White Noise Functional Solutions for Wick-type Stochastic Fractional KdV-Burgers-Kuramoto Equations

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1 CHINESE JOURNAL OF PHYSICS VOL. 5, NO. August 1 White Noise Functional Solutions for Wick-type Stochastic Fractional KdV-Burgers-Kuramoto Equations Hossam A. Ghany 1,, and M. S. Mohammed 1,3, 1 Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia Department Mathematics, Helwan University, Cairo, Egypt 3 Department Mathematics, Al Azhar University, Cairo, Egypt Received December 15, 11) This paper is devoted to give white noise functional solutions for the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with space-fractional derivatives. Using the homotopy analysis method HAM) that was developed for integer-order differential equations, we can find new approximations to the exact solutions of the fractional KdV-Burgers-Kuramoto equations. Moreover, the Hermit transform and the inverse Hermit transform are employed to find the Wick-type stochastic generalized fractional KdV-Burgers- Kuramoto equations with space-fractional derivatives. PACS numbers:.3.jr, 5.1.-a I. INTRODUCTION This paper is devoted to give white noise functional solutions for the Wick-type stochastic generalized fractional KdV-Burgers-Kuramoto equations with space-fractional derivatives. The generalized fractional KdV-Burgers-Kuramoto equation with spacefractional derivatives is given by u u u u u t +u α x α +at) x +bt) 3 x 3 +ct) =, t >, < α 1, 1) x where at), bt), and ct) are bounded measurable or integrable functions on R +. In the past decades, both mathematicians and physicists have devoted considerable effort to the study of explicit solutions to nonlinear integer-order differential equations. In recent years, important progress has been made in the research of the exact solutions of nonlinear partial differential equations PDEs). To seek various exact solutions of multifarious physical models described by nonlinear PDEs, various methods have been proposed. Recently, many researchers pay more attention to the study of random waves, which are important subjects in stochastic partial differential equations SPDE). Wadati [1] first answered the interesting question of how external noise affects the motion of solitons, and studied the diffusion of a soliton of the KdV equation under Gaussian noise, which satisfies a diffusion equation in transformed coordinates. Wadati and Akutsu also studied the behavior of Electronic address: h.abdelghany@yahoo.com Electronic address: M-S-Mohammed@yahoo.com c 1 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

2 6 WHITE NOISE FUNCTIONAL SOLUTIONS... VOL. 5 solitons under Gaussian white noise of the stochastic KdV equations with and without damping []. In addition, a nonlinear partial differential equation which describes wave propagation in random media was presented by Wadati [3]. Debussche and Printems [, 5], de Bouard and Debussche [6, 7], Konotop and Vazquez [8], Printems [9], and others also researched stochastic KdV-type equations. Xie first introduced Wick-type stochastic KdV equations on white noise space and showed the auto-backlund transformation and the exact white noise functional solutions in [1], furthermore, Chen and Xie [11, 1] and Xie [13 16] researched some Wick-type stochastic wave equations using the white noise analysis method. Eq. 1) plays a significant role in many scientific applications such as solid state physics, nonlinear optics, chemical kinetics, etc. If Eq. 1) is considered in a random environment, we can get random fractional KdV-Burgers-Kuramoto equations with space-fractional derivatives. In order to give the exact solutions of random fractional KdV-Burgers-Kuramoto equations with space-fractional derivatives, we only consider this problem in a white noise environment. Wick-type stochastic generalized fractional KdV- Burgers-Kuramoto equations with space-fractional derivatives are given by U t +U U x α +At) U x +Bt) U x 3 +Ct) U x =, U x β = β U x β, β = α,,3,. ) where at), bt), and ct) are integrable or bounded measurable functions on R +, is the Wick product on the Kondratiev distribution space S) 1 and At), Bt), and Ct) are white noise functionals. Eq. ) can be seen as the perturbation of the coefficients at), bt), and ct) of Eqn. 1) by white noise functionals. Our first interest in this work is implementing new strategies that give white noise functional solutions of the Wick-type two-dimensional stochastic fractional KdV-Burgers- Kuramoto equations. The strategies that will be pursued in this work rest mainly on the homotopy analysis method and Hermite transform, both of which are employed to find white noise functional solutions of Eq. ). The proposed schemes, as we believe, are entirely new and introduce new solutions in addition to the well-known traditional solutions. The ease of using these methods, to determine shock or solitary type of solutions, shows its power. II. WHITE NOISE FUNCTIONAL SOLUTIONS OF EQ. ) Taking the Hermite transform of Eq. ), we get the deterministic equation Ũ t x,t,z)+ũx,t,z)ũxαx,t,z)+ãt,z)ũxx,t,z)+ Bt,z)Ũx 3x,t,z) + Ct,z)Ũx x,t,z) =, 3) where z = z 1,z,...) C N ) c is a vector parameter. For the sake of simplicity we denote At,z) = Ãt,z), Bt,z) = Bt,z), Ct,z) = Ct,z) and ux,t,z) = Ũx,t,z). Using the homotopy analysis method HAM) that was developed for integer-order differential equations, we can find the solution of Eq. 3). Firstly, we will introduce the homotopy

