ItsApplication To Derivative Schrödinger Equation

Transcription

1 IOSR Journal of Mahemaics (IOSR-JM) e-issn: , p-issn: X. Volume 1, Issue 5 Ver. II (Sep. - Oc.016), PP The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger Equaion Alaaeddin Amin Moussa, Lama Abdulaziz Alhakim MIS deparmen, College of Business & Economics/Qassim Universiy, Buraidah, K.S.A Absrac: In his paper, an efficien generalized of cosh expansion mehod is proposed o seek raveling wave soluions of he derivaive Schrödinger equaion. The raveling wave soluions are expressed in erms of he hyperbolic and rigonomeric funcions. I is shown ha he mehod is sraighforward and effecive for solving nonlinear evoluion equaions in mahemaical physics. Keywords: Generalized of cosh Expansion Mehod; Exac Soluions; Derivaive Schrödinger Equaion. I. Inroducion Nonlinear evoluion equaions (NLEEs) are widely used as models o describe many imporan complex physical phenomena in various fields of science, such as plasma physics, nonlinear opics, solid sae physics, chemical kinemaics, fluid mechanics, chemisry, biology and so on. Thus, esablishing exac raveling wave soluions of NLEEs is very imporan o beer undersand nonlinear phenomenas as well as oher real-life applicaions. In he recen years, a wide range of mehods have been developed o generae analyical soluions of nonlinear parial differenial equaions. Among hese mehods are he G G expansion mehod [1,], he expansion mehod [], he exp φ expansion mehod [4,5], he generalized of exp φ expansion mehod [6,7], he Jacobi ellipic funcion expansion mehod [8], he generalized Riccai equaion mehod [9,10] he Sine-Cosine Mehod [11], he F expansion mehod [1], and various oher mehods [1-16]. This paper presens an efficien generalized of cosh expansion mehod for obaining novel and more general exac raveling wave soluions for he derivaive Schrödinger equaion. The remaining of he paper is organized as follows. Secion explains he cosh expansion mehod. Secion applies his mehod for solving derivaive Schrödinger equaion and presens some special soluions, which are shown graphically in Secion 4. Secion 5 concludes he paper. II. The Generalized of cosh expansion mehod Suppose ha we have a nonlinear PDE in he following form from he Inroducion F(u, u, u x, u, u x, u xx, u xx,.... ) = 0 (.1) where, u = u(x, ) is an unknown funcion, F is a polynomial in u = u(x, ) and is parial derivaives, in which he highes order derivaives and nonlinear erms are involved.the main seps of his mehod are as follows: Sep 1: Use he raveling wave ransformaion: u(x, ) = u(), = k 1 x + k (.) where k 1, k are a consans o be deermined laer, permis us reducing (.1) o an ODE for u = u in he form P u, k 1 u, k u, k 1 k u,.... = 0(.) where P is a polynomial of u = u() and is oal derivaives. Sep :Balancing he highes derivaive erm wih he nonlinear erms in (.),we find he value of he posiive ineger m. Sep :Suppose ha he soluion of (.) can be expressed as follows: G G, 1 G u = α0 + m i=0 A 1 sech + A cosh + A cosh + A 4 B 1 sech + B cosh + B cosh + B 4 i (.4) where α 0, A i, B i i = 1,,,4 are consans o be deermined laer. Sep 4: Subsiuing (.4) ino Eq. (.) and hen seing all he coefficiens of cosh i sysems o zero, yields a sysem of algebraic equaions for k 1, k, A i, B i i = 1,,,4 and α 0. of he resuling DOI: / Page

