Solution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods

Transcription

1 Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, HIKARI Ltd, Solution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods F. Fonseca Universidad Nacional de Colombia Grupo de Ciencia de Materiales y Superficies Departamento de Física Bogotá-Colombia Copyright c 2017 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 The Hirota equation is solved using the lattice-boltzmann technique and a d1q3 lattice velocity scheme. Also, we use a solitary wave method, called exponential function, finding several families of solutions. Those nonlinear solutions show the superposition soliton-like behaviour. Keywords: Hirota equation, lattice-boltzmann, Exponential function 1 Introduction In nonlinear dispersive media, Hirota equation (HEq) [1], has become in a very important analytical tool [2]-[3]. On the one hand, lattice Boltzmann (LB) has revealed itself as a very innovative and versatile technique. LB has been applied to fields of research as diverse as quantum theory till complex flows [4]-[5]. Also, LB has proved success in order to find solutions, among others, to KdV equation [6] and Poisson equation [7]. On the other hand, solitary wave methods have been extensively applied successfully to nonlinear differential equations. In particular, we are going to use the exp method [8].

2 308 F. Fonseca We apply the lattice Boltzmann method in order to solve the HEq. In section (2), we present the lattice-boltzmann technique. In section (3), we show the moment relations of the distribution functions in order to deduce the HEq. In section (4), we get the equilibrium distribution functions for the d1q3 lattice velocity scheme. In section (5), we obtain some families of solutions applying the exp function method to the HEq. At last, in section (6), we give results and conclusions. 2 The lattice Boltzmann model The lattice Boltzmann equation is [4]-[5]: f,l (x + v,l t, t + t) f,l (x, t) = Ω,l (x, t) + ω,l (x, t) (1) Here f is the distribution function, v the velocity, x the position, t time, t is the time step, and the subscripts i and accounts for the number of vectors velocity basis and the number of distribution functions, respectively. ω,l (x, t) [9], is known as the source term and the Ω,l (x, t)) is called the B.G.K. approximation [10], which is: Ω,l (x, t)) = 1 ( f,l (x, t) f eq,l τ (x, t)) (2) The equilibrium distribution is given by f eq,l (x, t) and τ is a nondimensional relaxation time. Expanding in a Taylor series, the distribution functions, up to order third, are: ( ) f,l (x + v,l ɛ, t + ɛ) f,l (x, t) = ɛ t + v f,l (3) x ( ) 2 ( ) 3 + ɛ2 2 t + v f,l + ɛ3 x 6 t + v f,l + O(ɛ 4 ) x Doing a perturbative expansion of the equilibrium distribution, in powers of ɛ, we get: And assuming: f,l = f (0),l + ɛf (1),l + ɛ 2 f (2),l + ɛ 3 f (3),l (4) f (0),l Where the temporal scales are defined as: = f (eq),l (5) t 0 = t ; t 1 = ɛt ; t 2 = ɛt 2 ; t 3 = ɛt 3 (6)

3 Solution of the Hirota equation 309 And the perturbative expansion in parameter ɛ of the temporal derivative operator t = + ɛ 1 + ɛ 2 + ɛ 3 (7) t 0 t 1 t 2 t 3 The extra term ω,l is chosen at second order in ɛ 2 to be at the same temporal scale of the diffusive processes [9], therefore: ω,l = ɛ 2 S,l (8) Replacing eqs. (4) and (7) in eq. (3), and expanding perturbatively in orders of the ɛ parameter. We get at first ɛ the next set of equations: At second order in ɛ f 0,l t 0 + v,l x f 0,l = 1 τ f 1,l (9) And the third order in ɛ, f,l 0 τ(1 1 ( ) 2 t 1 τ ) + v,l f,l = 1 t 0 x τ f,l 2 + S,l (10) f,l 0 ( + (1 2τ) + v,l t 2 t 0 x (τ 2 τ + 1 ( ) 3 6 ) + v,l f,l 0 = 1 t 0 x τ f,l 3 (τ) ( t 0 + v,l x 3 The moments of the distributions The moments of the distribution are: f (0),l = φ fl = ) f 0,l + (11) t 1 ) S,l f (eq),l (12) v,l f (0),l = 0 (13) v,m,1 v,n,1 f (0),1 = λ 1 φ f2 δ m,n αλ 2 φ f1 x δ m,n (14) v,m,2 v,n,2 f (0),2 = λ 1 φ f1 δ m,n + αλ 2 φ f1 x δ m,n (15)

