Solitary Wave Solution of the Plasma Equations

Transcription

1 Applied Mathematical Sciences, Vol. 11, 017, no. 39, HIKARI Ltd, Solitary Wave Solution of the Plasma Equations F. Fonseca Universidad Nacional de Colombia Grupo de Ciencia de Materiales y Superficies Departamento de Física Bogotá, Colombia Copyright c 017 F. Fonseca. This article is distributed under the Creative Commons Attribution License, which permits unrestrikd use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we solved the temporal nonlinear one-dimensional Plasma equations using the Tanh solitary wave method and the Jacobi elliptic functions. We find several families of solutions. Keywords: Plasma equations, Tanh solitary wave, Jacobi elliptic functions 1 Introduction From the famous Benjamin Franklin s kite experiment, through the boreal aurora, till the plasma at the begining of the universe, plasma physics has played a central role in the physical description of such phenomena, [1]-[5]. In order to give an analytical solution to one hydrodynamic plasma model [5], we used a solitary wave method, the Tanh technique [6], one of the most popular, and the Jacobi elliptic function [7]. The Plasma Equations We start from the ion acoustic model in a collisionless plasma equations, [5]: dn dt + d(nv) dx = 0 (1)

2 1934 F. Fonseca dv dt + d dx (v + ψ) = 0 () d ψ dx eψ + n = 0 (3) Where ψ is the electric potential, n the electron concentration and v the velocity. Now, defining the variables Then, the first and second derivatives are w = e ψ, lnw = ψ (4) dψ dx = 1 dw w dx, and replacing in eqs. (1)-(3) d ψ dx = 1 w (dw dx ) + 1 d w (5) w dx ( dw dx ) + w d w dx w3 + nw = 0 (6) w dv dt + w d dx (v ) + dw dx = 0 (7) Introducing ξ dn dt + dnv dx = 0 (8) The temporal and spatial derivatives are: ξ = x ct (9) t = c d dξ ; x = d dξ ; x = d dξ (10) Then, equations (6)-(8) w d w dξ (dw dξ ) w 3 + nw = 0 (11) cw dv dξ + w d dξ (v ) + dw dξ = 0 (1) c dn dξ + d(nv) dξ = 0 (13)

3 Solitary wave solution of the plasma equations The Tanh Function Method Now, we introduce a new independent variable, [6]: Then, the derivatives of ξ, are: Y = tanh (ξ) (14) d dξ = (1 Y ) d dy, d dξ = Y (1 Y ) d dy + (1 Y ) d (15) dy The solutions are postulated as: m p w = a i Y i ; n = b i Y i (16) i=1 i=1 Then, replacing in eqs. (11)-(13) ((1 Y ) dw dy ) + w( Y (1 Y ) dw dy + (1 Y ) d w dy ) w3 + nw = 0 (17) cw(1 Y ) dv v dy + w(1 Y d( ) ) dy + (1 Y ) dw dy = 0 (18) c(1 Y ) dn dy + (1 Y ) d(nv) dy = 0, d( cn + nv) dy Where k is an integration constant. Using eqs. (18) and (19) = 0, v = k + cn n (19) wk dn dy + dw dy = 0 (0) Now, we balance the highest-order linear derivative with the highest order nonlinear terms in eq. (17). So: wy 4 d w dy w3 m m = 3m m = (1) wy 4 d w dy nw m m = m + p p = ()

4 1936 F. Fonseca Then w = a 0 + a 1 Y + a Y, n = b 0 + b 1 Y + b Y (3) Replacing eq. (3) in eq. (17) ((1 Y )(a 1 + a Y )) + (a 0 + a 1 Y + a Y )( Y (1 Y ) (4) (a 1 + a Y ) + (1 Y ) a ) (a 0 + a 1 Y + a Y ) 3 +(b 0 + b 1 Y + b Y )(a 0 + a 1 Y + a Y ) = 0 and in eq. (0) (a 0 + a 1 Y + a Y )k (b 1 + b Y ) + (a 1 + a Y ) = 0 (5) Then, doing some algebra, we get: a 0,1 = 1 k, a 1,1 = k, a,1 = 4 3, b 1,1 = k, b,1 = 3 (6) a 0, = 1 k, a 1, = k, a, = 4 3, b 1, = k, b, = 3 (7) So b 0,1 = a3 0 + a 1 a 0 a a 0 = ( 1 k k )k4 (8) b 0, = a 0a 1 + 3a 0a 1 + a 1 a a 0b 1 a 0 a 1 = ( 4 k k k ) 1 k 3 (9) b 0,3 = 3a 0a 1 + 8a 0 a + 3a 0a + a a 0 a 1 b 1 a 0b a 1 + a 0 a (30) = 3 0 ( k k )k b 0,4 = a 0a 1 + a a 1 a + 6a 0 a 1 a a 1b 1 a 0 a b 1 a 0 a 1 b (31) a 1 a = 3 16 ( 8 3k k )k

5 Solitary wave solution of the plasma equations 1937 b 0,5 = a 1 + 6a 0 a + 3a 1a + 3a 0 a a 1 a b 1 a 1b a 0 a b a = 43 4k (3) Therefore, the families of solutions are: f 1 (a 0,1, a 1,1, a,1, b 0,1, b 1,1, b,1 ) (33) f (a 0,1, a 1,1, a,1, b 0,, b 1,1, b,1 ) (34) f 3 (a 0,1, a 1,1, a,1, b 0,3, b 1,1, b,1 ) (35) f 4 (a 0,1, a 1,1, a,1, b 0,4, b 1,1, b,1 ) (36) f 5 (a 0,1, a 1,1, a,1, b 0,5, b 1,1, b,1 ) (37) f 6 (a 0,, a 1,, a,, b 0,1, b 1,, b, ) (38) f 7 (a 0,, a 1,, a,, b 0,, b 1,, b, ) (39) f 8 (a 0,, a 1,, a,, b 0,3, b 1,, b, ) (40) f 9 (a 0,, a 1,, a,, b 0,4, b 1,, b, ) (41) f 10 (a 0,, a 1,, a,, b 0,5, b 1,, b, ) (4)

