Exact Solutions for the Nonlinear PDE Describing the Model of DNA Double Helices using the Improved (G /G)- Expansion Method

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1 Math Sci Lett 3, No, Mathematical Sciences Letters An International Journal Exact Solutions for the Nonlinear PDE Describing the Model of DNA Double Helices using the Improved /G- Expansion Method Elsayed M E Zayed, Yasser A Amer Reham M A Shohib Mathematics Department, Faculty of Sciences, Zagazig University, Zagazig, Egypt Received: 0 Nov 013, Revised: Feb 014, Accepted: 3 Feb 014 Published online: 1 May 014 Abstract: The objective of this article is to apply the improved /G- expansion method for constructing many exact solutions with parameters of the nonlinear partial differential equation PDE describing the model of DNA double helices The stretch of the hydrogen bonds is considered as a nonlinear chain with cubic quadratic potential When the parameters take special values, many solitary wave solutions periodic wave solutions can be found Comparison between some of our new results the well-known results are given Keywords: DNA double helices, Exact solutions, Solitary solutions, Periodic solutions, Improved /G- expansion method 1 Introduction Nonlinear phenomena play crucial roles in applied mathematics physics Exact solutions for nonlinear partial differential equations PDEs play an important role in many phenomena in such as fluid mechanics, hydrodynamics, optics, plasma physics so on With the development of solitary theory many powerful methods for obtaining the exact solutions of the nonlinear evolution equations are presented can be found in Refs [1 0] An attractive nonlinear model of the nonlinear science is the deoxynbonucleic acid DNA The DNA molecule encodes the information that organisms need to live reproduce themselves The DNA structure has been studied last decades see for example [ 3] The discovery of the double-helix structure of the DNA molecule has been established a strong relationship between its structure function Zhang et al [31] have studied the following nonlinear partial differential equation describing the model of DNA double helices using the homogeneous balance method: y tt c y xx + ω 0 y 1 y + y 3 + ηy t = 0, 1 where c, ω0, 1, are well-known constants, which can be found in [31], η is the damping constant The objective of this article is to apply the improved /G- expansion method to find many exact traveling wave solutions of the model 1, namely the hyperbolic, trigonometric rational function solutions Comparison between some of our new results the well-known results obtained in [31] will be given later The rest of this article is organized as follows: In Sec, the derivation of the model 1 is given In Sec 3 the description of the improved /G- expansion method is obtained In Sec 4 some conclusions are given Derivation of the model of DNA double helices 1 With reference to [31], the helical axis for the Watson-Crick model of DNA is taken in the x-direction The base its complementary base may vibrate each other along the direction of hydrogen bonds in a base pair Thus, it is assumed that the displacements of the nth base its complementary base along the direction of H-bonds are given by y n y n respectively Since the H-bond interaction has a remarkable nonlinearity due to its charge-transfer interaction, then the H- bond can be approximately described by the formula Vr n = V 0 + Ar n Br 3 n+cr 4 n, Corresponding author emezayed@hotmailcom c 014 NSP Natural Sciences Publishing Cor

