2+1)-dimensional fractional Schrödinger equation, Superlattices Microstruct. 107 (2017), 234–239. 10.1016/j.spmi.2017.04.029Search in Google Scholar

[13] N. Raza, S. Arshed, F. Salman, J. F. Gómez-Aguilar, and J. Torres-Jiménez, Phase characterization and new optical solitons of a pulse passing through nonlinear dispersive media, Modern Phys. Lett. B 36 (2022), 2250098. 10.1142/S0217984922500981Search in Google Scholar

[14] M. Sadaf, S. Arshed, G. Akram, and Igra, Exact soliton and solitary wave solutions to the fokas system using two variables G′G,1G-expansion technique and generalized projective Riccati equation method, Optik 268 (2022), 169713. 10.1016/j.ijleo.2022.169713Search in Google Scholar

[15] P. R. Kundu, M. R. A. Fahim, M. E. Islam, and M. A. Akbar, The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis, Heliyon 7 (2021), e06459. 10.1016/j.heliyon.2021.e06459Search in Google Scholar PubMed PubMed Central

[16] M. Peyrard and A. R. Bishop, Statistical mechanis of a nonlinear model for DNA denaturation, Phys. Rev. Lett. 62 (1989), 2755. 10.1103/PhysRevLett.62.2755Search in Google Scholar PubMed

[17] B. Ghanbari and J.F. Gómez-Aguilar, The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative, Rev. Mex. Fís. 65 (2019), no. 5, 503–518. 10.31349/RevMexFis.65.503Search in Google Scholar

[18] J. Manafian, O. A. Ilhan, and S. A. Mohammed, Forming localized waves of the nonlinearity of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model, AIMS Math. 5 (2020), no. 3, 2461–2483. 10.3934/math.2020163Search in Google Scholar

[19] A. Zafar, K. K. Ali, M. Raheel, N. Jafar, and K.S. Nisar, Soliton solutions to the DNA Peyrard-Bishop equation with beta-derivative via three distinctive approaches, Eur. Phys. J. Plus 135 (2020), no. 9, 1–17. 10.1140/epjp/s13360-020-00751-8Search in Google Scholar

[20] L. Najera, M. Carrillo, and M. A. Aguero, Non-classical solitons and the broken hydrogen bonds in DNA vibrational dynamics, Adv. Studies Theor. Phys. 4 (2010), 495–510. Search in Google Scholar

[21] K. K. Ali, C. Cattani, J. F. Gómez-Aguilar, D. Baleanu, and M. S. Osman, Analytical and numerical study of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model, Chaos Solitons Fractals 139 (2020), 110089. 10.1016/j.chaos.2020.110089Search in Google Scholar

[22] E. Capelas de Oliveira and J. Vanterler da C. Sousa, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Int. J. Anal. Appl. 16 (2018), no. 1, 83–96. Search in Google Scholar

[23] S. Salahshour, A. Ahmadian, S. Abbasbandy, and D. Baleanu, M-fractional derivative under interval uncertainty: Theory, properties and applications, Chaos Solitons Fractals 117 (2018), 84–93. 10.1016/j.chaos.2018.10.002Search in Google Scholar

[24] S. Liu, Z. Fu, S. Liu, and Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289 (2001), no. 1–2, 69–74. 10.1016/S0375-9601(01)00580-1Search in Google Scholar

[25] E. J. Parkes, Observations on the tanh-coth expansion method for finding solutions to nonlinear evolution equations, Appl. Math. Comput. 217 (2010), no. 4, 1749–1754. 10.1016/j.amc.2009.11.037Search in Google Scholar

[26] S. T. R. Rizvi, A. R. Seadawy, M. Younis, I. Ali, S. Althobaiti, and S. F. Mahmoud, Soliton solutions, Painleve analysis and conservation laws for a nonlinear evolution equation, Results Phys. 23 (2021), 103999. 10.1016/j.rinp.2021.103999Search in Google Scholar

[27] G. Akram, S. Arshed, M. Sadaf, and Zainab, Extraction of new exact soliton solutions and Painlevé-test for fractional Cahn-Allen equation, Opt. Quantum Electron. 54 (2022), no. 46, 1–16. 10.1007/s11082-021-03407-8Search in Google Scholar

[28] G. Akram, S. Arshed, and Zainab, Soliton solutions for fractional DNA Peyrard-Bishop equation via the extended G′G2-expansion method, Phys. Scr. 96 (2021), 1–12. 10.1088/1402-4896/ac0955Search in Google Scholar

Received: 2023-03-11

Revised: 2024-08-11

Accepted: 2025-05-22

Published Online: 2025-07-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0142

Keywords for this article

soliton solutions; fractional Peyrard-Bishop DNA model; β and M-fractional derivatives; Jacobi elliptic function; the tanh-coth expansion method; Painlevé-test

Creative Commons

BY 4.0