1-Soliton Solution of the Coupled Nonlinear Klein-Gordon Equations

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1 Studies in Mathematical Sciences Vol. 1, No. 1, 010, pp ISSN [Print] ISSN [Online] 1-Soliton Solution of the Coupled Nonlinear Klein-Gordon Equations Ryan Sassaman 1 Matthew Edwards Fayequa Majid 3 Anjan Biswas 1, Abstract: This paper studies the coupled Klein-Gordon equations in dimensions. The cubic law of nonlinearity arbitrary power law nonlinearity are considered in this paper. The 1-soliton solution of the coupled system, for both cases, is obtained. The solitary wave ansatz is used to carry out the integration. Key Words: Solitons; Integrability; Coupled Klein-Gordon Equations 1. INTRODUCTION The nonlinear Klein-Gordon equation NKGE is an important equation in the area of nonlinear evolution equations NLEE [1-10]. NKGE arises in theoretical physics, particularly in the area of relativistic quantum mechanics. There has been various methods that has been applied to carry out the integration of this equation. They are variational iteration method, exponential function method, Adomian decomposition method, G /G method of integration, semi-inverse variational principle. In this paper, the focus is going to be on the integration of -coupled NKGE with cubic nonlinearity. The solitary wave ansatz method will be used to carry out the integration. Finally, the results will be extended to the case of -coupled NKGE with arbitrary power law nonlinearity. In both cases, the results will be studied in dimensions [3].. CUBIC NONLINEARITY 1 Department of Mathematical Sciences, Center for Research Education in Optical Sciences Applications, Delaware State University, Dover, DE, , USA. Department of Physics, School of Arts Sciences, Alabama A & M University, Normal, AL-3576, USA. 3 Department of Mathematics, Alabama A & M University, Normal, AL-3576, USA. Corresponding author. address: biswas.anjan@gmail.com. Received 5 November 010; accepted 4 November

2 In this section, the coupled NKGE will be studied with cubic law of nonlinearity. The study will be split into the following two subsections which are on dimensions respectively Dimensions The dimensionless form of the -coupled NKGE in 1+1-dimensions with cubic nonlinearity is given by [3] q tt k q xx + a 1 q+b 1 q 3 + c 1 qr = 0, 1 r tt k r xx + a r+b r 3 + c q r=0, where in 1, the dependent variables q r are the wave fields while x t are the independent variables that respectively represent the spatial temporal variables. This coupled equation was already studied in 005 where doubly periodic solutions were obtained [3]. In this paper, the search is going to be for the non-topological 1-soliton solution to 1. Thus, the solitary wave ansatze are taken to be [6] where qx, t= rx, t=, 3 A, 4 = Bx vt, 5 in 3 4, A represent the soliton amplitudes while B is the inverse width of the soliton v is the soliton velocity. The unknown exponents p 1 p that are to be determined that will be discovered by the balancing method during the soliton solution derivation process. Thus, from 3 4 q tt = p 1 v B p 1p 1 + 1v B cosh p1+, 6 q xx = p 1 B p 1p B cosh p1+, 7 r tt = p v A B p p + 1v A B cosh p+, 8 r xx = p A B p 1+1A B cosh p+. 9 Substituting 6-9 into 1 respectively gives p 1 v k B p 1p v k B cosh p1+ + a 1 + b 1A 3 1 cosh 3p 1 + c 1 A = 0, 10 +p 31

3 p v k A B p p + 1 v k A B + + a A + b A 3 cosh 3p + c A 1 A = cosh p 1+p From 10 equating the exponents 3p 1 p 1 +, by the aid of balancing principle, gives [7, 8] which yields similarly, from 11, equating the exponents 3p p + also yields 3p 1 = p 1 +, 1 p 1 = 1, 13 p = Now, from 10, the linearly independent functions are 1/ + j for j=1, hence setting their respective coefficients to zero yields a1 B= v k, 15 b 1 A 1 + c 1A + a 1= Similarly from 11, the linearly independent functions are 1/ + j for j=1, hence setting their respective coefficients to zero yields [6] a B= v k, 17 From equating the two values of the soliton width B gives c A 1 + b A + a 1= a 1 = a, 19 finally solving the coupled system of equations with the soliton amplitudes yield a1 c 1 b =, 0 b 1 b c 1 c A = Hence, the 1-soliton solution to 1 are respectively given by qx, t= rx, t= a1 b 1 c c 1 c b 1 b. 1 cosh[bx vt], A cosh[bx vt], 3 where the amplitudes A are respectively given by 0 1, while the soliton width B is given by 15 or 17. These introduce the solvability condition given by 19. 3

4 . 1+ Dimensions The 1+ dimensional extension of the NKGE with cubic law of nonlinearity is given by q tt k q xx + q yy + a1 q+b 1 q 3 + c 1 qr = 0, 4 r tt k r xx + r yy + a r+ b r 3 + c q r=0, 5 where in 4 5, the dependent variables q r are the wave fields. The independent variables x y are both spatial variables, in this case t stays as the temporal variables. The solitary wave ansatze are same as given by 3 4 respectively within this case being given by = B 1 x+ B y vt. 6 Here, in 6, B 1 B are the inverse widths of the two solitons in the x y directions respectively again v is the soliton velocity. The unknown exponents p 1 p will again be computed later. Thus, from 3 6 q tt = p 1 v p 1p 1 + 1v, 7 cosh p1+ q xx = p 1 B 1 p 1p B 1, 8 cosh p1+ q yy = p 1 B p 1p B, 9 cosh p1+ r tt = p v A p p + 1v A, 30 cosh p+ r xx = p A B p 1+1A B cosh p+, 31 r yy = p A B p 1+1A B. 3 cosh p+ Substituting 7-3 into 4 5 respectively yields p 1 v k B 1 k B A1 p 1p v k B 1 k B cosh p1+ A1 B + a 1 + b 1A 3 1 cosh 3p 1 + c 1 A = 0, 33 +p p v k B 1 k b A p p + 1 v k B 1 k B cosh p+ A 33

