A New Generalized Riccati Equation Rational Expansion Method to Generalized Burgers Fisher Equation with Nonlinear Terms of Any Order

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1 Commun. Theor. Phys. Beijing China) ) pp c International Academic Publishers Vol. 46 No. 5 November A New Generalized Riccati Equation Rational Expansion Method to Generalized Burgers Fisher Equation with Nonlinear Terms of Any Order ZHANG Xiao-Ling 1 WANG Jing 1 and ZHANG Hong-Qing 1 1 Department of Applied Mathematics Dalian University of Technology Dalian China Department of Mathematics and Informatics Henan Polytechnic University Jiaozuo China Received December 0 005) Abstract In this paper based on a new more general ansätz a new algebraic method named generalized Riccati equation rational expansion method is devised for constructing travelling wave solutions for nonlinear evolution equations with nonlinear terms of any order. Compared with most existing tanh methods for finding travelling wave solutions the proposed method not only recovers the results by most known algebraic methods but also provides new and more general solutions. We choose the generalized Burgers Fisher equation with nonlinear terms of any order to illustrate our method. As a result we obtain several new kinds of exact solutions for the equation. This approach can also be applied to other nonlinear evolution equations with nonlinear terms of any order. PACS numbers: 0.30.Jr Yv Key words: generalized Riccati equation rational expansion method generalized Burgers Fisher equation with nonlinear terms of any order symbolic computation 1 Introduction As is well known nonlinear evolution equations are closely related to nonlinear science such as physics mechanics biology etc. To further explain some physical phenomena searching for exact solutions of nonlinear evolution equations is very important. Up to now except for the traditional methods such as inverse scattering method [1] Bäcklund transformation [3] Darboux transformation [] Hirota s method [4] there exist some direct and unified algebraic methods such as tanh function method [5] various extended tanh-function methods [6 9] Jacobi elliptic function expansion method [10] and various extended Jacobi elliptic function expansion methods. [111] These algebraic methods have the power to give a clear picture of the relation between different terms of nonlinear wave equations and they can simplify the routine calculation of the method and obtain more general solutions. This owes to that the success of the symbolic mathematical computation discipline is striking and the availability of computer symbolic system like Maple or Mathematica which allows us to perform complicated and tedious algebraic calculation that these algebraic methods include on a computer. Recently Chen and Wang first proposed a new algebraic method named Riccati equation rational expansion RERE) method in Ref. [13] which further exceeded the applicability of the tanh methods in obtaining a series of travelling wave solutions including rational form solitary wave solutions triangular periodic wave solutions and rational wave solutions. In recent years the nonlinear evolution equations with nonlinear terms of any order stimulated considerable interests. For example the generalized Burgers Fisher equation with nonlinear terms of any order u t + αu δ u x m u u x u xx = βu1 u δ ). 