2 108 E M E Zayed et al : Exact Solutions for the Nonlinear PDE Describing the Model where r n = y n y n A,B,C are constants The stacking energy between two neighboring base pairs is indicated by L n = 1 Ly n y n1 + 1 Ly n+1 y n, 3 where L is a parameter Let M be the average mass of a base, then the Hamiltonian of the DNA system can be written as: } 1 H = n Mẏ n+ ẏ n+vy n y n + 1 [ L y n y n1 + ] y n y n1 }, n The differential equations may be written as: M ÿ n = H =A y n y n + 3Byn y y n n 4Cy n y n 3 + Ly n+1 y n + y n1 ηẏ n, M ÿ n = H y = A y n y n 3Byn y n + n 4Cy n y n 3 + Ly n+1 y n+ y n1ηẏ n, where η is the damping constant We shall limit us to study the relative motion of the base such that y n =y n 7 Consequently, we have the differential equation M ÿ n =4Ay n + 1By n 3Cy 3 n+ Ly n+1 y n + y n1 ηẏ n 8 The base spacing d equals 34 A 0 for B-DNA, it is obtained using the continuum approximation y n t yx, t, 1 dx 9 n d Hence, the equation of motion can be reduced to the form of the model 1 where c = L M d, ω0 = 4A M, 1 = 1B M, = 3c M 3 Description of the improved /G-expansion method Suppose that we have the following nonlinear evolution equation: Fu, u t, u x, u tt, u xx,=0, 10 where F is a polynomial in ux, t its partial derivatives, in which the highest order derivatives the nonlinear terms are involved In the following, we give the main steps of this method [0, 1] as follows: Step 1 We use the traveling wave transformation ux, t=u, =lx+βt+ k, 11 to reduce Eq 10 to the following ordinary differential equation ODE: Pu,u,u,=0, 1 where l,β, k are constants while P is a polynomial in u its total derivatives, while the dashes denote the derivatives with respect to Step We assume that Eq 1 has the formal solution i u= a i, 13 G m i=m where a i i = m,,m are constants to be determined, G satisfies the following linear ODE: + + µg=0, 14 where µ are constants Step 3The positive integer m in Eq 13 can be determined by balancing the highest-order derivatives with the nonlinear terms appearing in Eq 1 Step 4 We substitute 13 along with Eq 14 into Eq 1 to obtain polynomials in G i,i=0,±1,±, Equating all the coefficients of these polynomials to zero, yields a set of algebraic equations, which can be solved using the Maple to find a i, β,l Step 5 Since the solutions of Eq14 are well-known for us, then we have the following ratios: i If 4µ > 0, we have G = + 4µ sinh 4µ + c cosh 4µ, cosh 4µ + c sinh 4µ ii If 4µ < 0, we have 4µ G = + sin 4µ + c cos 4µ cos 4µ + c sin 4µ c 014 NSP Natural Sciences Publishing Cor

3 Math Sci Lett 3, No, / wwwnaturalspublishingcom/journalsasp 109 iii If 4µ = 0, we have G = + + c, 17 where c are arbitrary constants Step 6 We substitute the values of a i, β,l as well as the ratios into 13 to obtain many exact solutions of Eq10 4 Exact solutions of the DNA model 1 In this section, we apply the proposed method of Sec 3, to construct the exact solutions of the DNA double helices modeling 1 To this end, we use the wave transformation 11 to reduce Eq 1 to the following ODE: l β c y + ω 0 y 1 y + y 3 + ηlβy = 0, 18 where β c 0 Balancing y with y 3 we get m=1 Consequently, we have 1 y=a 1 + a 0 + a 1 19 G G where a 1,a 0,a 1 are constants to be determined, such that a 1 0 or a 1 0 Substituting 19 along with Eq 14 into Eq 18 equating all the coefficients of G i,i = 0,±1,±,±3 to zero, we obtain the following algebraic equations: G 3 : a1 µ l β c + a 3 1 = 0, G 3 : a1 l β c + a 3 1 = 0, G : 3a1 µl β c 1 a 1 + 3a 0a 1 + a 1 µηlβ = 0, G : 3a1 l β c 1 a 1 + 3a 0a 1 a 1 ηlβ = 0, 1 G : a1 µ+ a 1 l β c +ω0 a 1 a 0 a a 1 a 1 + 3a 1a 0 +a 1ηlβ = 0, G : a 1 + a 1 µl β c +ω 0 a 1 a 0 a a 0 a 1+ 3a 1 a 1 a 1ηlβ = 0, G 0 : a1 µ + a 1 l β c +ω 0 a 0 1 a 0 + a 1a 1 + a a 0a 1 a 1 + a 1 a 1 µηlβ = 0, On solving the above algebraic equations using the Maple or Mathematica, we obtain the following cases: Case 1 a 0 = 1, a 3 1 = 9ω 0 1 4, a 1µ 1 1 = µ 1, η = 0, 3 1 = 1, c=c, l=l, 1 β = [ 18a 1 3 l 34a 1 l c ω0 81 ω4 0 ], 4 0 Case a 0 = ω 0+l c β, a 1 = l c β, a 1 = 0, µ = l β c +ω 0 4β c l,η = 0, 1 = 3 ω 0, c=c, β = β,l=l, 1 Case 3 a 0 = 1 4+ηlβ 1, a 1 = ηβl, a 1 1 = 0, µ = ω η β l, 16η β l 4 1 = 1, β = β, l=l, η = η, =, c = β 1 4+ η Exact solutions of the DNA model 1 for case 1 Substituting 0 into 19 using 15-17, we have the following exact solutions for the model 1: iif 4µ > 0 Hyperbolic function solutions, we have the exact solution y= 1 9ω µ 1 + 4µ sinh 4µ +c cosh 4µ cosh 4µ +c sinh 4µ + µ µ sinh 4µ +c cosh 4µ cosh 4µ +c sinh 4µ 3 Substituting the formulas 8, 10, 1 14 obtained in [33] into 3 we have respectively the following kinktype traveling wave solutions: 1 If > c, then y 1 = 1 1µ 1 + 4µ tanh } 4µ+ sgn c ψ 1 + µ ω µ tanh 1 4µ+ sgn c ψ 1 } 1 4 c 014 NSP Natural Sciences Publishing Cor

4 110 E M E Zayed et al : Exact Solutions for the Nonlinear PDE Describing the Model If c > 0, then y = 1 1µ 1 + 4µ coth } 4µ+ sgn c ψ + µ ω µ coth 3 If c > =0, then 4µ+ sgn c ψ } 1 y 3 = 1 1µ 1 + 4µ coth } 4µ + µ µ coth } 1 4µ 3 9ω If = c, then y 4 = 1 9ω µ 1 } 1, + µ µ } + 4µ where ψ 1 = tanh 1 c, ψ = coth 1 c sgn c is the sign function iiif 4µ < 0 Trigonometric function solutions, we have the exact solution y= 1 9ω µ 1 sin 4µ +c cos 4µ cos 4µ +c sin 4µ + 4µ + µ µ sin 4µ +c cos 4µ cos 4µ +c sin 4µ 8 Now, simplify 8 to get the following periodic solutions: y 1 = 1 + 4µ tan 9ω µ 1 + µ µ tan 0 4µ } 1 } 1 4µ, 1, 9 y = 1 + 4µ cot 9ω µ 1 + µ µ cot 0 4µ } + } 1 4µ, 30 where 1 = tan 1 c, = cot 1 c c 1 + c 0 iii If 4µ = 0 Rational function solutions,then we have } y= 1 9ω µ c } µ c 4 Exact solutions of the DNA model 1 for case Substituting 1 into 19 using 15-17, we have the following exact solutions for the model 1: iif 4µ > 0 Hyperbolic function solutions,then we have the exact solution y= 1 ω 0 + l c β + l c β + 4µ sinh 4µ +c cosh 4µ cosh 4µ +c sinh 4µ 3 Substituting the formulas 8, 10, 1 14 obtained in [33] into 3 we have respectively the following kink-type traveling wave solutions: 1 If > c, then y 1 = 1 ω 0 + l c β + l c β + 4µ tanh } 4µ+ sgn c ψ 1, If c > 0, then 33 y = 1 ω 0 + l c β + l c β + 4µ coth } 4µ+ sgn c ψ 3 If c > =0, then y 3 = 1 ω 0 + l c β + l c β + 4µ coth 4µ } 4 If = c, then y 4 = 1 ω 0 + l c β + l c β } + 4µ c 014 NSP Natural Sciences Publishing Cor