2 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. Sep 5: Suppose ha he value of he consans k 1, k, A i, B i i = 1,,,4 and α 0 can be found by solving he algebraic equaions which are obained in sep 4. Since he general soluions of (.) have been well known for us, subsiuing k 1, k, A i, B i i = 1,,,4 and α 0 ino (.4), we have he exac soluions of he nonlinear PDEs (.1). III. The Exac Soluions of Derivaive Schrödinger Equaion In his secion, we will apply he he proposed mehod o find he exac soluions of he derivaive Schrödinger equaion. Le us consider he derivaive Schrödinger equaion: iw = 1 W xx + ik W W x ; W = W x, (.1) we make he following ransformaion W x, = e iδ e iψ h ; = x v; δ, v R h, ψ: R (.) R; h > 0 Subsiuing (.) ino Eq.(.1) and making he real par and imaginary par equal o zero, we have δh + vh ψ + h h ψ + kh ψ = 0, (.) h ψ + h ψ vh 6kh h = 0, (.4) Le ψ = A + Bh (.5) Subsiuing (.5) ino Eq. (.4) and equaing he coefficiens of hese erms h h, h o zero, we ge A = v, B = k. Therefore, when ψ = v + k h (.6) Eq. (.4) is idenical o zero. Subsiuing (.6) ino Eq. (.) we ge he following equaion: δ + v h + vk h + k 4 h 5 + h = 0, (.7) In order o solve Eq. (.7), we make he following ransformaion h = u (.8) Then, u saisfies 4 δ + v u + 8vku + k u 4 + uu u = 0, (.9) By balancing he erm u wih he erm u 4 in (.9), gives m = 1.Therefore, he cosh expansion mehod allows us o use he soluion in he following form: u = α 0 + A 1sech + A cosh + A cosh + A 4 B 1 sech + B cosh + B cosh + B 4 (.10) Subsiuing (.10) ino (.9), he lef-hand side is convered ino polynomials in cosh j, j = 0,1,,.... We collec each coefficien of hese resuled polynomials o zero, yields a se of simulaneous algebraic equaions (for simpliciy,which are no presened) for A 1, A, A, A 4, B 1, B, B, B 4, δ, v and α 0. Solving hese algebraic equaions wih he help of algebraic sofware Maple, we obain he following resuls: DOI: / Page

3 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. Case 1: α 0 = α 0, v = 1, δ = 1 8, A 1 = 1 4 A, A = A, A = A α 0 k 1 A 4 = 1 4 A, B 1 = 0, B = 0, B = ka, B 4 = 1 4 ka (.11) Subsiuing (.11) ino (.10), we have : where = x + Subsiuing (.1) ino (.8) and (.6) yields u = cosh 1 kcosh (.1) h = cosh 1 kcosh ψ = 1 + k cosh 1 kcosh d (.1) = kα kα ln anh + 1 anh 1 arcan anh Consequenly, he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.1) and Eq. (.) are obained in he following form: kα 0 1 W x, = cosh 1 kcosh + kα ln anh + 1 anh 1 (.14) arcan anh 1 8 = x + In paricular seing α 0 = 0, k = 6 we find : W 1 x, = cosh 1 6cosh + ln anh + 1 anh 1 arcan anh 1 8 (.15) = x + See Figure (.1) Case : α 0 = α 0, v =, δ = 1, A 1 = B 1 4 α 0 k, A k = B 1 α 0 k k A = 0, A 4 = 0, B 1 = B 1, B = B 1, B = 0, B 4 = 0, (.16) Subsiuing (.16) ino(.10), we have : DOI: / Page

4 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. 4 sech cosh u = k sech cosh (.17) where = x + Subsiuing (.17) ino (.8) and (.6) yields h = 4 k sech cosh sech cosh ψ = + 6 = sech cosh sech cosh d α 0k + α 0k ln anh +arcan anh anh arcan anh 1 (.18) Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.18) and Eq. (.) are obained in he following form: 4 k sech cosh sech cosh W x, = α 0k + α 0k ln anh 1 anh + 1 (.19) +arcan anh arcan anh 1 1 = x + In paricular seing α 0 = 0, k = 4 we find : sech cosh sech cosh W x, = ln anh 1 anh + arcan anh (.0) +arcan anh 1 1 = x + See Figure (.) Case : DOI: / Page