6 1938 F. Fonseca 4 Jacobi Elliptic Function Method We suppose a solution in eqs. (17) and (0) given by, [7]: r w(ξ) = f i sn i (ξ), s n(ξ) = g i sn i (ξ) (43) i=0 i=0 Where the Jacobi identities are: sn (ξ, m) + cn (ξ, m) = 1, dn (ξ, m) + m sn (ξ, m) = 1 (44) (sn(ξ)) = cn(ξ, m)dn(ξ, m), (cn(ξ)) = sn(ξ, m)dn(ξ, m) (dn(ξ)) = m sn(ξ, m)cn(ξ, m), (sn(ξ)) = m sn(ξ, m)cn (ξ, m) sn(ξ, m)dn (ξ, m), (cn(ξ)) = m sn (ξ, m)cn(ξ) dn (ξ, m)cn(ξ, m), (dn(ξ)) = m sn (ξ, m)dn(ξ, m) m cn (ξ, m)dn(ξ, m) Balancing the highest derivative with the nonlinear term we get f = g =. Then, we get: w(ξ) = f 0 + f 1 sn(ξ) + f sn (ξ), n(ξ) = g 0 + g 1 sn(ξ) + g sn (ξ) (45) Replacing in eqs. (17) and (0) and doing some algebra, and defining l 1 = 1 4k 4k m + 40k 4 m, we get g = 0, f = 10m, and g 1,1 = 5 m m l 1 k 10 m k m m l 1 5 l k k 1 m l 1 k k (46) k + k m 60k 4 m f 0,1 = 1 l 1 k, g 0,1 = 1 ( 1 k l ) 1 f k 1,1 = m 10 k m l 1 (47) k g 1, = 5 m m l m m l l m k k k k 1 m l 1 k k (48) k + k m 60k 4 m f 0, = 1 l 1 k, g 0, = 1 ( 1 k l ) 1, f k 1, = m 10 k m l 1 (49) k

7 Solitary wave solution of the plasma equations 1939 g 1,3 = 5 m + m l 1 10 m + m l l m k k k k 1 + m l 1 k k (50) k + k m 60k 4 m f 0,3 = 1 + l 1 k, g 0,3 = 1 ( 1 k + l ) 1, f k 1,3 = m 10 k + m l 1 (51) k g 1,4 = 5 m + m l 1 k + 10 m k m + m l 1 5 l k k 1 + m l 1 k k (5) k + k m 60k 4 m f 0,4 = 1 + l 1 k, g 0,4 = 1 ( 1 k + l ) 1, f k 1,4 = m 10 k + m l 1 (53) k The families of solutions are: f 11 (f 0,1, f 1,1, f, g 0,1, g 1,1, g, ) (54) f 1 (f 0,, f 1,, f, g 0,, g 1,, g, ) (55) f 13 (f 0,3, f 1,3, f, g 0,3, g 1,3, g, ) (56) f 14 (f 0,4, f 1,4, f, g 0,4, g 1,4, g, ) (57) 5 Conclusions We solved the Plasma Equations using the solitary wave methods known as the Tanh solitary wave method and the Jacobi elliptic functions. We find 14 families of solutions. The solutions are: ψ(x, t) = ln (a 0 + a 1 tanh (x ct) + a tanh (x ct)) (58)

8 1940 F. Fonseca n(x, t) = b 0 + b 1 tanh (x ct) + b tanh (x ct) (59) And v(x, t) = k + c(b 0 + b 1 tanh (x ct) + b tanh (x ct)) b 0 + b 1 tanh (x ct) + b tanh (x ct) (60) ψ(x, t) = ln (f 0 + f 1 sn((x ct), m) + f sn ((x ct), m) (61) n(x, t) = g 0 + g 1 sn((x ct), m) + g sn ((x ct), m) (6) v(x, t) = k + c(g 0 + g 1 sn((x ct), m) + f sn ((x ct), m) g 0 + g 1 sn((x ct), m) + g sn ((x ct), m) (63) As a future work, we can extend the method to higher dimensions and high frecuency solitons in isotropic plasma and strong magnetic fields [5]. Acknowledgements. This research was supported by Universidad Nacional de Colombia in Hermes project (3501). References [1] R. J. Goldston and P. H. Rutherford, Introduction to Plasma Physics, Taylor & Francis, [] F. E. Chen, Introduction to Plasma Physics and Controlled Fusion, Plenum Press, New York, [3] J.A. Bittencourt, Fundamentals of Plasma Physics, Springer Science+Business Media, New York, [4] A. L. Peratt, Evolution of the Plasma Universe: II. The Formation of Systems of Galaxies, IEEE Transactions on Plasma Science, 14 (1986), [5] E.A. Kuznetsov, A.M. Rubenchik and V.E. Zakharov, Soliton stability in plasmas and hydrodynamics, Physics Reports (Review Section of Physics Letters), 14 (1986), no. 3,

9 Solitary wave solution of the plasma equations 1941 [6] W. Malfliet and W. Hereman, The Tanh Method: I. Exact solutions of Nonlinear Evolution and Wave Equations, Physica Scripta, 54 (1996), [7] Z.T. Fu, S.K. Liu, S.D. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 90 (001), Received: July 1, 017; Published: July 5, 017