5 Math Sci Lett 3, No, / wwwnaturalspublishingcom/journalsasp 111 where ψ 1 = tanh 1 c, ψ = coth 1 c sgn c is the sign function Remark: On comparing our result 33 with the well-known result 18 of Ref [31] we deduce the following results: 1 Setting ω 0 = l c β sgn c = 0, the solitary wave solution 33 can be simplified to become l y 1 = Γ +Γ tanh x+βt+ k where Γ = l c β 37 If we set α = 0 η = 0 in the formulas of Ref [31] we have the same formula 37 while Γ is given by Γ = ± 3l c β Here Γ is called the amplitude When Γ > 0 we get the kink solution, while if Γ < 0, we get the antikink solution iiif 4µ < 0 Trigonometric function solutions,then we have the exact solution y= 1 ω 0 + l c β + l c β + 4µ sin 4µ +c cos 4µ cos 4µ +c sin 4µ 38 Now, simplify 38 to get the following periodic solutions: y 1 = 1 ω 0 + l c β + l c β + 4µ tan 1 4µ }, y = 1 ω 0 + l c β + l c β + 4µ cot + 4µ }, where 1 = tan 1 c c 1 + c 0, = cot 1 c iiiif 4µ = 0 Rational function solutions,then we have y= 1 ω 0 + l c β + l c β } + +c Exact solutions of the DNA model 1 for case 3 Substituting into 19 using 15-17, we have the following exact solutions for the model 1: iif 4µ > 0 Hyperbolic function solutions,then we have the exact solution y= ηβl 1 + 4µ lηβ sinh 4µ +c cosh 4µ cosh 4µ +c sinh 4µ 4 Substituting the formulas 8, 10, 1 14 obtained in [33] into 4 we have respectively the following kinktype traveling wave solutions: 1 If > c, then y 1 = lηβ + ηβl 1 + 4µ tanh } 4µ+ sgn c ψ 1, If c > 0, then 43 y = lηβ + ηβl 1 + 4µ coth } 4µ+ sgn c ψ 3 If c > =0, then y 3 = lηβ + ηβl 1 + 4µ coth 4µ } 4 If c =, then y 4 = lηβ + ηβl } 1 + 4µ where ψ 1 = tanh 1 c, ψ = coth 1 c sgn c is the sign function iiif 4µ < 0 Trigonometric function solutions,then we have the exact solution y= ηβl 1 + 4µ lηβ sin 4µ +c cos 4µ cos 4µ +c sin 4µ 47 c 014 NSP Natural Sciences Publishing Cor

6 11 E M E Zayed et al : Exact Solutions for the Nonlinear PDE Describing the Model Now, simplify 47 to get the following periodic solutions: 0 y 1 = lηβ + ηβl 1 + 4µ tan 1 4µ }, y = lηβ + ηβl 1 + 4µ cot + 4µ }, where 1 = tan 1 c, = cot 1 c c 1 +c 5 Some conclusions discussions In this article, we have employed the improved /G- expansion method described in Sec 3, to find many exact solutions as well as many solitary wave solutions many periodic solutions 3-49 of the nonlinear PDE 1 describing the DNA double helices modeling which look new This model has been discussed in [31] using the homogeneous balance method, where some solitary wave solutions have been found On comparing our new results 3-49 with that obtained in [31] we conclude that the improved /G- expansion method used in this article is more effective giving more exact solutions than the homogeneous balance method used in [31] According to our knowledge, we deduce that the DNA model 1 its solutions 3-49 have not been discussed elsewhere using the improved /G- expansion method Finally, all the solutions of Eq 1 obtained in this article have been checked with the Maple by putting them back into the original Eq 1 6 Acknowledgment The first author thanks his student Ahmed H Arnous for his improvement typing this paper References [6] M J Ablowitz P A Clarkson, Solitons, nonlinear evolution equation inverse scattering, Cambridge University press, New York, 1991 [7] F Tascan A Bekir, Appl Math Comput, 0, [8] S A EL-Wakil, Phys Lett A, 369, [9] J Weiss, M Tabor G Carnevalle, J Math Phys, 4, [10] E M E