Stochastic Effects for the Reaction-Duffing Equation with Wick-Type Product

We construct new explicit solutions of theWick-type stochastic reaction-Duffing equation arising frommathematical physics with the help of the white noise theory and the system technique. Based on these exact solutions, we also discuss the influences of stochastic effects for dynamical behaviors according to functions h1(t), h2(t), and Brownian motion B(t) which are the solitary wave group velocities.


Introduction
We consider the following generalized Wick-type stochastic reaction-Duffing equation with the variable coefficients and the white noise environment: where ♢ is the Wick product on the Hida distribution space (R  ) * such as the white noise functional space and  1 () = ℎ 1 () +  1 () and  2 () = ℎ 2 () +  1 () [1,2].Here  1 and  2 are constants, () is the Gaussian white noise, ℎ 1 () and ℎ 2 () are two arbitrary functions, and , , , and  are real constants.The concern of the Wick-type stochastic equations is very common because many authors discussed the Wick-type stochastic equations with variable coefficients and fractional derivatives and so forth [3][4][5][6].
Constructing new explicit solutions of nonlinear evolution equations arising from the field of mathematical physics and engineering is an important topic.However, solving the nonlinear evolution equations is much more difficult than solving the linear ones.Most of physical phenomena coming from the fields of mathematical physics and engineering can be described by the closed-form solutions of nonlinear partial differential equations.So, a great deal of attention has been paid towards both exact and numerical solutions of these equations.Thus we need many efforts for studying mathematical algorithms for determining the exact solutions of nonlinear evolution equations.Recently, there are many effective and powerful methods for obtaining exact solutions to know physical phenomena in mathematical physics and engineering and also we need the study for the development of some techniques, for example, the stochastic linearization method [7], the wavelet-based method [8], the quasi-static method [9], differential transform method [10,11], factorization method [12], extended Jacobi elliptic function expansion method [13,14], tanh-expansion method [15,16], (  /)expansion method [17,18], and Kudryashov method [19][20][21][22].Moreover, through these methods, we know that explicit solutions of nonlinear partial differential equations can reveal the internal mechanism of physical phenomena.As well as physical importance, the closed-form solutions of nonlinear partial differential equations can assist the numerical solvers to compare the correctness of their results and can help them in the stability analysis.
We would like to use the system technique that contains exponential function based on suitable choices of parameters.We show that the system technique has its validity and potential for obtaining new explicit solutions of (1), which contain kink-type motions and soliton-type motions of physical phenomena and other new types of exact solutions.