5 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. Subsiuing (.1) ino(.10), we have : where = x + 1+α 0 k 4α 0 k Subsiuing (.) ino (.8) and (.6) yields u = α 0 α 0 k + 1 cosh + α 0 k + 1 α 0 k + 1 cosh + α 0 k 1 (.) h = α 0 α 0 k + 1 cosh + α 0 k + 1 α 0 k + 1 cosh + α 0 k 1 ψ = v + kα 0 = 4 α 0k 1 4α 0 k α 0 k + 1 cosh + α 0 k + 1 α 0 k + 1 cosh + α 0 k 1 + arcan an α 0 k d(.) Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.) and Eq. (.) are obained in he following form: α 0 α 0 k + 1 cosh + α 0 k + 1 α 0 k + 1 cosh + α 0 k 1 W x, = = x α 0 k 4α 0 k 4 α 0k 1 4α 0 k + arcan an α 0 k + 1 α 0 k 1 α 0 k α 0 k (.4) In paricular seing α 0 = 1, k = 1 we find : W x, = cosh + 1 cosh 1 + arcan an 1 8 (.5) = x + See Figure (.) Case 4: Subsiuing (.6) ino(.10), we have : u = 9sech vk sech 4k 4v + 9 cosh + k 4v + 9 (.7) where = x v Subsiuing (.7) ino (.8) and (.6) yields DOI: / Page

6 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. h = 9sech vk sech 4k 4v + 9 cosh + k 4v + 9 ψ = v + k 9sech d vk sech 4k 4v + 9 cosh + k 4v + 9 = v + 7k ε ε 4 ε + 1 ln anh ε 6 S + S S 1 ε S 1 + S S ε + S S 1 S ε S 1 = vk, S = 4k 4v + 9, S = k 4v + 9 (.8) 0 = S + S S 1 ε 6 + S 1 + S S ε 4 + S S 1 S ε + S 1 + S + S Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.8) and Eq. (.) are obained in he following form: W x, = e = x v 9 8 +v i e iψ 9sech vk sech 4k 4v + 9 cosh + k 4v + 9 In paricular seing α 0 = 0, v = 0, k = 1 we find : (.9) sech 4cosh W 4 x, = arcan anh arcan anh + (.0) +arcan anh + 8 = x See Figure (.4) Case 5: α 0 = α 0, v = k α 0 B + A + 4B, A 4kB α 0 B + A = A, A = 0, A 4 = 0, 4α 0 B A 1 = k α 0 B + A + 4B, δ = k α 0 B + A 4B k α 0 B + A + 4B k B α 0 B + A,(.1) 4B B 1 = k α 0 B + A + 4B, B = B, B = 0, B 4 = 0 Subsiuing (.1) ino(.10), we have : α 0 B + A k α 0 B + A + 4B cosh u = 4B sech + k B α 0 B + A + 4B cosh (.) where = x + k α 0 B +A +4B 4kB α 0 B +A Subsiuing (.) ino (.8) and (.6) yields DOI: / Page

7 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. h = ψ = v + k = α 0 B + A k α 0 B + A + 4B cosh 4B sech + k B α 0 B + A + 4B cosh kl 1 4L L L + L ln +v + kl 1 L α 0 B + A k α 0 B + A + 4B cosh 4B sech + k B α 0 B + A + 4B cosh L + L anh L anh + L + L L + L anh + L anh + L + L ln anh + 1 anh 1 d (.) L 1 = α 0 B + A k α 0 B + A + 4B ; L = 4B ; L = k B α 0 B + A + 4B Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.) and Eq. (.) are obained in he following form: In paricular seing α 0 = 1, k =, A =, B = 1 we find : W 5 x, = 0cosh 10cosh sech 6ln an h +1 an h 1 arcan an h arcan an h (.5) = x + 14 See Figure (.5) Case 6 v = k α 0 B + A + 4B 4kB α 0 B + A, δ = k α 0 B + A + 4B k α 0 B + A 4B k α 0 B + A B, 4α 0 B B A 1 = k α 0 B + A + 4B, A = A B 4α 0 B, A B = A, A 4 = k α 0 B + A + 4B, α 0 = α 0 (.6) 4B B B 1 = k α 0 B + A + 4B, B 4B = B, B = B, B 4 = k α 0 B + A + 4B Subsiuing (.6) ino(.10), we have : u = α 0B + A k α 0 B + A + 4B cosh 4B + k B α 0 B + A + 4B cosh (.7) where = x + k α 0 B +A +4B 4kB α 0 B +A DOI: / Page