4 310 F. Fonseca f (k),l = 0, if k 1 (16) Where φ fl correspond to the microscopic fields of the physical system and δ i is Kronecker s delta. 4 The Hirota Equation Then, summing on in eq. (9), we obtain: f 0,l t 0 + x v,l f 0,l = 1 τ Taking into account eqs. (12)-(13), we have: f,l 1 (17) f 0,l t 0 And now summing on in eq. (10) and multiplying by ɛ = φ f l t 0 = 0 (18) ɛ f 0,l t 1 ɛ 1 τ f (2),l ɛτ(1 1 τ ) + ɛ S,l ( ) 2 + v,l f,l = (19) t 0 x And using the equations (12)-(13) and (18), we have: ɛ φ f l t 1 ɛτ(1 1 τ )( v,m,l v,n,l f,l ) = 1 x m x n τ f (2),l + ɛ S,l (20) Using eq. (16) we have in eq. (20) ɛ φ f l t 1 ɛτ(1 1 τ ) v,m,l v,n,l f,l = ɛ x m x n l Summing eqs. (18) and (21), we obtain: S l, (21) ( + ɛ )φ fl = ɛτ(1 1 t 0 t 1 τ ) v,m,l v,n,l f,l + ɛ x m x n S,l (22) And using eq. (7), at second order, we obtain:

5 Solution of the Hirota equation 311 φ fl t Now using eqs.(14-15) = ɛτ(1 1 2τ ) v,m,l v,n,l f,l + ɛ x m x n S,l (23) φ f1 t = ɛτ(1 1 2τ )λ 2 φ f2 1 ɛτ(1 1 x 2 2τ )αλ 3 φ f1 2 x 3 + ɛ S,1 (24) φ f2 t = λ 1 ɛτ(1 1 φ f1 2τ )2 + αλ x 2 2 ɛτ(1 1 φ f2 2τ )3 x 3 + ɛ S,2 (25) If we define: λ 1 1 = λ 1 2 = ɛ(τ 1 ); m = (b + 1)ɛ (26) 2 Where (b + 1) is the number of vectors in the vector basis. Also, if we define (φ f1, φ f2 ) (φ r, φ c ), their real and imaginary components, ɛ S,1 = ms r, and ɛ S,2 = ms c, the real and complex sources terms, respectively. Then, eqs. (24) and (25) are: φ r t = φ c 2 x φ r 2 α3 x + ms 3 r (27) φ c t = φ r 2 x + φ c 2 α3 x + ms 3 c (28) Defining the real and complex sources terms ms r and ms c as: S r = 1 m ( 2( φ 2 r + φ 2 c)φ c 6α( φ 2 r + φ 2 c) φ r x ) (29) S c = 1 m ( 2( φ 2 r + φ 2 c)φ r + 6α( φ 2 r + φ 2 c) φ c x ) (30) Then, putting it all together in eqs. (27-28), and defining φ = φ r + iφ c, it can be shown that: Which is the Hirota equation [2]. i φ t + 2 φ x φ 2 φ + iα 3 φ φ + 6iα φ 2 x3 x = 0 (31)

6 312 F. Fonseca Figure 1: The lattice velocity scheme D1Q3. Figure 2: Solitary wave solution for φ(x, t), eq. (44), using two, sech(x), initial pulses. 5 The distribution function We use a d1q3, see figure (1), one-dimensional velocity scheme with e α = {0, c, c}, [4], [6]-[7]. Then, the one particle equilibrium distribution function is defined as: f (eq) i = 6 Exp-Function method λ φ m 2c 2 l i = 0 φ λ φ m 2c 2 l i = 1 φ λ φ m 2c 2 l i = 2 (32) Let us consider solitary wave solutions of Hirota equation, eq. (31), and we are going to apply the Exp-function method, presented in reference [8]. It is assumed that the solution can be given as:

