# New Fractal Soliton Solutions and Sensitivity Visualization for Double-Chain DNA Model

Authors: Zara Hassan; Nauman Raza; Abdel-Haleem Abdel-Aty; Mohammed Zakarya; Riaz Ur Rahman; Adeela Yasmeen; Abdisalam Hassan Muse; Emad E. Mahmoud
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

This article discusses dynamics of the fractal double-chain deoxyribonucleic acid model. This structure contains two long elastic homogeneous strands that serve as two polynucleotide chains of deoxyribonucleic acid molecules, bounded by an elastic membrane indicating hydrogen bonds between the base pairs of two chains. The semi-inverse variational principle and auxiliary equation method are employed to extricate soliton solutions. The collection of retrieved exact solutions includes bright, dark, periodic, and other solitons. The constraint conditions emerge naturally which ensure the presence of these solutions. Additionally, 2D and 3D graphs showing the impact of fractals on solutions are included. These plots use appropriate parameter values. Furthermore, sensitivity analysis of the considered model is also acknowledged. The outcomes reveal that these techniques are reliable, effective, and applicable to various biological systems.


---

## Body

## 1. Introduction

Deoxyribonucleic acid (DNA) is an interesting nonlinear model of biological sciences [1, 2]. Since it is needed for protein-coding, inheritance, and genetic instruction manual for life, it contains instructions for cell growth, reproduction, and death of a human. DNA molecules are the foundation of life so their dynamics are one of the interesting problems in biophysics. Researchers have been studying this structure during the last decades [3, 4]. The study of DNA mechanism predicts the presence of significant nonlinear structures. It has been established that localized waves are caused by nonlinearity and these waves are fascinating because they can transmit power without causing power loss [5–7].Nonlinear partial differential equations (NLPDEs) have been considered for studying several nonlinear physical phenomena. Many physicists and mathematicians have worked hard to develop further precise alternatives to NLPDEs for a better understanding of these processes. Therefore, exact solutions of NLPDEs are essential for exploring physical explanations and qualitative aspects of different mechanisms [8–18]. These solutions demonstrate the dynamics of several nonlinear complex models symbolically and physically. Numerous methods were implemented to attain exact and wave solutions of the nonlinear governing model [19–26].This paper introduces the fractal double-chain DNA model to scrutinize the double-helix structure. Fractal calculus has been a flourishing subject of biology, mathematics, and physics because it deals with the modeling of distinct nonlinear procedures [27–34]. Since the fractal model covers many powerful properties, which the traditional system fails to explain. The field of biophysics greatly benefits from a unique class of solitary wave solutions referred to as solitons of proposed problems [3]. Wave packets known as solitons propagate at a constant pace and maintain their shape despite nonlinearities and dispersion [35–43]. Semi-inverse scheme and auxiliary equation method (AEM) are two effective techniques implemented to derive a set of solitons in this manuscript.The Ritz-like approach linked with the variational principle termed as He’s semi-inverse variational method [44] is applied to attain the bright solitons of fractal DNA model which may aid biologists to comprehend its physical significance. An effective and straightforward algebraic method for finding soliton solutions is the semi-inverse scheme [45]. Many authors contributed to develop this technique to analyze fractal models in distinct scientific fields [33, 46, 47]. Another method adopted here is AEM that retrieves dark, periodic, bright, and other shaped solitons. This reliable strategy is employed to obtain dual-mode solutions of various equations found in literature [48, 49]. It is the generalization of many existing techniques. By using various values of the parameters and fractal dimension, the nonlinear dynamics of DNA strands can be addressed. The sensitivity analysis assesses how different uncertainties affect the overall level of uncertainty in a mathematical model. Specific boundaries that are dependent on one or more parameters have been applied using this technique.The rest of the article is organize as follows: The governing model is included in Section2. In section 3, soliton solutions are extracted along with geometrical analysis by employing semi-inverse method. Section 4 comprises solitons obtained via AEM with graphs. Section 5 of the report discusses the findings. Section 6 provides a sensitivity analysis of the suggested system. The article’s conclusion is provided in Section 7.