1) Some kink profile solitary wave solutions bell profile solitary wave solutions rational solutions periodic solutions and combined formal solutions are obtained in Refs. [14] [16]. When β = m = 0 δ = 1 the generalized Burgers Fisher equation becomes the Burgers equation u t + αuu x u xx = 0. ) This equation arises in a variety of physical contexts and has been studied by many authors. [17 19] For example in Ref. [0] many explicit exact solutions which contain new solitary wave solutions periodic wave solutions and the combined formal solitary wave solutions are obtained. When α = m = 0 and β = δ = 1 the generalized Burgers Fisher equation reads u t = u xx + u1 u). 3) Kolmogorov Petrovskii and Piskunov studied this equation in Ref. [0]. They showed if initial datum satisfies some conditions then the solution of Eq. 3) approaches a travelling wave of speed C 0 =. Exact solution of Eq. 3) was found by Ablowitz and Zeppetella in Ref. [1] at C 0 = ±5/ 6. However this special solution has only one arbitrary constant. Solitary waves of Eq. 3) with two arbitrary constants and more general class of special solutions have been shown by Nikolai A. Kudryashov in Ref. [4]. Using new variables β ) 1/x t = tβ x = γ The project partially supported by the State Key Basic Research Program of China under Grant No. 004CB xlzhang0614@yahoo.com.cn

2 780 ZHANG Xiao-Ling WANG Jing and ZHANG Hong-Qing Vol. 46 below primes are omitted) equation 3) becomes the Fisher equation u t = γu xx + βu1 u). 4) Nonlinear evolution equation 4) was proposed as a model for the propagation of a mutant gene with an advantageous selection intensity β. [3] The same equation occurs in flame propagation in branching Browian motion process and in nuclear reactor theory. [1] On the lines of the rational expansion thought in this paper we propose a new algebraic method named generalized Riccati equation rational expansion GRERE) method to consider the generalized Burgers Fisher equation with nonlinear terms of any order. And we successfully construct some new and more general rational formal solutions of the equation. This paper is organized as follows. In Sec. we summarize the new GRERE method. In Sec. 3 we apply the new GRERE method to the generalized Burgers Fisher equation with nonlinear terms of any order and obtain many new form solutions. Conclusions will be presented in Sec. 4. Summary of Generalized Riccati Equation Rational Expansion Method In the following we would like to outline the main steps of our method: Step 1 For a given nonlinear evolution system with some physical fields u i x y t) in three variables x y t i u i u it u ix u iy u itt u ixt u iyt u ixx u iyy u ixy...) = 0 5) by using the wave transformation u i x y t) = U i ξ) ξ = kx + ly λt) 6) k l and λ are constants to be determined later the nonlinear partial differential system 5) is reduced to a nonlinear ordinary differential equation ODE): = d/dξ). i U i U i U i...) = 0 7) Step We introduce a new ansätz in terms of finite rational formal expansion in the following forms: m i a ij ψ j ξ) U i = a i µψξ)) j 8) j=1 the new variable ψ = ψξ) satisfying ψ h 1 + h ψ ) = dψ d ξ h 1 + h ψ ) = 0 9) a i0 a ij and µ i = 1...) are constants to be determined later. Step 3 The parameter m i can be found by balancing the highest order derivative term and the nonlinear terms in Eq. 5) or Eq. 7): i) If m is a positive integer then Step 4; ii) If m = q/p we make the transformation Uξ) = φ q/p ξ) and