7 Solution of the Hirota equation 313 φ(x, t) = Therefore, we suppose a solution like: mi=0 a i exp (iξ) n=0 ; ξ = kx + wt + δ (33) b exp (iξ) φ(ξ) = The temporal and spatial derivatives are: a 1 exp (ξ) ; ξ = kx + wt + δ (34) 1 + b 1 exp (ξ) t = w d dξ ; x = k d dξ ; 2 d2 = k2 x2 dξ ; 3 2 x 3 = k3 d3 dξ 3 (35) Then eq. (31) become: iw dφ dξ + k2 d2 φ dξ φ 2 φ + iαk 3 d3 φ dξ 3 + 6iα φ 2 k dφ dξ = 0 (36) And replacing in eq. (34) in eq. (36) e ξ iwa 1 (1 + b 1 e ξ ) + e ξ (1 b 1 e ξ ) 2 k2 a 1 (1 + b 1 e ξ ) + e 3ξ 3 2a3 1 (37) (1 + b 1 e ξ ) 3 +2iαk 3 e ξ (b 1 e ξ 2) a 1 b 1 (1 + b 1 e ξ ) + e ξ e 2ξ 4 6iα φ 2 ka 1 (1 + b 1 e ξ ) 2 a2 1 (1 + b 1 e ξ ) = 0 2 Then, we obtain iwa 1 + a 1 k 2 4αk 3 a 1 b 1 = 0 (38) 2iwa 1 b 1 + 2αk 3 a 1 b 2 1 = 0 (39) iwa 1 b 2 1 a 1 k 2 b a iαka 2 1 = 0 (40) 2a 3 1b 1 = 0 b 1,1 = 0, a 1,1 0 (41)

8 314 F. Fonseca Replacing eq. (41) in eq. (38) Replacing eq. (41) in eq. (40) iwa 1 + a 1 k 2 = 0 k 2 = iw (42) 2a 1 + 6iαk = 0 a 1,1 = 3iαk (43) On the other hand, from eq. (39), and from eq. (38) b 1,2 = iw k 3 α, Replacing eq. (44) in eq. (40) b 1,3 = iw + k2 4k 3 α (44) a 12,3 = 3α3 ik 7 ± 9α 6 i 2 k 14 2α 2 k 8 w 2 + 2α 2 ik 6 w 3 2α 2 k 6 (45) And now replacing eq. (??) in eq. (40) a 14,5 = 96α3 ik 7 ± 9216α 6 i 2 k α 2 k 6 ( k 6 + ik 4 w k 2 w 2 + iw 3 ) (46) 64α 2 k 6 Therefore, we get five sets of solitary wave solutions for equations (34). φ(ξ) 1 = (a 1,1, b 1,1 ), φ(ξ) 2 = (a 1,2, b 1,2 ), φ(ξ) 3 = (a 1,3, b 1,2 ), (47) 7 Conclusions φ(ξ) 4 = (a 1,4, b 1,3 ), φ(ξ) 5 = (a 1,5, b 1,3 ) Figure (2) shows the temporal evolution of two initial colliding exp pulses, for the φ field, using the lattice-boltzmann technique and presenting the appearing and reappearing process, characteristic in nonlinear solitary wave phenomena. φ(ξ) = a 1 exp (ξ) ; ξ = kx + wt + δ (48) 1 + b 1 exp (ξ) Then, this work presents two new solutions provided by the lattice-boltzmann and the Exp-Function method, applied to the HEq in one dimension. We find five families of solutions using the solitary wave method. Acknowledgements. This research was supported by Universidad Nacional de Colombia in Hermes proect (32501).

9 Solution of the Hirota equation 315 References [1] R.J. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), [2] Wang Qi, Chen Yong, Li Biao and Zhang Hong-Qing, New Exact Travelling Wave Solutions to Hirota Equation and (1+1)-Dimensional Dispersive Long Wave Equation, Commun. Theor. Phys., 41 (2004), [3] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, [4] D. A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Springer, Berlin, [5] S. Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Oxford University Press, [6] G. Yan, J. Zhang, A higher-order moment method of the lattice Boltzmann model for the Korteweg-de Vries equation, Mathematics and Computers in Simulation, 79 (2009), [7] Zhenhua Chai, Baochang Shi, A novel lattice Boltzmann model for the Poisson equation, Applied Mathematical Modelling, 32 (2008), [8] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006), [9] A. A. Mohamad, A. Kuzmin, A critical evaluation of force term in lattice Boltzmann method, natural convection problem, International Journal of Heat and Mass Transfer, 53 (2010), [10] P. L. Bathnagar, E. P. Gross, M. Krook, A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One- Component Systems, Phys. Rev., 94 (1954), Received: May 12, 2017; Published: May 23, 2017