8 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. Subsiuing (.7) ino (.8) and (.6) yields h = ψ = v + k = α 0 B + A k α 0 B + A + 4B cosh 4B + k B α 0 B + A + 4B cosh α 0 B + A k α 0 B + A + 4B cosh 4B + k B α 0 B + A + 4B cosh kt 1 T ln T + T anh T anh + T + T 4T T + T T + T anh + T anh + T + + T + kt 1 4T ln anh + 1 anh 1 + v T 1 = α 0 B + A k α 0 B + A + 4B, T = 4B, T = k B α 0 B + A + 4B d (.8) Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.8) and Eq. (.) are obained in he following form: In paricular seing α 0 = 0, k = 1, A =, B = we find : W 6 x, = 5cosh 5cosh 4 arcan anh 5 arcan anh 5 + +( )ln(anh + 1 anh 1 ) (.40) = x See Figure (. 6) Case 7: α 0 = α 0, v = k A 1 + α 0 B 1 4B 1, δ = k A 1 + α 0 B 1 + 4B 1 4kB 1 A 1 + α 0 B 1 k B 1 A 1 + α 0 B, 1 A 1 = A 1, A = α 0 k A 1 + α 0 B 1 + 4B 1, A 4B = 0, A 4 = 0, (.41) 1 B 1 = B 1, B = α 0 k A 1 + α 0 B 1 + 4B 1 4B 1, B = 0, B 4 = 0 DOI: / Page

9 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. Subsiuing (.41) ino(.10), we have : u = 4B 1 α 0 B 1 + A 1 sech 4B 1 sech k α 0 B 1 + A 1 + 4B 1 cosh (.4) where = x + k A 1 +α 0 B 1 4B 1 4kB 1 A 1 +α 0 B 1 Subsiuing (.4) ino (.8) and (.6) yields h = 4B 1 α 0 B 1 + A 1 sech 4B 1 sech k α 0 B 1 + A 1 + 4B 1 cosh ψ = v + k 4B 1 α 0 B 1 + A 1 sech 4B 1 sech k α 0 B 1 + A d 1 + 4B 1 cosh = v + kβ 1 ln β + β anh + β anh + β + β 4 β β + β β + β anh β anh + β + β β 1 = 4B 1 α 0 B 1 + A 1, β = 4B 1, β = k α 0 B 1 + A 1 + 4B 1 (.4) Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.4) and Eq. (.) are obained in he following form: 4B 1 α 0 B 1 + A 1 sech 4B 1 sech k α 0 B 1 + A 1 + 4B 1 cosh W x, = = x + k A 1 + α 0 B 1 4B 1 4kB 1 A 1 + α 0 B 1 kβ 1 ln β + β anh + β anh + β + β 4 β β + β β + β anh β anh + β + β +v + k A 1 + α 0 B 1 + 4B 1 k B 1 A 1 + α 0 B 1 In paricular seing α 0 = 1, k = 6, A 1 = 1, B 1 = 1 we find : W 7 x, = 8sech 4sech 148cosh arcan 6anh arcan 6anh (.45) (.44) = x See Figure (. 7) Case 8: α 0 = α 0, v = k A 4 + α 0 B 4 + B 4, δ = k A 4 + α 0 B 4 B 4 k A 4 + α 0 B 4 + B 4 4kB 4 A 4 + α 0 B 4 k A 4 + α 0 B 4 + B 4 A 1 = A 4k A 4 + α 0 B 4 + B 4 A 4 + α 0 B 4 k A 4 + α 0 B 4 + B 4, A = 0, A = 0, A 4 = A 4 B 1 = B 4 k A 4 + α 0 B 4 B 4 k A 4 + α 0 B 4 + B 4, B = 0, B = 0, B 4 = B 4 (.46) DOI: / Page