3 VOL. 5 HOSSAM A. GHANY AND M. S. MOHAMMED 61 analysis method: Definition. A real function ht), such that ht) = t p h 1 t), t >, is said to be in the space C µ, µ R if there exists a real number p > µ, where h 1 t) C[, ), and it is said to be in the space C n µ if and only if h n) C µ, n N. Definition. The Riemann-Liouville fractional integral operator J α ) of order α, of a function h C µ, µ 1, is defined as J α ht) = 1 Γα) J ht) = ht), t t τ) α 1 hτ)dτ α > ), ) Γα) is the well- known Gamma function. Some of the properties of the operator J α, which we will need here, are as follows: 1) J α J β ht) = J α+β ht), ) J α J β ht) = J β J α ht), 3) J α t γ = Γγ+1) Γα+γ+1) tα+γ, where β, and γ 1. Definition. The fractional derivative D α ) of ht) in Caputo s sense is defined as t D α 1 ht) = t τ) n α 1 h n) τ)dτ, Γn α) for n 1 < α n, n N, t >, h C 1. n 5) The following are two basic properties of Caputo s fractional Derivative [17]: 1) Let h C n 1, n N. Then Dα h, α n is well defined and D α h C 1. ) Let n 1 < α n, n N and h C n µ, µ 1. Then n 1 J α D α )ht) = ht) h k) + ) tk k!. 6) k= Consider the fractional differential equation in the following general form N ux, t, z)) =, 7) where N is a fractional differential operator, x and t denote independent variables, ux, t, z) is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in the same way. Based on the constructed zero-order deformation equation by Liao [18], we give the following zero-order deformation equation in the similar way, 1 q)lφx, t, z; q) u x, t, z)) = qhn [φx, t, z; q)], 8)