## 2. Governing System

Consider the following two general nonlinear dynamical equations which describe double-chain model of DNA:(1)uττ−β12uxx=σ1u+ξ1uv+η1u3+α1uv2,(2)vττ−β22vxx=σ2v+ξ2u2+η2u2v+α2v3+a,whereu is the difference between the top and bottom strands’ longitudinal displacements, i.e., the deviations of the bases from their equilibrium positions along the direction of the phosphodiester bridge, which joins the two bases of the same strands, v is the difference between the bottom and top strands’ transverse displacements, i.e., the bases displacement from its equilibrium point with the pathway of hydrogen bond which joins two bases of base pair where
(3)β1=±Gν,β2=±Hν,σ1=−2ηνbhh−mo,σ2=−2ηνb,ξ1=2ξ2=22ηmoνbh2,η1=η2=−2ηmoνbh3,α1=α2=4ηmoνbh3,a=2ηνbh−mo,whereH, b, G, and ν represent the tension density, cross-sectional area, Young’s modulus, and the mass density of each strand, h is the distance between the strands, while η is the stiffness and mo is the membrane height in positive equilibrium. The difference between the longitudinal displacements of the bottom and top strands is u in equation (1), as opposed to v, which represents the difference between the transverse displacements of the lower and higher strands.Noq using a transformation:(4)v=eu+f,wheree and f are constants, to simplify Equation (1) into the system of equations as follows:
(5)uττ−β12uxx=u3η1+α1e2+u22α1ef+ξ1e+uσ1+fξ1+α1f2,(6)uττ−β22uxx=u3η2+α2e2+u23α2ef+ξ2e+η2fe+uσ2+3α2f2+σ2fe+α2f3e+ae.Comparing Equations (4) and (5), we infer that f=h/2 and H=G. So, Equation (5) can be written as
(7)uττ−β12uxx=Su3+Tu2+Vu,where
(8)S=ζh3−2+4e2,T=62eζh2,V=−2ζmo+6ζh,ζ=ηmoνb,β1=Gν.


## 3. Mathematical Analysis

The wave transformationux,τ=uδ,δ=lx+κτ reduces Equation (7) to the following ODE:
(9)κ2−l2β12u′′−Su3−Tu2−Vu=0,whereκ2−l2β12≠0.According to [50, 51], a fractal DNA model can be written as
(10)κ2−l2β12ddδγdudδγ−Su3−Tu2−Vu=0,whereγ and du/dδγ are the fractal dimensional value and derivative, respectively, stated as
(11)dudδγ=Γ1+κlimδ−δo⟶Δδuδ−uδoδ−δoκ,Δδ≠0.The variational principle [44] can be used to produce the following trial-functional:
(12)J=∫Ldδ=∫K−Edδ.The variational formulation of Equation (10) is given as
(13)J=∫0∞κ2−l2β12dudδγ2+Su3+Tu2+Vudδγ,whereK=κ2−l2β12du/dδγ2 is the kinetic energy and E=−Su3−Tu2−Vu is the potential energy.
(14)L=κ2−l2β12dudδγ2+Su3+Tu2+Vu,H=κ2−l2β12dudδγ2−Su3−Tu2−Vu.The above equations are the Lagrangian and Hamiltonian. Using the two scale transformation,(15)A=δγ.Equation. (13) can be written as
(16)J=∫0∞κ2−l2β12dudA2+Su3+Tu2+VudA.


### 3.1. Soliton Solutions of Fractal Model

Using Ritz technique, one can construct the solitary wave solution as(17)u=CsechDA,whereC and D are constants to be further calculated. Putting Equation (17) into Equation (16), we have
(18)J=C212DπCT+6V+2SC2+C2D6κ2−l2β12.SettingJ stationary with respect to C and D, it results,
(19)∂J∂C=C12D3πCT+12V+8SC2+CD3κ2−l2β12,(20)∂J∂D=−C212D2πCT+6V+2SC2+C26κ2−l2β12.From Equations (19) and (20), we have
(21)C=±−5πT+25π2T2−1152SV24S,D=±5πT−25π2T2−1152SVπT−288SV288l2β12−κ2S.Now, Equation (17) can be described as
(22)ux,τ=±−5πT+25π2T2−1152SV24Ssech±5πT−25π2T2−1152SVπT−288SV288l2β12−κ2SA.Inserting the value ofu in Equation (4), then, we have
(23)vx,τ=±e−5πT+25π2T2−1152SV24Ssech±5πT−25π2T2−1152SVπT−288SV288l2β12−κ2SA+f,where A=lx+κτγ.Additionally, we look another soliton solution in the form:(24)u=Psech4QA,whereP and Q are constants to be further calculated. Substituting Equation (24) in Equation (16), we have
(25)J=P2135135Q16640PT+30888V+10752SP2+128P2Q315κ2−l2β12.When we keepJ stationary with respect to P and Q, it gives
(26)∂J∂P=P45045Q16640PT+20592V+14336SP2+256PQ315κ2−l2β12,(27)∂J∂Q=−P2135135Q216640PT+30888V+10752SP2+128P2315κ2−l2β12.From Equations (26) and (27), we have
(28)P=±−325T+105625T2−486486SV504S,Q=±10T325T−105625T2−486486SV−18711SV99792l2β12−κ2S.Equation (24) becomes
(29)ux,τ=±−325T+105625T2−486486SV504Ssech4±10T325T−105625T2−486486SV−18711SV99792l2β12−κ2SA.Plugging the value ofu in Equation (4), we have
(30)vx,τ=±e−325T+105625T2−486486SV504Ssech4±10T325T−105625T2−486486SV−18711SV99792l2β12−κ2SA+fwhereA=lx+κτγ.


