Spiral Wave Solutions of One-dimensional Ginzburg-Landau Equation By Extended F-expansion Method

Spiral W a ve S olutions of One-dimensional Ginzbur g-Landau Equation By Extended F-expansion Method Xurong Chen 1 , ∗ 1 Physics Department, University of South Car olina, Columbia, SC29208, USA (Dated: Augu st 25, 2021 ) Abstract The one-dimension al Ginzb urg-Lan dau (GL) Equation is consid ered. W e use the recently de velo ped ext ended F -ex pansion method to obtain spiral wa ve sol ution of one-dime nsional GL Equation. P A CS nu mbers: 02.30 .Jr , 02.30 .Gp 1 INTR ODUCT ION The Ginzburg-La ndau (GL) Equation i s one of the most-st udied nonlinear p artial differential equations (P DE) [1]. Spiral wa ves a re important patterns in many systems, such as physics, chem- istry , m aterials, biology , etc. Ex ploring exact solutions of n onlinear PDE i s a hot and difficult topic in mathematical physics . Many methods were de veloped for studying e x act solutions of nonlinear PDEs. Re cently , a new method named e xt ended F-e xpansion was proposed [2][3]. It’ s useful to obtain more Jacobi elliptic function solutions. In this paper it is used to study the one-dimensio nal GL Equation. W e obtain a spi ral w av e solution, which is exactly same with Hagan’ s solution [4]. SPIRAL W A VES OF ONE-DIMENSIONAL GINZ BURG-LANDA U EQ U A TION The Ginzb urg-Landau Equation is gi ven by ∂ t A = A + (1 + ib )∆ A − (1 + ic ) | A | 2 A, (1) where A is a com plex function of time t and space x ; b and c are real parameters d escribing linear and nonlinear dispersion, r espectively . It’ s useful to represent the complex function A in the following way A = R e iθ . (2) Then the Eq. ( 1) becomes ∂ t R = [ △ − ( ▽ θ ) 2 ] R − b (2 ▽ θ · ▽ R + R △ θ ) + (1 − R 2 ) R, R∂ t θ = b [ △ − ( ▽ θ ) 2 ] R + 2 ▽ θ · ▽ R + R △ θ − cR 3 . (3) If b = 0 , t hen it becomes a class of reaction-di f fusion equations called λ − ω sy stems which hav e the following gene ral form ∂ t R = [ △ − ( ▽ θ ) 2 ] R + Rλ ( R ) , R∂ t θ = 2 ▽ θ · ▽ R + R △ θ + q Rω ( R ) . (4) For one-dimensional systems, let us assume the analogues of spiral w ave solutions ha ve the form R = ρ ( x ) , θ = − c (1 − k 2 ) T + ψ ( x ) , (5) 2 where k is an arbitrary c onstant and T is a long-tim e scale [4]. Substi tution into Eq. ( 4) yields [4] ρ xx + ρ (1 − ρ 2 − ψ 2 x ) = 0 , ψ xx + 2 ρ x ψ x /ρ = − c (1 − k 2 − ρ 2 ) . (6) Let us assume, ξ = f x, ρ = a 0 + a 1 F ( ξ ) + ... + a m F ( ξ ) m , ψ x = b 0 + b 1 F ( ξ ) + ... + b n F ( ξ ) n . (7) where a i and b j ( i = 0 , 1 , m ; j = 0 , 1 , ...n ) are constants to be determined later . F ( ξ ) is a solutio n of the first-oder nonlinear ODE F ′ 2 = s 4 F 4 + s 2 F 2 + s 0 , f 2 ρ xx + ρ (1 − ρ 2 − ψ 2 x ) = 0 , f ψ xx + 2 f ρ x ψ x /ρ = − c (1 − k 2 − ρ 2 ) . (8) According to the homogeneous balance method [2], we know m = n = 1 , hence ρ = a 0 + a 1 F ( ξ ) , ψ x = b 0 + b 1 F ( ξ ) . (9) So we get the equations, a 1 f 2 F ′′ + ( a 0 + a 1 F ) − ( a 0 + a 1 F ) 3 − ( a 0 + a 1 F )( b 0 + b 1 F 2 ) 2 = 0 , ( a 0 + a 1 F ) b 1 f F ′ + 2 a 1 f F ′ ( b 0 + b 1 F 2 ) = − c (1 − k 2 )( a 0 + a 1 F ) + c ( a 0 + a 1 F ) 3 , 2 a 1 s 4 f 2 F 3 + a 1 s 2 f 2 F + a 0 + a 1 F − a 3 0 − 3 a 2 0 a 1 F − 3 a 0 a 2 