4 6 WHITE NOISE FUNCTIONAL SOLUTIONS... VOL. 5 where q [, 1] is the embedding parameter, h is a non-zero auxiliary parameter, L is an auxiliary linear integer-order operator and it possesses the property LC) =, u x, t, z) is an initial guess of ux, t, z), Ux, t, z; q) is a unknown function of the independent variables x, t, z, q. It is important that one has great freedom to choose the auxiliary parameter h in HAM. If q = and q = 1, the following holds: φx, t, z; ) = u x, t, z), φx, t, z, 1) = ux, t, z), 9) Thus as q increases from to 1, the solution φx, t, z; q) varies from the initial guess u x, t, z) to the solution ux, t, z). Expanding φx, t, z; q) in a Taylor series with respect to q, one has where φx, t, z; q) = u x, t, z) + u m x, t, z)q m, 1) m=1 u m x, t, z) = 1 m φx, t, z; q) m! q m q=. 11) If the auxiliary linear integer-order operator, the initial guess, and the auxiliary parameter h are so properly chosen, the series 1) converges at q = 1, and one has ux, t, z) = u x, t, z) + u m x, t, z). 1) m=1 According to 11), the governing equation can be deduced from the zero-order deformation equation 8). Define the vector ux, t, z) = {u x, t, z), u 1 x, t, z), u x, t, z),..., u n x, t, z)}. 13) Differentiating Eq. 8) m times with respect to the embedding parameter q and then setting q = and finally dividing them by m!, we have the so-called mth-order deformation equation Lu m x, t, z) κ m u m 1 x, t, z)) = where κ m = { m 1, 1 m > 1. h m 1 [φ 1 x, t, z; q),..., φ n x, t, z; q)] m 1)! q m 1 q=, 1) The mth-order deformation equation 1) is linear and thus can be easily solved, especially by means of symbolic computation software such as MATHEMATICA, MAPLE, MATH- LAB, and so on. To demonstrate the effectiveness of the method, we consider Eq. 1) with the following initial condition: ux, ) = k cos x, k C. 16) 15)

5 VOL. 5 HOSSAM A. GHANY AND M. S. MOHAMMED 63 We choose the linear integer-order operator L[Ux, t, z; q)] = Ux, t, z; q). 17) t Furthermore, Eq. 3) suggests we define the nonlinear fractional differential operator N [Ux, t, z; q)] = Ux, t, z; q) t + Ux, t, z; q) α Ux, t, z; q) x α + At) Ux, t, z; q) x +Bt) 3 Ux, t, z; q) x 3 + Ct) Ux, t, z; q) x. 18) Using the above definition, we construct the zeroth-order deformation equation 1 q)lux, t, z; q) u x, t, z)) = qhn [Ux, t, z; q)]. 19) Obviously, when q = and q = 1, Ux, t, z; ) = u x, t, z), Ux, t, z; 1) = ux, t, z). ) According to 1) 15), we gain the mth-order deformation equation { um 1 x, t, z) Lu m x, t, z) κ m u m 1 x, t, z)) = h t m 1 + i= Now, the solution of Eq. 1) for m 1 becomes u m x, t, z) = κ m u m 1 x, t, z) + hl 1 { um 1 x, t, z) t m 1 + i= u i x, t, z) α u m 1 i x, t, z) x α +At) u m 1 x, t, z) x + Bt) 3 u m 1 x, t, z) } x 3 +Ct) u m 1 x, t, z) x. 1) u i x, t, z) α u m 1 i x, t, z) x α + At) u m 1 x, t, z) x +Bt) 3 u m 1 x, t, z) x 3 + Ct) u m 1 x, t, z) x From 16), ), and ), we now successively obtain }. ) u = ux,, z) = k exp{z} cos x, k C

6 6 WHITE NOISE FUNCTIONAL SOLUTIONS... VOL. 5 { u 1 = hl 1 u t + u α } u x α + u At) x + u Bt) 3 x 3 + u Ct) x, { htk u 1 = exp{z} ez cos x + απ ) +hk [γ 1 t) α 1 t)] cos x + hkβ 1 t) sin x + htk ez cos απ ) }, u = u 1 { +hl 1 u1 t + u α u 1 x α + u α } u 1 x α + u 1 At) x + u 1 Bt) 3 x 3 + u 1 Ct) x, { u = exp{z} kλ t)e z Λ1 t) cos3x + απ) + +kλ t)e z cosx + απ) + Λ t)e z sin x + απ ) + 1Λ t) ) +Λ 3 t) + kλ t))e z cos x + Λ 5 t) sin x + Λ 1t) ez cos e z cos απ ) }, x + απ where Λ 1 t) = k h1 + h)t + kγ t) α t)); Λ t) =.5h k t ; Λ t) = k h β t) + 3t ); Λ 3 t) = kh1 + h)γ 1 t) α 1 t))kh β ; Λ 5 t) = kh1 + h)β 1 t) + h kγ t) α t)); α t) = α 1 t)dt = At)dtdt; β t) = β 1 t)dt = Bt)dtdt; γ t) = γ 1 t)dt = Ct)dtdt. ) Special Case γ n t) = n α n t); Bt) = ; α = 1; h = 1 u = k exp{z} cos x, u 1 = tk exp{z} sinx), u = t k 3 exp{3z} cos3x), u 3 = t3 k 8 exp{z} sinx),... u m = ) m t m k m+1 exp{m + 1)z} cos[m + 1]x + 1 [ 1]m+ π). So, the solution of Eq. 3) can be written in the form Ũx, t, z) = m= + exp { m 1 t) m k m+1 exp i {[m + 1]x + 1 [ 1]m+ }) π 3) i {[m + 1]x + 1 })} [ 1]m+ π exp{m + 1)z}. )