## 3.1. Soliton Solutions of Fractal Model

Using Ritz technique, one can construct the solitary wave solution as(17)u=CsechDA,whereC and D are constants to be further calculated. Putting Equation (17) into Equation (16), we have
(18)J=C212DπCT+6V+2SC2+C2D6κ2−l2β12.SettingJ stationary with respect to C and D, it results,
(19)∂J∂C=C12D3πCT+12V+8SC2+CD3κ2−l2β12,(20)∂J∂D=−C212D2πCT+6V+2SC2+C26κ2−l2β12.From Equations (19) and (20), we have
(21)C=±−5πT+25π2T2−1152SV24S,D=±5πT−25π2T2−1152SVπT−288SV288l2β12−κ2S.Now, Equation (17) can be described as
(22)ux,τ=±−5πT+25π2T2−1152SV24Ssech±5πT−25π2T2−1152SVπT−288SV288l2β12−κ2SA.Inserting the value ofu in Equation (4), then, we have
(23)vx,τ=±e−5πT+25π2T2−1152SV24Ssech±5πT−25π2T2−1152SVπT−288SV288l2β12−κ2SA+f,where A=lx+κτγ.Additionally, we look another soliton solution in the form:(24)u=Psech4QA,whereP and Q are constants to be further calculated. Substituting Equation (24) in Equation (16), we have
(25)J=P2135135Q16640PT+30888V+10752SP2+128P2Q315κ2−l2β12.When we keepJ stationary with respect to P and Q, it gives
(26)∂J∂P=P45045Q16640PT+20592V+14336SP2+256PQ315κ2−l2β12,(27)∂J∂Q=−P2135135Q216640PT+30888V+10752SP2+128P2315κ2−l2β12.From Equations (26) and (27), we have
(28)P=±−325T+105625T2−486486SV504S,Q=±10T325T−105625T2−486486SV−18711SV99792l2β12−κ2S.Equation (24) becomes
(29)ux,τ=±−325T+105625T2−486486SV504Ssech4±10T325T−105625T2−486486SV−18711SV99792l2β12−κ2SA.Plugging the value ofu in Equation (4), we have
(30)vx,τ=±e−325T+105625T2−486486SV504Ssech4±10T325T−105625T2−486486SV−18711SV99792l2β12−κ2SA+fwhereA=lx+κτγ.