then return to Step 1; iii) If m is a negative integer we make the transformation Uξ) = φ m ξ) and then return to Step 1. Step 4 Substitute Eq. 8) into Eq. 7) along with Eq. 9) and then set all coefficients of ψ i ξ) i = 1...) of the resulting system s numerator to be zero to get an overdetermined system of nonlinear algebraic equations with respect to λ a i0 a ij and µ i = 1...). Step 5 Solving the over-determined system of nonlinear algebraic equations by use of Maple we would end up with the explicit expressions for λ a i0 a ij and µ i = 1...). Step 6 It is well known that the general solutions of Eq. 9) are i) When h 1 = 0 and h 0 ii) When h 1 = h = 1 iii) When h 1 = h = 1 1 ψξ) = ; 10) h ξ + h 0 iv) When h 1 = 1 and h = 1 ψξ) = tanξ) ; 11) ψξ) = cotξ) ; 1) ψξ) = tanhξ) ψξ) = cothξ) ; 13) v) When h 1 = h = ±1/ ψξ) = secξ) ± tanξ) ψξ) = cscξ) ± cotξ) 14) vi) When h 1 = 1/ and h = 1/ ψξ) = tanhξ) ± isechξ) ψξ) = cothξ) ± cschξ) ; 15) ξ = kx + ly λt) i = 1 and h 0 is an arbitrary constant. Remark 1 The ansätz proposed here which is based on Riccati equation rational expansion method [4] is more general than the ansätz in the tanh function method [5] and various extended tanh-function methods. [6 9] If we set the parameters in Eqs. 8) and 9) to different values the above methods can all be recovered by this new method. 3 Applications to Generalized Burgers Fisher Equation with Nonlinear Terms of Any Order Let us consider Eq. 1) by using the method above. First we take the form of the required solution as follows: ux t) = Uξ) ξ = kx λt). 16) We change Eq. 1) to the form λkuu + αku δ+1 U mk U k UU

3 No. 5 A New Generalized Riccati Equation Rational Expansion Method to Generalized Burgers Fisher Equation 781 βu 1 U δ ) = 0 17) = d/d ξ. By balancing the highest order partial derivative term and the nonlinear term in Eq. 17) we get the value of m m =. Therefore we make the following transformation: Uξ) = φ ξ). 18) Then substituting Eq. 18) into Eq. 17) reads βδ 1 + φ)φ + k mk + δk )φ δφ [ λk αkφ)φ + k φ ] = 0. 19) According to the proposed method we expand the solution of Eq. 19) in the form φξ) = a 0 + a 1ψξ) 1 + µψξ) 0) ψξ) satisfies Eq. 9). With the aid of Maple substituting Eq. 0) along with Eq. 9) into Eq. 19) yields a set of algebraic equations for ψ i ξ) i = ). Setting the coefficients of these terms ψ i ξ) to zero yields a set of over-determined algebraic equations with respect to a 0 a 1 µ λ k α β and m Because the over-determined algebraic equations are so many they are omitted). Then we get the following results. Case 1 Case Case 3 a 0 = a 0 a 1 = a 0 h µ = ± h λ = λ k = k h 1 h 1 βδ α = ± β = β m = δλkh 1 h /h 1 + βδ h /h 1 kh 1 4k 1. 1) h 1 h a 0 = a 0 a 1 = ± h h a 0 ) a 0 µ = h h a 0 ) λ = α ± βδ + 4k h 1 h k h /h 1 h 1 k = k α = α β = β m = δ + 1) δα h k a 0 = a 1 = ±4 h µ = ±h 1 h /h h 1 h h 1 h 1 h h 1. ) Case 4 a 0 = a 0 µ = ± λ = ±k h /h 1 h 1 h + kα h 1 h /h 1 h1 h ± β δ h 1 h ± h 1 h k h 1 h h h + ) k = k h /h 1 h1 h k α = α β = β m = β δ h + β δ h /h 1 h1 h ± 4δ h /h 1 h 1 h kα 4h 1 h k ±h + h /h 1 h1 h ) 1 δ. 3) a 1 = h 1 a a a 0 ) h 1 9 a a a 0 ) h 1 a a a 0 ) h 1 9 a a a 0 ) a 0) 18 a a a 0 8 a 0 + 1)h k λ = ± h1 a03 16 a 0 +9 a 0 ) h 19 a a 0 +9 a 0 ) 6a 0 4a 0 + 1)a 0 1) 6 a0 4 a ) 1 9 a a a 0 6 a ) a 0 h k α = ± 9 a a a 0 ) 6 a 0 4 a 0 + 1) h1 a03 16 a 0 +9 a 0 ) h 19 a a 0 +9 a 0 ) β δ 90 a a a a a 0 ) ± 9 a 0 7 a 0 + ) 6 a 0 4 a 0 + 1) h 1 k 3βδa 4 0 h1 a03 16 a 0 +9 a 0 ) h 19 a a 0 +9 a 0 ) 6a 0 4a 0 + 1) h 1 k h1 a03 16 a 0 +9 a 0 ) h 19 a a 0 +9 a 0 ) k = k