7 VOL. 5 HOSSAM A. GHANY AND M. S. MOHAMMED 65 Obviously, the solution given by Eq. ) belongs to the infinite class of solutions for the deterministic Eq. 3). Each solution belonging to this class can be reached by supposing some initial condition for u x, t) for Eq. 1), and then by following the above steps for this initial condition we get another solution for Eq. 3), and so on. In order to get exact solutions of Eq. ), we will give the following condition: A): Suppose that At), Bt), and Ct) satisfy the condition that there exists a bounded open set G R R +, m <, n > such that ux, t, z), u t x, t, z), u x x, t, z), u xx x, t, z), u xxx x, t, z), and u xxxx x, t, z) are uniformly) bounded for x, t, z) G K m n), continuous with respect to x, t) G for all z K m n), and analytic with respect to z K m n), for all x, t) G. Under condition A) Theorem.1 of Xie [13] implies that there exists Ux, t) S) 1 such that ux, t, z) = Ũx, t)z) for all x, t, z) G K mn) and that Ux, t) solves ). From the above, we have that Ux, t) is the inverse Hermite transformation of ux, t, z). Hence, Eq. ) yields stochastic single solitary solutions of Eq. ) in the following form: Ux, t) = m 1 k m+1 t) m exp i {[m + 1]x + 1 }) [ 1]m+ π m= + m 1 k m+1 t) m exp i {[m + 1]x + 1 }) [ 1]m+ π. 5) m= Clearly, applying the ratio test implies that the above summations are convergent. III. EXAMPLE Since Wick versions of functions are usually difficult to evaluate, we will give some non-wick versions of solutions of Eq. ) in a special case: At) = ft) + δ 1 W t), Bt) = gt) + δ W t), 6) Ct) = ht) + δ 3 W t), with ft), gt), and ht) being integrable or bounded measurable functions on R + and δ i i = 1,, 3) being constants, where W t) is Gaussian white noise, i.e., W t) = Ḃt), Bt) is a Brownian motion. We have the Hermite transforms: At, z) = ft)+δ 1 W t, z), Bt, z) = gt) + δ W t, z), and Ct, z) = ht) + δ3 W t, z), where W t, z) = i=1 z t i η is)ds, η i t) is defined in the second section of [1]. In this case, we obtain the solution of Eq. 3) as follows: { ux, t, z) = m 1 t) m k m+1 exp i {[m + 1]x + 1 }) [ 1]m+ π m= + exp i {[m + 1]x + 1 })} [ 1]m+ π exp{ψ m x, t, z)}, 7)