## 4. Illustration of the AEM

The following statement illustrates the general NLPDE structure:(31)Qu,ut,ux,uut,utuxx,uutt,⋯=0,whereQ is polynomial function of u and its derivatives in relation to two independent variables t and x. Use the single variable conversion δ=lx−κt to reduce Equation (31) into ODE of the form:
(32)Ru,u′,u′′,uu′′,⋯=0.Here,R is a polynomial function with both linear and nonlinear terms and the superscripts of u show its ordinary derivative with respect to δ. The algorithm of AEM suggests the initial solution of Equation (32) as
(33)uδ=∑i=0Mdiβiϕδ,satisfying the auxiliary equation
(34)ϕ′δ=1lnβaβ−ϕδ+c+bβϕδ,whered0, d1, d2, …, dM are coefficients to be evaluated such that dM≠0. The value of M is determined by balancing the highest order derivative and nonlinear term involved in Equation (9).Now, putting Equation (33) into Equation (9) and performing few steps of algebra yields a system of algebraic equations in βϕδ.The family of solutions of Equation (34) can be obtained as follows:Family 1.
Whenc2−4ab<0 and b≠0,
(35)βϕδ=−c2b+4ab−c22btan4ab−c22δ,βϕδ=−c2b−4ab−c22bcot4ab−c22δ.Family 2.
Whenc2−4ab>0 and b≠0,
(36)βϕδ=−c2b−c2−4ab2btanhc2−4ab2δ,βϕδ=−c2b−c2−4ab2bcothc2−4ab2δ.Family 3.
Whenc2+4a2<0 and b≠0 and b=−a,
(37)βϕδ=c2a−−4a2−c22atan−4a2−c22δ,βϕδ=c2a+−4a2−c22acot−4a2−c22δ.Family 4.
Whenc2+4a2<0 and b≠0 and b=−a,
(38)βϕδ=c2a+4a2+c22atanh4a2+c22δ,βϕδ=c2a+4a2+c22acoth4a2+c22δ.Family 5.
Whenc2−4a2<0 and b=a,
(39)βϕδ=−c2a+4a2−c22atan4a2−c22δ,βϕδ=−c2a−4a2−c22acot4a2−c22δ.Family 6.
Whenc2−4a2>0 and b=a,
(40)βϕδ=−c2a−−4a2+c22atanh−4a2+c22δ,βϕδ=−c2a−−4a2+c22acoth−4a2+c22δ.Family 7.
Whenc2=4ab,
(41)βϕδ=−2+cδ2bδ.Family 8.
Whenab<0, c=0 and b≠0,
(42)βϕδ=−−abtanh−baδ,βϕδ=−−abcoth−baδ.Family 9.
Whenc=0 and a=−b,
(43)βϕδ=1+e−2bδ−1+e−2bδ.Family 10.
Whena=b=0,
(44)βϕδ=coshcδ+sinhcδ.Family 11.
Whena=c=K and b=0,
(45)βϕδ=eKδ−1.Family 12.
Whenb=c=K and a=0,
(46)βϕδ=eKδ1−eKδ.Family 13.
Whenc=a+b,
(47)βϕδ=−1−aea−bδ1−bea−bδ.Family 14.
Whenc=−a+b,
(48)βϕδ=a−ea−bδb−ea−bδ.Family 15.
Whena=0,
(49)βϕδ=cecδ1−becδ.Family 16.
Whenc=a=b≠0,
(50)βϕδ=123tan32aδ−1.Family 17.
Whena=b and c=0,
(51)βϕδ=tanaδ.Family 18.
Whenb=0,
(52)βϕδ=ecδ−mn.