4 78 ZHANG Xiao-Ling WANG Jing and ZHANG Hong-Qing Vol. 46 β = β Note 3βδ a 5 m = 0a 0 1) 4 6 a 0 4 a 0 + 1) 1 a a a 0 ) h h 1 k + δ 1. 4) From Eqs. 16) 18) 0) and 10) 15) we obtain the following solutions for Eq. 1). When h 1 = 0 and h 0 we can easily obtain the solutions for Eq. 1) so we omit them in this paper. Family 1 According to Eq. 1) we obtain the following solutions of the generalized Burgers Fisher equation. When h 1 = h = 1 { a0 { i tan [kx λt)]} u 11 = ; 1 i tan [kx λt)] 5) When h 1 = h = 1 { a0 { i cot [kx λt)]} u 1 = ; 1 i cot [kx λt)] 6) When h 1 = 1 h = 1 { a0 { tanh [kx λt)]} u 13 = 1 tanh [kx λt)] 7) { a0 { coth [kx λt)]} u 14 = ; 1 coth [kx λt)] 8) When h 1 = h = ±1/ When h 1 = 1/ h = 1/ a 0 λ and k are arbitrary constants. { a0 { i sec[kx λt)] i tan[kx λt)]} u 15 = 1 i sec[kx λt)] ± i tan[kx λt)] 9) { a0 { i csc[kx λt)] i cot[kx λt)]} u 16 = 1 i csc[kx λt)] ± i cot[kx λt)] 30) { a0 { tanh[kx λt)] i sech[kx λt)]} u 17 = 1 tanh[kx λt)] ± i sech[kx λt)] 31) { a0 { coth[kx λt)] csch[kx λt)]} u 18 = 1 coth[kx λt)] ± csch[kx λt)] 3) Family According to Eq. ) we obtain the following solutions of the generalized Burgers Fisher equation. When h 1 = h = 1 { a 0 ± ia 0 tan [ kx ± 1 u 1 = iβ δ ± k i + kα ) t ] 1 i tan [ kx ± 1 i β δ ± k i + kα ) t ] ± i tan [ kx ± 1 i β δ ± k i + kα ) t ] ; 33) a 0 When h 1 = h = 1 { a 0 ± i a 0 cot [ kx ± 1 u = i β δ ± k i + kα ) t ] 1 i cot [ kx ± 1 i β δ ± k i + kα ) t ] ± i cot [ kx ± 1 i β δ ± k i + kα ) t ] ; 34) a 0 When h 1 = 1 h = 1 { a 0 ± a 0 tanh [ kx ± 1 u 3 = β δ ± k + kα ) t ] 1 tanh [ kx ± 1 β δ ± k + kα ) t ] ± tanh [ kx ± 1 β δ ± k + kα ) t ] 35) a 0 { a 0 ± a 0 coth [ kx ± 1 u 4 = β δ ± k + kα ) t ] 1 coth [ kx ± 1 β δ ± k + kα ) t ] ± coth [ kx ± 1 β δ ± k + kα ) t ] ; 36) a 0 When h 1 = h = ±1/ { { [ a 0 1 i sec kx ±i βδ ± ik + kα)t ] ± i tan [ kx ±iβδ ± ik + kα)t ]} u 5 = 1 + i 1 + a 0 ) { sec [kx ±iβδ ± ik + kα)t] ± tan [kx ±iβδ ± ik + kα)t]} 37) { { [ a 0 1 i csc kx ±i βδ ± ik + kα)t ] ± i cot [ kx ±i βδ ± ik + kα)t ]} u 6 = ; 1 + i 1 + a 0 ) { csc [kx ±i βδ ± ik + kα)t] ± cot [kx ±iβδ ± ik + kα)t]} 38) When h 1 = 1/ h = 1/ { { [ a 0 1 tanh kx ±i βδ ± ik + kα)t ] ± i sech [ kx ±i βδ ± ik + kα)t ]} u 7 = 39) a 0 ) { tanh [kx ±i βδ ± ik + kα)t] ± i sech [kx ±iβδ ± ik + kα)t]}

5 No. 5 A New Generalized Riccati Equation Rational Expansion Method to Generalized Burgers Fisher Equation 783 { { [ a 0 1 coth kx ±i βδ ± ik + kα)t ] ± csch [ kx ±i βδ ± ik + kα)t ]} u 8 = 40) a 0 ) { coth [kx ±i βδ ± ik + kα)t] ± csch [kx ±i βδ ± ik + kα)t]} a 0 and k are arbitrary constants. Family 3 According to Eq. 3) we obtain the following solutions of the generalized Burgers Fisher equation: When h 1 = h = 1 { 4i tan [ kx ± 1 u 31 = ± i βδ ± k i + kα)t ] ± 1)i tan [ kx ± 1 i βδ ± k i + kα)t ] ; 41) When h 1 = h = 1 When h 1 = 1 h = 1 u 3 = u 33 = u 34 = { 4i cot [ kx ± 1 ± i βδ ± k i + kα)t ] ± 1)icot [ kx ± 1 i βδ ± k i + kα)t ] ; 4) { 4 tanh [ kx ± 1 ± + kα)t ] ± 1) tanh [ kx ± 1 βδ ± k + kα)t ] 43) { 4 coth [ kx ± 1 ± + kα)t ] ± 1) coth [ kx ± 1 βδ ± k + kα)t ] ; 44) When h 1 = h = ±1/ { 4i { sec [ kx ± 1 u 35 = ± βδ ± k + kα)t ] ± tan [ kx ± 1 βδ ± k + kα)t ]} ± 1)i { sec [ kx ± 1 βδ ± k + kα)t ] ± tan [ kx ± 1 βδ ± k + kα)t ]} 45) u 36 = { 4i { csc [ kx ± 1 ± βδ ± k + kα)t ] ± cot [ kx ± 1 βδ ± k + kα)t ]} ± 1)i { csc [ kx ± 1 