8 66 WHITE NOISE FUNCTIONAL SOLUTIONS... VOL. 5 ψ m x, t, z) = α x + β mz γ σ t [gs) + δ W s, z) ] ds + x + z, where α and σ are arbitrary constants and β m depends only on m. From 7) and the definition of W t, z), it is easy to prove that condition A) is tenable for At), Bt), and Ct) in the case 6). Hence, Eq. 7) yields an exact solution of Eq. ) as follows: Ux, t) = exp {Φx, t)} m 1 k m+1 t) m m= + exp i {[m + 1]x + 1 [ 1]m+ {exp i {[m + 1]x + 1 [ 1]m+ }) π })} π, 8) with { } t Φx, t) = α x γ σ gs)ds + δ 1 Bt) + x. In terms of the equality exp {Bt)} = exp {Bt) t /} [19], from 8), we have Ux, t) = exp{φ 1 x, t)}..5k) m+1 t +.5t ) m m= + exp { exp i {[m + 1]x + 1 [ 1]m+ }) π i {[m + 1]x + 1 })} [ 1]m+ π, 9) with φ 1 x, t) = α x γ σ { t } gs)ds + δ 1 Bt) t /) + x. IV. SUMMARY AND DISCUSSION In general, the solution of SPDE will be a stochastic distribution, and we have to interpret possible products that occur in the equation, as one cannot in general take the product of two distributions, in our paper, products are considered to be Wick products which overcome this difficulty through the white noise functional approach. Subsequently, we take the Hermite transform of the resulting equation and obtain an equation that we try to solve, where the random variables have been replaced by complex-valued functions of infinitely many complex variables. Finally, we use the inverse Hermit transform to obtain a solution of the regularized, original equation []. Since Φ x) = Φx) for any non-random function Φx), hence 17) are solutions of the variable coefficients fractional KdV-Burgers-Kuramoto Equation 1), where at), bt), and ct) are bounded measurable

9 VOL. 5 HOSSAM A. GHANY AND M. S. MOHAMMED 67 or integrable functions on R +. And noting that there exists a unitary mapping between the Wiener white noise space and the Poisson white noise space, we can obtain the solution of the Poisson SPDE simply by applying this mapping to the solution of the corresponding Gaussian SPDE. A nice and concise account of this connection was given by Benth and Gjerde [1]. We can see it in [, Section.9] as well. Hence, we can attain stochastic soliton solutions as we do in Section II if the coefficients At), Bt), and Ct) are Poisson white noise functions in Eq. ). References [1] M. Wadati, J. Phys. Soc. Jpn. 5, ). [] M. Wadati and Y. Akutsu, J. Phys. Soc. Jpn. 53, ). [3] M. Wadati, J. Phys. Soc. Jpn. 59, 1 199). [] A. Debussche and J. Printems, Comput. Anal. Appl. 3, 183 1). [5] A. Debussche and J. Printems, Physica D: Nonlinear Phenomena 13, 1999). [6] A. de Bouard and A. Debussche, J. Funct. Anal. 169, ). [7] A. de Bouard and A. Debussche, J. Funct. Anal. 15, ). [8] VV. Konotop and L. Vzquez, Nonlinear Random Waves World Scientific, 199). [9] J. Printems, J. Differen. Equat. 153, ). [1] Y. C. Xie, Phys. Lett. A 31, 161 3). [11] B. Chen and Y. C. Xie, Chaos, Solitons & Fractals 3, 81 5). [1] B. Chen and Y. C. Xie, J. Phys. A 38, 815 5). [13] Y. C. Xie, Chaos, Solitons & Fractals 1, 73 ). [1] Y. C. Xie, Chaos, Solitons & Fractals, 337). [15] Y. C. Xie, Chaos, Solitons & Fractals 19, 59 ). [16] Y. C. Xie, J. Phys. A: Math. Gen. 37, 59 ). [17] I. Podlubny, Fractional Differential Equations Academic Press, San Diego. 1999). [18] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method Chapman & Hall/CRC Press, Boca Raton, 3). [19] H. Hölden, B. Øsendal, J. Ubøe, T. Zhang, Stochastic partial differential equations Springer Science+Business Media, LLC, 1). [] H. A. Ghany, Chinese J. Phys. 9, 96 11). [1] E. Benth and J. Gjerde, Potential Anal. 8, ).