### 4.1. Application of AEM

The balancing principle employed to Equation (9) yields the value of index M=1. Hence, Equation (33) takes the form:
(53)uδ=d0+d1βϕδ.Now, invoking Equation (53) into Equation (9) gives a system of equations which is further evaluated via Maple, it generates
(54)d0=3VqΘ−12T,d1=3bVΘT,S=2T29V,l=−4abκ2−c2κ2/Θ−Vβ1,where
(55)Θ=−14ab−c2.Insertion of Equation (54) into Equation (53) results to
(56)uδ=3VcΘ−12T+3bVΘTβϕδ.By substituting the solutions specified by Equation (34) into Equation (58), the solutions retrieved areFor Family1, when c2−4ab<0 and b≠0,
(57)u1,1x,t=3VcΘ−12T+3VΘT−c2+4ab−c2tan4ab−c22δγ,(58)u1,2x,t=3VcΘ−12T+3VΘT−c2−4ab−c2cot4ab−c22δγ,(59)v1,1x,t=3eVcΘ−12T+3eVΘT−c2+4ab−c2tan4ab−c22δγ+f,(60)v1,2x,t=3eVcΘ−12T+3eVΘT−c2−4ab−c2cot4ab−c22δγ+f.For Family2, when c2−4ab>0 and b≠0,
(61)u2,1x,t=3VcΘ−12T+3VΘT−c2−c2−4abtanhc2−4ab2δγ,u2,2x,t=3VcΘ−12T+3VΘT−c2−c2−4abcothc2−4ab2δγ,v2,1x,t=3eVcΘ−12T+3eVΘT−c2−c2−4abtanhc2−4ab2δγ+f,v2,2x,t=3eVcΘ−12T+3eVΘT−c2−c2−4abcothc2−4ab2δγ+f.For Family3, when c2+4ab<0, b≠0 and b=−a,
(62)u3,1x,t=3VcΘ−12T−3VΘTc2−4a2−c2tan4a2−c22δγ,u3,2x,t=3VcΘ−12T−3VΘTc2+4a2−c2cot4a2−c22δγ,v3,1x,t=3eVcΘ−12T−3eVΘTc2−4a2−c2tan4a2−c22δγ+f,v3,2x,t=3eVcΘ−12T−3eVΘTc2+4a2−c2cot4a2−c22δγ+f.For Family4, when c2+4a2>0, b≠0 and b=−a,
(63)u4,1x,t=3VcΘ−12T−3VΘTc2+4a2+c2tanh4a2+c22δγ,u4,2x,t=3VcΘ−12T−3VΘTc2+4a2+c2coth4a2+c22δγ,v4,1x,t=3eVcΘ−12T−3eVΘTc2+4a2+c2tanh4a2+c22δγ+f,v4,2x,t=3eVcΘ−12T−3eVΘTc2+4a2+c2coth4a2+c22δγ+f.For Family5, when c2−4a2<0 and b=a,
(64)u5,1x,t=3VcΘ−12T+3VΘT−c2+4a2−c2tan4a2−c22δγ,u5,2x,t=3VcΘ−12T+3VΘT−c2−4a2−c2cot4a2−c22δγ,v5,1x,t=3eVcΘ−12T+3eVΘT−c2+4a2−c2tan4a2−c22δγ+f,v5,2x,t=3eVcΘ−12T+3eVΘT−c2−4a2−c2cot4a2−c22δγ+f.For Family6, when c2−4a2>0 and b=a,
(65)u6,1x,t=3VcΘ−12T+3VΘT−c2−−4a2+c2tanh−4a2+c22δγ,u6,2x,t=3VcΘ−12T+3VΘT−c2−−4a2+c2coth−4a2+c22δγ,v6,1x,t=3eVcΘ−12T+3eVΘT−c2−−4a2+c2tanh−4a2+c22δγ+f,v6,2x,t=3eVcΘ−12T+3eVΘT−c2−−4a2+c2coth−4a2+c22δγ+f.For Family7, when c2=4ab,
(66)u7x,t=3VcΘ−12T+3VΘT−2+cδγ2δγ,v7x,t=3eVcΘ−12T+3eVΘT−2+cδγ2δγ+f.For Family8, when ab<0, c=0 and b≠0,
(67)u8,1x,t=−3V2T−3bVΘT−abtanh−abδγ,u8,2x,t=−3V2T−3bVΘT−abcoth−abδγ,v8,1x,t=−3eV2T−3abVΘT−abtanh−abδγ+f,v8,2x,t=−3eV2T−3abVΘT−abcoth−abδγ+f.For Family9, when c=0 and a=−b,
(68)u9x,t=−3V2T+3bVΘT1+e−2bδγ−1+e−2bδγ,v9x,t=−3eV2T+3ebVΘT1+e−2bδγ−1+e−2bδγ+f.For Family12, when b=c=K and a=0,
(69)u12x,t=3VKΘ−12T+3KVΘTeKδγ1−eKδγ,v12x,t=3eVKΘ−12T+3eKVΘTeKδγ1−eKδγ+f.For Family13, when c=a+b,
(70)u13x,t=3Va+bΘ−12T−3bVΘT1−aea−bδγ1−bea−bδγ,v13x,t=3eVa+bΘ−12T−3ebVΘT1−aea−bδγ1−bea−bδγ+f.For Family14, when c=−a+b,
(71)u14x,t=−3Va+bΘ−12T+3bVΘTa−ea−bδγb−ea−bδγ,v14x,t=−3eVa+bΘ−12T+3ebVΘTa−ea−bδγb−ea−bδγ+f.For Family15, when a=0,
(72)u15x,t=3VcΘ−12T+3bVΘTcecδγ1−becδγ,v15x,t=3eVcΘ−12T+3ebVΘTcecδγ1−becδγ+f.For Family16, when c=a=b≠0,
(73)u16x,t=3VcΘ−12T+3bVΘ2T3tan32aδγ−1,v16x,t=3eVcΘ−12T+3ebVΘ2T3tan32aδγ−1+f.For Family18, when c=a=0,
(74)u18x,t=−3V2T−3VΘT1δγ,v18x,t=−3eV2T−3eVΘT1δγ+f.For Family17, when a=b and c=0,
(75)u19x,t=−3V2T+3bVΘTtanbδγ,v19x,t=−3eV2T+3ebVΘTtanbδγ+f.