10 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. Subsiuing (.46) ino (.10), we have : u = α 0 B 4 + A 4 B 4 + k α 0 B 4 + A 4 sech + 1 k B 4 α 0 B 4 + A 4 B 4 sech + k B 4 α 0 B 4 + A 4 + B 4 (.47) where = x + k A 4 +α 0 B 4 +B 4 4kB 4 A 4 +α 0 B 4 Subsiuing (.47) ino (.8) and (.6) yields h = ψ = v + k α 0 B 4 + A 4 B 4 + k α 0 B 4 + A 4 sech + 1 k B 4 α 0 B 4 + A 4 B 4 sech + k B 4 α 0 B 4 + A 4 + B 4 = v + ke 1 E E E E E α 0 B 4 + A 4 B 4 + k α 0 B 4 + A 4 sech + 1 k B 4 α 0 B 4 + A 4 B 4 sech + k B 4 α 0 B 4 + A 4 + B 4 arcan E E anh E E E + ke 1 E E 1 = α 0 B 4 + A 4 B 4 + k α 0 B 4 + A 4 ; E = k B 4 α 0 B 4 + A 4 B 4 ; E = k B 4 α 0 B 4 + A 4 + B 4 ln anh + 1 anh 1 d (.48) Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.48) and Eq. (.) are obained in he following form: In paricular seing α 0 = 0, k = 1, A 4 =, B 4 = 1 we find : 10 sech + 1 W 8 x, = sech + 5 arcan anh ln anh 1 anh (.50) = x 1 8 See Figure (. 8) Case 9: α 0 = α 0, v = v, δ = 1 v + 1 4, A 1 = 4α 0k v A 4 + α 0 B 4 kv A 4 + α 0 B 4 + B 4, A = α 0 k A 4 + α 0 B v, A = 0, (.51) k 1 + 4v A 4 = A 4, B 1 = v vk A 4 + α 0 B 4 B 4, B = k A 4 + α 0 B v, B = 0, B 4 = B v Subsiuing (.51) ino (.10), we have : DOI: / Page

11 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. u α 0 vb 4 + va 4 B 4 sech α 0 B 4 + A v = α 0 v B 4 vb 4 + 4v A 4 sech α 0 B 4 + 4α 0 v B 4 + A 4 + 4v A 4 cosh B v (.5) where = x v Subsiuing (.5) ino (.8) and (.6) yields Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.5) and Eq. (.) are obained in he following form: In paricular seing α 0 = 1, v = 1, k =, A 4 = 1, B 4 = 1 we find : 7 5 sech sech 0 5 cosh 5 W 9 x, = 4 + arcan + 5 anh +ln anh 1 anh (.55) = x See Figure (. 9) Case 10: α 0 = α 0, v = k A + α 0 B + 9B, δ = k4 A + α 0 B 4 + 6B k A + α 0 B 4 7B 4kB A + α 0 B B k A + α 0 B, A 1 = A k A + α 0 B + 9B A + α 0 B 4k A + α 0 B + 6B, A = 0, A = A, A 4 = 4 A B 1 = B k A + α 0 B 9B 4 k A + α 0 B + 9B, B = 0, B = B, B 4 = 4 B (.56) Subsiuing (.56) ino (.10), we have : DOI: / Page

12 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. u = α 0 + A k A +α 0 B +9B A +α 0 B 4k A +α 0 B +6B sech + A cosh A 4 (.57) sech + B cosh B 4 B k A +α 0 B 9B 4 k A +α 0 B +9B where = x + k A +α 0 B +9B 4kB A +α 0 B Subsiuing (.57) ino (.8) and (.6) yields Consequenly,he exac soluion of he derivaive Schrödinger equaion (.1) wih he help of Eq. (.58) and Eq. (.) are obained in he following form: W x, = v + k α 0 + A k A +α 0 B +9B A +α 0 B 4k A +α 0 B +6B sech + A cosh 4 A B k A +α 0 B 9B 4 k A +α 0 B +9B sech + B cosh 4 B (.59) ψ k4 A + α 0 B 4 + 6B k A + α 0 B 7B 4 B k A + α 0 B = x + k A + α 0 B + 9B 4kB A + α 0 B In paricular seing α 0 = 1, k = 1, A = 4, B = we find : 1 + sech + 4cosh cosh W 10 x, = + arcan anh arcan anh + + arcan anh + ln anh + 1 anh (.60) = x + See Figure (. 10) DOI: / Page

13 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. IV. Table Graphics Figure (.1) Figure (.) Figure (.) The D plo of W 1 The D plo of W The D plo of W = x + = x + = x + 5 x x x Figure (.4) Figure (.5) Figure (.6) The D plo of W 4 The D plo of W 5 The D plo of W 6 = x = x + 14 = x x x x Figure (.7) Figure (.8) Figure (.9) The D plo of W 7 The D plo of W 8 The D plo of W 9 = x = x 1 8 = x 5 x x 0 5 x DOI: / Page