Zayed K A Gepreel, J Math Phys, 50, [11] A J M Jawad, M D Petkovic A Biswas, Appl Math Comput, 17, [1] E M E Zayed, Appl Math Comput, 18, [13] E M E Zayed, S A Hoda Ibrahim, Chin Phys Lett, 9, [14] N A Kudryashov N B Loguinova, Appl Math Comput, 05, [15] P N Ryabov, Appl Math Comput, 17, [16] P N Ryabov, D I Sinelshchikov M B Kochanov, Appl Math Comput, 18, [17] M Wang, Y Zhou Z Li, Phys Lett A, 16, [18] N A Kudryashov, Commun Nonlin Sci Numer Simul A, 17, [19] E M E Zayed A H Arnous, Chin Phys Lett, 9, [0] E M E Zayed K A Gepreel, In J Non Sci Numer Simula, 11, [1] J Zhang, F Jiang X Zhao, Int J Compu Math, 87, [] M Aguero, M Najera M Carrillo, Int J Modern Phys B,, [3] G Gaeta, J Biol Phys, 4, [4] G Gaeta, C Reiss, M Peyrard T Dauxois, Rivista del Nuovo Cimento, 17, [5] L V Yakushevich, Nonlinear physics of DNA, Studio Biophys, 11, [6] L V Yakushevich, Phys Lett A, 136, [7] L V Yakushevich, Nonlinear physics of DNA Engl, Wiley Sons, 1998 [8] M Peyrard A Bishop, Phys Rev Lett, 6, [9] D X Kong, S Y Lou J Zeng, Commu Thero Phys, 36, [30] W Alka, A Goyal C N Kumar, Phys Lett A, 375, [31] L Y Zhang S Hong, Chin Phys Lett, 15, [3] E M E Zayed A H Arnous, Sci Res Essays, 8, [33] Z Peng, Commu Theor Phys, 5, [1] M Wang, X Li J Zhang, Phys Lett A, 37, [] D Lu Q Shi, Int J Nonlinear Sci, 10, [3] M R Miura, Backlund Transformation, Springer, Berlin, Germany, 1978 [4] E J Parkes B R Duffy, Comput Phys Commu, 98, [5] R Hirot, Phys Rev Lett, 7, c 014 NSP Natural Sciences Publishing Cor

7 Math Sci Lett 3, No, / wwwnaturalspublishingcom/journalsasp 113 Elsayed M E Zayed Professor of Mathematics from 1989 till now at Zagazig University His interests are the nonlinear PDEs in Mathematical Physics, Inverse problems in differential equations, Nonlinear difference equations He Published about 31 articles in famous International Journals around the world He was the head of Mathematics Department at Zagazig University, Egypt from 001 till 006 He is an Editor of WSEAS Transaction on Mathematics He has got some Mathematical prizes by the Egyptian Academy of scientific research Technology He got the Medal of Science Arts of the first class from the president of Egypt He got the Medal of Excellent of the first class from the president of Egypt He is one of the Editorial Board of the Punjab University Journal of Mathematics He have reviewed many articles for many international Journals He has got his BSC of Mathematics from Tanta University, Egypt 1973 He has got two MSC degrees in Mathematics The first one from Al-Azher University, Egypt 1977, while the second one from Dundee University,Scotl, UK 1978 He has got his PHD in Mathematics from Dundee University, Scotl, UK 1981 He has got the Man of the Year award, 1993, by the American Biographical, Institute, USA He has got the Research Fellow Medal, 1993, by the American Biographical Institute, USA The Curriculum Vitae of Professor EMEZayed has been published by Men of achievement of the International Biographical center, Cambridge, vol 16, 1995, p534 The Curriculum Vitae of Professor EMEZayed has been published by Dictionary of International Biography of the International Biographical center, Cambridge, vol 3, 1995, p714 He was a Professor of Mathematics at Taif University, Saudi Arabia c 014 NSP Natural Sciences Publishing Cor