## 4.1. Application of AEM

The balancing principle employed to Equation (9) yields the value of index M=1. Hence, Equation (33) takes the form:
(53)uδ=d0+d1βϕδ.Now, invoking Equation (53) into Equation (9) gives a system of equations which is further evaluated via Maple, it generates
(54)d0=3VqΘ−12T,d1=3bVΘT,S=2T29V,l=−4abκ2−c2κ2/Θ−Vβ1,where
(55)Θ=−14ab−c2.Insertion of Equation (54) into Equation (53) results to
(56)uδ=3VcΘ−12T+3bVΘTβϕδ.By substituting the solutions specified by Equation (34) into Equation (58), the solutions retrieved areFor Family1, when c2−4ab<0 and b≠0,
(57)u1,1x,t=3VcΘ−12T+3VΘT−c2+4ab−c2tan4ab−c22δγ,(58)u1,2x,t=3VcΘ−12T+3VΘT−c2−4ab−c2cot4ab−c22δγ,(59)v1,1x,t=3eVcΘ−12T+3eVΘT−c2+4ab−c2tan4ab−c22δγ+f,(60)v1,2x,t=3eVcΘ−12T+3eVΘT−c2−4ab−c2cot4ab−c22δγ+f.For Family2, when c2−4ab>0 and b≠0,
(61)u2,1x,t=3VcΘ−12T+3VΘT−c2−c2−4abtanhc2−4ab2δγ,u2,2x,t=3VcΘ−12T+3VΘT−c2−c2−4abcothc2−4ab2δγ,v2,1x,t=3eVcΘ−12T+3eVΘT−c2−c2−4abtanhc2−4ab2δγ+f,v2,2x,t=3eVcΘ−12T+3eVΘT−c2−c2−4abcothc2−4ab2δγ+f.For Family3, when c2+4ab<0, b≠0 and b=−a,
(62)u3,1x,t=3VcΘ−12T−3VΘTc2−4a2−c2tan4a2−c22δγ,u3,2x,t=3VcΘ−12T−3VΘTc2+4a2−c2cot4a2−c22δγ,v3,1x,t=3eVcΘ−12T−3eVΘTc2−4a2−c2tan4a2−c22δγ+f,v3,2x,t=3eVcΘ−12T−3eVΘTc2+4a2−c2cot4a2−c22δγ+f.For Family4, when c2+4a2>0, b≠0 and b=−a,
(63)u4,1x,t=3VcΘ−12T−3VΘTc2+4a2+c2tanh4a2+c22δγ,u4,2x,t=3VcΘ−12T−3VΘTc2+4a2+c2coth4a2+c22δγ,v4,1x,t=3eVcΘ−12T−3eVΘTc2+4a2+c2tanh4a2+c22δγ+f,v4,2x,t=3eVcΘ−12T−3eVΘTc2+4a2+c2coth4a2+c22δγ+f.For Family5, when c2−4a2<0 and b=a,
(64)u5,1x,t=3VcΘ−12T+3VΘT−c2+4a2−c2tan4a2−c22δγ,u5,2x,t=3VcΘ−12T+3VΘT−c2−4a2−c2cot4a2−c22δγ,v5,1x,t=3eVcΘ−12T+3eVΘT−c2+4a2−c2tan4a2−c22δγ+f,v5,2x,t=3eVcΘ−12T+3eVΘT−c2−4a2−c2cot4a2−c22δγ+f.For Family6, when c2−4a2>0 and b=a,
(65)u6,1x,t=3VcΘ−12T+3VΘT−c2−−4a2+c2tanh−4a2+c22δγ,u6,2x,t=3VcΘ−12T+3VΘT−c2−−4a2+c2coth−4a2+c22δγ,v6,1x,t=3eVcΘ−12T+3eVΘT−c2−−4a2+c2tanh−4a2+c22δγ+f,v6,2x,t=3eVcΘ−12T+3eVΘT−c2−−4a2+c2coth−4a2+c22δγ+f.For Family7, when c2=4ab,
(66)u7x,t=3VcΘ−12T+3VΘT−2+cδγ2δγ,v7x,t=3eVcΘ−12T+3eVΘT−2+cδγ2δγ+f.For Family8, when ab<0, c=0 and b≠0,
(67)u8,1x,t=−3V2T−3bVΘT−abtanh−abδγ,u8,2x,t=−3V2T−3bVΘT−abcoth−abδγ,v8,1x,t=−3eV2T−3abVΘT−abtanh−abδγ+f,v8,2x,t=−3eV2T−3abVΘT−abcoth−abδγ+f.For Family9, when c=0 and a=−b,
(68)u9x,t=−3V2T+3bVΘT1+e−2bδγ−1+e−2bδγ,v9x,t=−3eV2T+3ebVΘT1+e−2bδγ−1+e−2bδγ+f.For Family12, when b=c=K and a=0,
(69)u12x,t=3VKΘ−12T+3KVΘTeKδγ1−eKδγ,v12x,t=3eVKΘ−12T+3eKVΘTeKδγ1−eKδγ+f.For Family13, when c=a+b,
(70)u13x,t=3Va+bΘ−12T−3bVΘT1−aea−bδγ1−bea−bδγ,v13x,t=3eVa+bΘ−12T−3ebVΘT1−aea−bδγ1−bea−bδγ+f.For Family14, when c=−a+b,
(71)u14x,t=−3Va+bΘ−12T+3bVΘTa−ea−bδγb−ea−bδγ,v14x,t=−3eVa+bΘ−12T+3ebVΘTa−ea−bδγb−ea−bδγ+f.For Family15, when a=0,
(72)u15x,t=3VcΘ−12T+3bVΘTcecδγ1−becδγ,v15x,t=3eVcΘ−12T+3ebVΘTcecδγ1−becδγ+f.For Family16, when c=a=b≠0,
(73)u16x,t=3VcΘ−12T+3bVΘ2T3tan32aδγ−1,v16x,t=3eVcΘ−12T+3ebVΘ2T3tan32aδγ−1+f.For Family18, when c=a=0,
(74)u18x,t=−3V2T−3VΘT1δγ,v18x,t=−3eV2T−3eVΘT1δγ+f.For Family17, when a=b and c=0,
(75)u19x,t=−3V2T+3bVΘTtanbδγ,v19x,t=−3eV2T+3ebVΘTtanbδγ+f.