14 The Generalized of cosh() Expansion Mehod And IsApplicaion To Derivaive Schrödinger.. Figure(.10) The D plo of W 10 = x + 5 x V. Conclusion In his aricle, we propose new echnique called The generalized of cosh expansion mehod. his mehod has been applied o find he exac raveling soluions of he Derivaive Schrödinger Equaion. hese soluions have rich local srucures, I may be imporan o explain some physical phenomena. This work shows ha, he generalized of cosh expansion mehod is direc, effecive and can be used for many oher NLPDEs in mahemaical physics. G References [1]. M. Wang, X. Li,and J.Zhang (008).The expansion mahod and ravelling wave soluions of nonlinear evoluion equaions in G mahemaical physics, Physics Leers A, Vol. 7, no. 4,: []. Mohammad Ali Bashir, Alaaedin Amin Moussa (014). New Approach of Journal of Mahemaics Research, Vol. 6, No. 1: 4-1. G G G Expansion Mehod. Applicaions o KdV Equaion, []. L.X.Li, E.q.Li and M. wang (010).The expansion mehod and is applicaion o ravelling wave soluions of he Zakharov G G equaion, Applied Mahemaics B, Vol. 5, No. 4: [4]. M Ali Akbar and Norhashidah Hj Mohd Ali (014).Soliary wave soluions of he fourh order Boussinesq equaion hrough exp φ he expansion mehod, Springer plus,:44. [5]. K. Khan and M.A. Akbar (01). Applicaion of exp φ expansion mehod o find he exac soluions of modified Benjamin-Bona-Mahony equaion, World Applied Sciences Journal, 4(10): [6]. Mohammad Ali Bashir, Alaaedin Amin Moussa, Lama Abdulaziz Alhakim (015). The new generalized of exp φ expansion mehod and is applicaion o some complex nonlinear parial differenial equaion, Journal of advanced in Mahemaics, Vol.9, No. 8: [7]. Khalil Hassan Yahya, Zelal Amin Moussa, (015). new approach of generalized exp φ expansion mehod and is applicaion o some nonlinear parial differenial equaion, Journal of Mahemaics research, Vol.7, No.: [8]. M.Inc and M, Ergu, (005).Periodic Wave Soluions for he Generalized Shallow Waer wave Equaion by he Improved Jacobi Ellipic Funcion Mehod, Appl. Mah. E-Noes.,5: [9]. C. A. Gomez, A. H. Salas, and B. A. Frias(010).Exac soluions o KdV6 equaion by using a new approach of he projecive Riccai equaion mehod,mahemaical Problems in Engineering, Aricle ID , 010, doi: /010/797084, (10 pages). [10]. S. Guo, L. Mei, Y. Zhou and C. Li, (011). The exended Riccai equaion mapping mehod for variable-coefficien diffusion-reacion and mkdv equaion, Applied Mahemaics and Compuaion, vol. 17: [11]. E. Yusufoglu and A, Bekir, (006).Solions and Periodic Soluions of Coupled Nonlinear Evoluion Equaions by Using Sine-Cosine Mehod, Inerna J. Compu. Mah., 8(1): [1]. Mohammad Ali Bashir, Lama Abdulaziz Alhakim (01).New F Expansion mehod and is applicaions o Modified KdV equaion, Journal of Mahemaics Research, Vol. 5, No [1]. M. J.Ablowiz, G.Biondini and S. De Lillo (1997).On he Well-posedness of he Eckhaus Equaion", physics Leers A 0., 19-. [14]. M. Mirzazadeh, S. Khaleghizadeh (01). Modificaion of runcaed expansion mehod o some complex nonlinear parial differnial equaions, Aca Universiais Apulensis, No : [15]. S.T. Mohyud-Din and M. A. Noor (008).Solving Schrödinger Equaions by Modified Variaional Ieraion Mehod", World Appl. Science J., 5():5-57. [16]. Mohammad Ali Bashir, Alaaedin Amin Moussa (014).The coh a Expansion Mehod and is Applicaion o he Davey-Sewarson Equaion, Applied Mahemaical Sciences, Vol. 8, no. 78,: DOI: / Page