βδ ± k + kα)t ] ± cot [ kx ± 1 βδ ± k + kα)t ]} ; 46) When h 1 = 1/ h = 1/ { 4 { tanh [ kx ±βδ k + kα)t ] ± i sech [ kx ±βδ k + kα)t ]} u 37 = ± ± 1) {tanh [kx ±βδ k + kα)t] ± i sech [kx ±βδ k + kα)t]} 47) { 4 { coth [ kx ±βδ k + kα)t ] ± csch [ kx ±βδ k + kα)t ]} u 38 = ± ± 1) {coth [kx ±βδ k + kα)t] ± csch [kx ±βδ k + kα)t]} 48) k is an arbitrary constant. Family 4 According to Eq. 4) we obtain the following solutions of the generalized Burgers-Fisher equation. When h 1 = h = 1 1 a03 16 a 0 +9 a 0 u 41 = a 9 a a 0 +9 a 0 6 a0 4 a ) tan [kx λt)] 0 ± 49) 1 a03 16 a 0 +9 a 0 9 a a 0 +9 a a 0 0) tan [kx λt)] k18a a a 0 8a 0 + 1) λ = ± 1a3 0 16a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1)a 0 1) 1a3 0 16a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1) k When h 1 = h = 1 u 4 = 6 a0 4 a ) cot [kx λt)] 1 a03 16 a 0 +9 a 0 9 a a 0 +9 a a 0 0) cot [kx λt)] 1 a03 16 a 0 +9 a 0 9 a a 0 +9 a 0 k18a a a 0 8a 0 + 1) λ = ± 1a3 0 16a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1)a 0 1) 1a3 0 16a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1) k 50)

6 784 ZHANG Xiao-Ling WANG Jing and ZHANG Hong-Qing Vol. 46 When h 1 = 1 h = 1 u 43 = u 44 = 1 a a 0 +9 a 0 6 a0 4 a ) tanh [kx λt)] 1 a a 0 +9 a 0 9 a a 0 +9 a a 0 0) tanh [kx λt)] 9 a a 0 +9 a 0 1 a a 0 +9 a 0 6 a0 4 a ) coth [kx λt)] 1 a a 0 +9 a 0 9 a a 0 +9 a a 0 0) coth [kx λt)] 9 a a 0 +9 a 0 k18a a a 0 8a 0 + 1) λ = ± ± 1a3 0 16a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1)a 0 1) 1a3 0 16a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1) k When h 1 = h = ±1/ 1 a03 16 a 0 +9 a 0 u 45 = a 9 a 0 0 ± 3 16 a 0 +9 a 0 6 a0 4 a ) {sec [kx λt)] ± tan [kx λt)]} 1 a03 16 a 0 +9 a 0 9 a a 0 +9 a a 0 0) {sec [kx λt)] ± tan [kx λt)]} 1 a03 16 a 0 +9 a 0 u 46 = a 9 a 0 0 ± 3 16 a 0 +9 a 0 6 a0 4 a ) {csc [kx λt)] ± cot [kx λt)]} 1 a03 16 a 0 +9 a 0 9 a a 0 +9 a a 0 0) {csc [kx λt)] ± cot [kx λt)]} k18a a a 0 8a 0 + 1) λ = ± 1a3 0 16a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1)a 0 1) 1a3 0 16a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1) k When h 1 = 1/ h = 1/ u 47 = u 48 = 1 a a 0 +9 a 0 9 a a 0 +9 a 0 6 a0 4 a ) {tanh [kx λt)] ± isech [kx λt)]} 1 a a 0 +9 a 0 9 a a 0 +9 a a 0 0) {tanh [kx λt)] ± i sech [kx λt)]} 1 a a 0 +9 a 0 6 a0 4 a ) {coth [kx λt)] ± csch [kx λt)]} 1 a a 0 +9 a 0 9 a a 0 +9 a a 0 0) {coth [kx λt)] ± csch [kx λt)]} 9 a a 0 +9 a 0 k18a a a 0 8a 0 + 1) λ = ± ± 1a a 0 +9a0 9a a 0 +9a0 6a 0 4a 1a 0 + 1)a 0 1) a 0 +9a0 9a a 0 +9a0 6a 0 4a 0 + 1) k Remarks as when 51) 5) 53) 54) 55) 56) It is easily to see that when µ = 0 the solutions include results in Ref. [14] [16] as special cases such a 0 = 1 k = ± αδ 1 + m + δ) λ = α 1 + m) + β1 + m + δ) α1 + m + δ) u 13 and u 14 are just the solution u 11 and u 1 in Ref. [16]. When a 0 = 1 k = R λ = α 1 + m) + β1 + m + δ) α1 + m + δ) u 17 and u 18 are just the solution u 1 and u in Ref. [15]. When a 0 k and λ are other values we can get other solutions in Refs. [14] [16]. Here we omit them. But to our knowledge the other solutions obtained here are not found earlier. Remark 3 Here we get more generalized exact solutions of nonlinear evolution equations via a generalized ansätz. Recently we notice that there are a lot of papers of subequation methods focusing their attention on generalizing