## 5. Results and Discussion

This section covers the graphical interpretation of the results and the impact of the fractal parameter on them. Two powerful integration approaches, namely, semi-inverse scheme and AEM are used to extract soliton solutions of governing model. It has been established that the approaches currently provided for the double-chain DNA model, which were utilized to create closed-form exact results, are novel and distinct from those currently in use. The innovative solitonic solution structure and the new equations that yielded distinct types of solutions are the observable characteristics for finding solutions from the method outlined. Graphics that elaborate the various novel exact solitons in the forms of dynamics and nonlinear waves are presented for a physical description of the solutions that have been achieved. The semi-inverse principle offers bright soliton solutions Equations (22), (23), (29), and (30) of the aforesaid system. The physical significance of these solitons is shown in terms of modulus of ux,t by assigning particular values of free parameters. Equations (23) and (30) show the same graphical behavior with just translation given in Equation (4) as in the figures. In Figures 1 and 2, these solitons in the form of 3D plots for fractal dimension value γ=0.8 and 2D graphs for γ=0.5,0.7,0.9 are provided. Dual-wave periodic, dark, bright solitons are raised by executing AEM. The dynamics of DNA strands ux,t and vx,t are presented in Figures 3–12. 3D sketches for fractal value γ=0.3 and 2D graphics for γ=0.5,0.7,0.9 are provided. The fractal impact is displayed by the irregularity in the curves of solutions. A few representative solutions are graphically illustrated to consider the appropriate connotation of dual-wave behaviors of the DNA system. The propagation of solitons and collisions of dual-mode pulses are examined using graphs. It is important to note that the proposed schemes may be used to generate soliton solutions for any NLPDE.Figure 1
We considerκ=2, l=1, β1=1, S=−1, T=2, V=1, and γ=0.5,0.7,0.9 for the solution described by Equation (22). (a) The 3D sketch of u2 taking γ=0.5. (b) 2D-plot of u2 with three distinct γ values.
(a)(b)Figure 2
We suggestκ=2, l=1, β1=1, S=−1, T=2, V=1, and γ=0.5,0.7,0.9 for the solution obtained in Equation (29). (a) The 3D-plot of u2 with γ=0.7. (b) The 2D-plot of u2 with distinct values of γ.
(a)(b)Figure 3
We suggestκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9. for the solution u1,1 . (a) The 3D graph of u1,1x,t with γ=0.3. (b) The 2D graph of u1,1x,t with distinct values of γ.
(a)(b)Figure 4
We takeκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9. for the solution u3,1. (a) The 3D graph of u3,1x,t with γ=0.3. (b) The 2D graph of u3,1x,t with distinct values of γ.
(a)(b)Figure 5
We takeκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9 for the solution u3,2. (a) The 3D graph of u3,2x,t with γ=0.3. (b) The 2D graph of u3,2x,t with distinct values of γ.
(a)(b)Figure 6
We takeκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9 for the solution u5,1. (a) The 3D graph of u5,1x,t with γ=0.3. (b) The 2D graph of u5,1x,t with distinct values of γ.
(a)(b)Figure 7
We considerκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9 for the solution u6,1. (a) The 3D graph of u6,1x,t with γ=0.8. (b) The 2D graph of u6,1x,t with distinct values of γ.
(a)(b)Figure 8
We suggestκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9 for the solution u6,2. (a) The 3D graph of u6,2x,t with γ=0.3. (b) The 2D graph of u6,2x,t with distinct values of γ.
(a)(b)Figure 9
We takeκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9 for the solution u8,1. (a) The 3D graph of u8,1x,t with γ=0.3. (b) The 2D graph of u8,1x,t with distinct values of γ.
(a)(b)Figure 10
We takeκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9 for the solution u8,2. (a) The 3D graph of u8,2x,t with γ=0.3. (b) The 2D graph of u8,2x,t with distinct values of γ.
(a)(b)Figure 11
In this figure, we takeκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9 for the solution u15. (a) The 3D graph of u15x,t with γ=0.3. (b) The 2D graph of u15x,t with distinct values of γ.
(a)(b)Figure 12
In this figure, we takeκ=−1, β1=1, T=2, V=1, c=3, a=1, b=1, and γ=0.5,0.7,0.9 for the solution u16. (a) The 3D graph of u16x,t with γ=0.3. (b) The 2D graph of u16x,t with distinct values of γ.
(a)(b)


## 6. Sensitivity Analysis

The sensitive analysis of the formulated soliton solutions is demonstrated in this section. There are several research publications on the methods and applications of sensitivity analysis of parameter uncertainty to mathematical problems. The goal of the study is to identify and classify the different types of uncertainty that can affect how well a mathematical equation or framework performs in relation to its inputs. Results are presented based on various parametric values, and sensitivity is investigated by taking into account how a little change in input can significantly alter output. A detailed analysis of Equation (9) is introduced in Figures 13 and 14.Figure 13
Sensitivity of model for initial conditionsu,δ=0.18,0.01 and u,δ=0.2,0.02 with free parameters S=−1, T=2, V=1, κ=2, l=1, and β1=1 while both conditions represent the curves in red and blue colors, respectively.Figure 14
Sensitivity of model for initial conditionsu,δ=0.5,0.001 and u,δ=0.15,0.001 with free parameters S=1, T=−2, V=1, κ=2, l=1, and β1=1 while both conditions represent the curves in red and blue, respectively.


## 7. Conclusion

In this manuscript, semi-inverse method and AEM have been successfully applied to the double-chain DNA model that is one of the interesting models of current biophysics since it is related to an organism’s life. It is likely that the well-known cubic nonlinear Klein-Gordon equation is the linearly reduced model to (1). The fractal DNA system has a high influence because it is used to describe the nonlinear dynamics of DNA molecules. The dark, periodic, bright, and other soliton solutions are derived which may help biologists for physical simulation of suggested equations. It should be highlighted that our results are novel and different from those of earlier investigations [3, 9]. The semi-inverse scheme is a fascinating integration tool to deduce variational principles for various differential models, whereas AEM is compelling to derive a family of dual-wave solitons of any NLPDE. The relevant choices of parameters enable us to discuss fractal behavior of the system. Also, the nature of attained solutions is reviewed by their 3D and 2D graphics. By using various initial conditions, the system is subjected to sensitivity analysis, which is then visualized using graphs. The acquired effects might help spark original suggestions for future biological applications.



---

*Source: 2297866-2022-08-25.xml*