1 F 2 − a 3 1 F 3 − a 0 b 2 0 − 2 a 0 b 0 b 1 F − a 0 b 2 1 F 2 − a 1 b 2 0 F − 2 a 1 b 0 b 1 F 2 − a 1 b 2 1 F 3 = 0 . (10) By requiring the coe ffic ients of each term F i in the third equati on of Eqs. (10) are zero, we obtain, F 3 : 2 a 1 s 4 f 2 − a 3 1 − a 1 b 2 1 = 0 , F 2 : − 3 a 0 a 2 1 − a 0 b 2 1 − 2 a 1 b 0 b 1 = 0 , F 1 : a 1 s 2 f 2 + a 1 − 3 a 2 0 a 1 − 2 a 0 b 0 b 1 − a 1 b 2 0 = 0 , F 0 : a 0 − a 3 0 − a 0 b 2 0 = 0 . (11) From the above equations we obtain a 0 = b 0 = 0 , f 2 = − 1 /s 2 , a 2 1 + b 2 1 = 2 s 4 f 2 . Then from the first two equations of Eqs. ( 10) we hav e 3 b 1 f F ′ = − c (1 − k 2 ) − a 2 1 q F 2 . (12) 3 Finally , we obtain 4 s 0 s 4 = s 2 2 , s 2 < 0 , a 2 1 = − s 2 (1 − k 2 ) / (2 s 0 ) , b 2 1 = − c 2 (1 − k 2 ) 2 s 2 / (9 s 0 ) . (13) s 4 s 2 s 0 F ′ 2 = s 4 F 4 + s 2 F 2 + s 0 F ( x ) m 2 − (1 + m 2 ) 1 F ′ 2 = (1 − F 2 )(1 − m 2 F 2 ) sn ( x ) − m 2 2 m 2 − 1 1 − m 2 F ′ 2 = (1 − F 2 )(1 + m 2 F 2 − m 2 ) cn ( x ) − 1 2 − m 2 m 2 − 1 F ′ 2 = (1 − F 2 )( F 2 + m 2 − 1) dn ( x ) 1 − (1 + m 2 ) m 2 F ′ 2 = (1 − F 2 )( m 2 − F 2 ) ns ( x ) 1 − m 2 2 m 2 − 1 − m 2 F ′ 2 = (1 − F 2 )[( m 2 − 1) F 2 − m 2 ] nc ( x ) m 2 − 1 2 − m 2 − 1 F ′ 2 = (1 − F 2 )[(1 − m 2 ) F 2 − 1] nd ( x ) − m 2 (1 − m 2 ) 2 − m 2 1 F ′ 2 = (1 + F 2 )[(1 − m 2 ) F 2 + 1] sc ( x ) − m 2 (1 − m 2 ) 2 m 2 − 1 1 F ′ 2 = (1 + m 2 F 2 )[( m 2 − 1) F 2 + 1) sd ( x ) 1 2 − m 2 1 − m 2 F ′ 2 = (1 + F 2 )(1 − m 2 + F 2 ) cs ( x ) 1 2 m 2 − 1 − m 2 (1 − m 2 ) F ′ 2 = ( m 2 + F 2 )( m 2 − 1 + F 2 ) ds ( x ) T ABLE I: Relations between the paramete r ( s 0 , s 2 , s 4 ) and F ( x ) , where F ( x ) is satisfied with ODE F ′ 2 = s 4 F 4 + s 2 F 2 + s 0 . According to the T able I, the only possible J acobi ell iptic functio n which is s atisfied wit h t he Eqs. (13) is sn ( ξ ) wi th m = 1 , i.e. ta nh( ξ ) [2]. So we obtain, a 0 = b 0 = 0 , a 1 = √ 1 − k 2 , b 1 = k , F ( ξ ) = tanh( ξ ) , f = 1 / √ 2 . (14) So the spiral wa ve soluti on for the one-dimensi onal GL Equation is ρ ( x ) = √ 1 − k 2 tanh( x/ √ 2) , ψ ( x ) = k tanh( x/ √ 2) , c = − 3 k / ( √ 2(1 − k 2 )) . (15) 4 So we finally get the one-dimensional GL Equation’ s solution A = √ 1 − k 2 tanh( x/ √ 2) e i [3 k (1 − k 2 ) T / ( √ 2(1 − k 2 ))+ k tanh( x/ √ 2)] . (16) This is the same as Hagan’ s solution [4]. CONCLU SIONS Based on balance mechanism in n onlinear PDEs, th e extended F-expansion method is widely used to obtain singl e and combined non-degenerati ve Jacobi e llipti c function solutions, as well as their corre spondin g de g enerati ve solutions, f or many kinds of PDEs. As example, we gi ve a spiral wa ve sol ution for the one-dimensi onal GL equation in this paper . From thi s example we can see that the extended F-expansion method is a p owe rful tool in the nonlinear PDE fi eld. W e can expect that solutions of the higher dimensional GL Equations also will be obtained in this way . ∗ Electroni c address : chen@physic s.sc.edu [1] I. Aranson, L. Kramer , Rev . Mod. Phys., 74 (2002), 99 [2] J. Liu, K. Y ang, Chaos, Solito ns and F ractals 22 (200 4) 111 [3] J. Zhang, M. W ang, Y . W ang, Z. Fang, Phys. Lett. A 350 (200 6) 103 [4] Hagan, P . S., SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 42 (1982), 762 5