7 No. 5 A New Generalized Riccati Equation Rational Expansion Method to Generalized Burgers Fisher Equation 785 auxiliary equations and more solutions of existing auxiliary equations getting more generalized solutions of target equations. But to our knowledge most of the solutions obtained by these methods can reduce to the known solutions. For example to some nonlinear evolution equations by using some generalized auxiliary equations some authors obtained the following solutions: As is well known there exist the following relationships { a0 + a 1 sech kx λt)) } 1/p u = µ 1 + µ sech 57) kx λt) { a0 + a 1 tanh kx λt)) } 1/p u = µ 1 + µ tanh. 58) kx λt) coshξ) = cosh ξ) 1 coshξ) = sinh ξ) ) so equations 8) and 9) can reduce to the following form: { }1/p A0 + A 1 sechkx λt)) Hξ) =. 60) ν 1 + ν sechkx λt)) It is easy to see that they are not new and can be easily obtained by the projective Riccati equation method which has been extensively studied by Chen and Li. [5] So we can conclude that generalized auxiliary equations or more solutions of auxiliary equations may not all generate new solutions of nonlinear evolution equations. And according to this we think we can get more generalized solutions of target equations by using of a more generalized ansätz a simple auxiliary equation and simple solutions of the auxiliary equation. Remark 4 In fact we naturally can present a more general ansätz which reads m a j φ j ξ) + b j φ j 1 ξ)φ ξ) + c j φ ξ)φ j ξ) + d j φ j ξ) uξ) = a 0 + µ j1 φξ) + µ j φ ξ) + µ j3 φ ξ)φ 1 ξ) + µ j4 φ 1 ξ) + 1) j 61) j=1 and the new vriable φ = φξ) satisfies φ ξ)) = dφξ) dξ ) r = h l φ l ξ) 6) a i0 a ij b ij c ij d ij µ j1 µ j µ j3 µ j4 h l and ξ i = 1...; j = 1... m i ; l = 1... r) are differentiable function to be determined. Therefore for some nonlinear equations more types of non-travelling solutions such as soliton-like solutions would be expected. 4 Summary and Conclusion In summary based on a new more general ansätz a generalized Riccati equation rational expansion method is put forward and applied to the generalized Burgers Fisher equation with nonlinear terms of any order. And many types of exact solutions have been derived by our improved method. The method can also easily be extended to treat other nonlinear evolution equations and is sufficient to seek more solitary wave solutions and other formal solutions of nonlinear evolution equations. In addition this method is also computerizable which allows us to perform complicated and tedious symbolic algebraic calculation on a computer. l=0 References [1] M.J. Ablowitz and P.A. Clarkson Soliton Nonlinear Evolution Equations and Inverse Scattering Cambridge University Press New York 1991). [] V.B. Matveev and M.A. Salle Darboux Transformation and Solitons Springer Berlin 1991). [3] Y. Chen Z.Y. Yan and H.Q. Zhang Theor. Math. Phys ) 970; Y. Chen B. Li and H.Q. Zhang Commun. Theor. Phys. Beijing China) ) 135; B. Li Y. Chen and H.Q. Zhang Phys. Lett. A ) 377. [4] R. Hirota and J. Satsuma Phys. Lett. A ) 407. [5] E.J. Parkes and B.R. Duffy Comput. Phys. Commun ) 88; W. Hereman Comput. Phys. Commun ) 143. [6] E.G. Fan Phys. Lett. A ) 1; E.G. Fan J. Zhang and Y.C. Hon Phys. Lett. A ) 376; E.G. Fan Z. Naturforsch. A ) 31. [7] Z.Y. Yan Phys. Lett. A 9 001) 100; Y. Chen Z.Y. Yan and H.Q. Zhang Phys. Lett. A ) 107; Y. Chen Z.Y. Yan B. Li and H.Q. Zhang Chin. Phys. 1 1) 003) 1. [8] S.A. Elwakil S.K. El-labany M.A. Zahran and R. Sabry. Phys. Lett. A 99 00) 179. [9] Y. Chen and Y. Zheng Int. J. Mod. Phys. C )

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