5 + a A + b A 3 cosh 3p + c A 1 A cosh p = p Similarly from 33 34, as before the same values of p 1 p are obtained as in the previous case. Identifying the linearly independent functions in yields the relations [7, 8] B 1 + B = v + a 1 k, 35 B 1 + B = v + a k, 36 along with the same coupled equations for the amplitudes that are given by Therefore the amplitudes A are the same as in 0 1 from 35 36, the same constraint condition as in 19 is obtained. Hence, finally, the 1-soliton solution to 4 5 is given by qx, y, t= coshb 1 x+ B y vt, 37 rx, y, t= A coshb 1 x+ B y vt, 38 where the amplitudes A are respectively given by 0 1, while the soliton widths B 1 B are given by 35 or 36. These solutions introduce the solvability condition given by POWER LAW NONLINEARITY In this section, the coupled NKGE will be studied with power law of nonlinearity. The study will be split into the following two subsections which are on dimensions respectively Dimensions The dimensionless form of the -coupled NKGE in 1+1-dimensions with cubic nonlinearity is given by [9] q tt k q xx + a 1 q+b 1 q m+n + c 1 q m r n = 0, 39 r tt k r xx + a r+ b r m+n + c q n r m = 0, 40 where in 39 40, the exponents m n are positive numbers. The starting hypothesis is going to be the same as 3 4. In this case respectively reduce to v k B p 1 p 1p v k B cosh p1+ + a 1 + b 1 A m+n 1 cosh m+np 1 + c 1 A m 1 An cosh mp 1+np = 0, 41 p v k A B p p + 1 v k A B + 34

6 + a A + b A m+n cosh m+np + c A n 1 Am cosh np 1+mp = 0. 4 From 41 equating the exponents m+np 1 p 1 + gives [7, 8] which gives m+np 1 = p 1 +, 43 p 1 = m+n 1, 44 similarly, from 4, equating the exponents m+np p + that yields the same value of p as in p 1 seen in 44. Now, from 4, the linearly independent functions are 1/ + j for j=1, hence setting their respective coefficients to zero yields B= m+n 1 a1 v k, 45 b 1 A m+n c 1 A m 1 1 A n + m+n 1a 1=0. 46 Similarly from 4, the linearly independent functions are 1/ + j for j=1, hence setting their respective coefficients to zero yields B= m+n 1 a v k, 47 c A n 1 Am 1 + b A m+n 1 + m+n 1a 1 = From 45 47, equating the two values of the soliton width B gives the same constraint condition as 19. The amplitudes of the soliton are obtained by solving the coupled system given by Hence, the 1-soliton solutions to are given by qx, t=, 49 cosh m+n 1 [Bx vt] A rx, t=, 50 cosh m+n 1 [Bx vt] where the amplitudes A are respectively given by the coupled system 46 48, while the soliton width B is given by 45 or 47. These lead to the solvability condition given by Dimensions The 1+ dimensional extension of the NKGE with cubic law of nonlinearity is given by q tt k q xx + q yy + a1 q+b 1 q m+n + c 1 q m r n = 0, 51 35

7 r tt k r xx + r yy + a r+ b r m+n + c r m q n = 0, 5 where in 51 5, the dependent variables q r are the wave fields while x, y t are the independent variables that respectively represent the spatial temporal variables. In order to solve 51 5 for soliton solution the starting hypothesis is the same as 3 4 withbeing given by 6. Substituting 7-3 into 51 5 respectively yields p 1 v k B 1 k B A1 p 1p v k B 1 k B A1 B cosh p1+ + a 1 + b 1 A m+n 1 cosh m+np 1 + c 1 A m 1 An cosh mp 1+np = 0, 53 p v k B 1 k b A p p + 1 v k B 1 k B A cosh p+ + a A + b A m+n cosh m+np + c A n 1 Am cosh np 1+mp = Equations yields as before the same values of p 1 p as in 44. Identifying the linearly independent functions in yields the relations B 1 + B = 4v + a 1 m+n 1 4k, 55 B 1 + B = 4v + a m+n 1 4k, 56 along with the same coupled equations for the amplitudes that are given by the same constraint condition as in 19 is obtained. Hence, finally, the 1-soliton solution to 51 5 is given by qx, y, t= cosh m+n 1 B1 x+ B y vt, 57 A rx, y, t= cosh m+n 1 B1 x+ B y vt, 58 where the amplitudes A are respectively given by the coupled system 46 48, while the soliton width B is given by 55 or 56. These give the solvability condition given by 19. Additionally, the solitons introduce the constraint on the exponents m n that is given by which must hold for the solitons to exist. ACKNOWLEDGEMENT m+n>1, 59 This research work for first RS last AB authors was fully supported by NSF-CREST Grant No: HRD this support is genuinely sincerely appreciated. 36

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