Descriptions of the Wick-Type Product and Algorithm for Finding Solutions
The space ()  −1 consists of all formal expansions () = ∑      () with   ∈ R  and  = ( 0 , . . .,   ) ∈ ((R)) * .Here ((R)) * is the Hida distribution space on R. The Wick product ♢ of two elements  = ∑      ,  = ∑      ∈ ()  −1 with   ,   ∈ R  is defined by For  = ∑      ∈ ()  −1 , with   ∈ R N , the Hermite transformation of , denoted by F(z), is defined by where z = (z 1 , z 2 , . ..) ∈ C N (the set of all sequences of complex numbers) and , where   ∈ N 0 and z 0  = 1.For ,  ∈ ()  −1 , by this definition, we have for all z such that F(z) and G(z) exist.The product on the right hand side of the above formula is the complex bilinear product between two elements of C  defined by Then the vector  0 = X(0) ∈ R  is called the generalized expectation of  and is denoted by ().Suppose that  :  → C  is an analytic function, where  is a neighborhood of  0 fl ().Assume that the Taylor series of  around  0 have coefficients in R  .Then the Wick version  ⬦ () = H −1 ( ∘ X) is an element of ()  −1 .In other words, if  has the power series expansion The Wick exponent of  ∈ ()  −1 is defined by exp ⬦ {} = ∑ ∞ =0  ⬦ /!.With the Hermite transformation, the Wick exponent shows the same algebraic properties as the usual one.For example, exp ⬦ { + } = exp ⬦ {} ♢ exp ⬦ {}.
We next outline the main processes of the compatibility method for the Wick-type stochastic partial differential equations.Suppose that modeling considerations lead us to consider the stochastic partial differential equation expressed formally as where P is a polynomial, in which the highest order derivatives and nonlinear terms are involved, and  = (x, , ) is the unknown (generalized) stochastic process and where the operators   = /, ∇ x = (/x 1 , . . ., /x  ) when x = (x 1 , . . ., x  ) ∈ R  .With the aid of the Hermite transformation, we transform the Wick-type equation into an ordinary product equation with variable coefficients where Ũ = H(U) is the Hermite transform of  and z 1 , z 2 , . . .are complex numbers.Suppose that we can get exact solutions  = Ũ(, x, z) of (7) for each z ∈ K  (), where K  () = {z = (z 1 , z 2 , . ..) ∈ C N and ∑  ̸ =0 |z  | 2 (2N)  <  2 } for some , .For a start, we would like to introduce the process for obtaining exact solutions of (7) as follows.
Combining the variables x and  into one traveling wave variable , we suppose that traveling wave transformation, permits us to reduce (7) to an ordinary differential equation, for u = u(), where u  = u/, u  =  2 u/ 2 , u  =  3 u/ 3 , . . . .We can get the -order pole solution of (9) that the positive integer  can be determined by considering the homogeneous balance property between the highest order derivatives of linear term and nonlinear term appearing in (9).Suppose that exact solution of ( 9) can be expressed by a polynomial in (F()/G()) as follows: where F() and G() are satisfying the system where  and  are arbitrary constants, and we will call it the system, and   ,  −1 , . . .,  1 ,  0 ,  and  will be determined later with   ̸ = 0. System (11) gives us the following function: where  and  are arbitrary constants with  ̸ =  in Remark 1.By substituting (10) into (9), collecting all terms with the same order of (F()/G()) together, the left-hand side of ( 9) is converted into another polynomial in (F()/G()) using derivatives in Remark 2. Equating each coefficient of this polynomial to zero, we have a set of algebraic equations for   ,  −1 , . . .,  1 ,  0 , , and .
Assuming that the coefficients   ,  −1 , . . .,  1 ,  0 and traveling wave coefficients ,  can be obtained by solving the algebraic equations in the above, since the general solutions of system (11) have been known for us, then by substituting   ,  −1 , . . .,  1 ,  0 ,  and  into (10), we have more new explicit solutions of ( 9) by traveling wave transformation (8).
Finally, we would end up with the explicit expressions for some undetermined coefficients and then we can obtain explicit solutions of ( 7) by substituting them into  = Ũ(, x, z).Moreover, under certain conditions, we take the inverse Hermite transformation  = H −1  ∈ () −1 and thereby can obtain a Wick-type solution  of (6) [23].
Remark 1.The integrability conditions of the system, given by (11), have been extensively discussed in the following literature, and so the relations among the coefficients of the system involve two constraints as follows:  ̸ =  and  = .Firstly, we consider the solution of (11) for  ̸ = .Assume F(0) = 0 and G(0) = 0 without loss of generality.From the first part of ( 11), the solution F() is given by Substituting ( 13) into the second equation of ( 11), we get a differential equation Then, we can find the solution of ( 14) as follows: Combining ( 13) and (15), we obtain the following function: where  and  are arbitrary nonzero constants with  ̸ = .On the other hand, when  = , we have another function: Remark 2. The following derivatives are very useful for equating the expressions at the same degrees of (F/G) to zero in ( 9): 4 , and so on.
To simplify (18), we take traveling wave transformation and H1 (, z) =  1 (, z) and H2 (, z) =  2 (, z).Then we have where u  = u/, u  =  2 u/ 2 , and   = /.Now, let us find exact solutions of (20) by the system technique.It can be easily seen that ( 20) has the first-order pole solution by the balancing property.Suppose that explicit solution of ( 20) can be expressed in the form where (F()/G()) is given by (12) and F() and G() satisfy system (11).By taking expression (21) and using Remark 2, we can obtain the derivatives u  and u  expressed via the function (F()/G()).By substituting u, u  , and u  into (20) and equating to zero the expressions with the same degrees of (F()/G()), we can obtain the set of algebraic equations with respect to the unknowns  1 (, z),  0 (, z), , and (, z) and, by solving the algebraic system by the help of Maple, we obtain the following five explicit solutions of ( 18) with traveling wave transformation (19). 2  be constants to reduce the forms of explicit solutions of (18).The first exact solution of (18) can be written as where The second exact solution of ( 18) can be expressed as where The third exact solution of ( 18) is given by where (, z) The fourth exact solution of ( 18) is where The last exact solution of (18) is where In order to obtain white functional solutions for (1), we use the Hermit transformation and Theorem 4.1.1 in [23].The property of generalized exponential functions yields that there exists a bounded open set G ⊂ R × R + ,  < ∞,  > 0 such that the exact solutions Ũ(x, , z) of ( 18) are uniformly bounded for (x, , z) ∈ G × K  (), continuous with respect to (x, ) ∈ G and analytic with respect to z ∈ K  (), for all (x, ) ∈ G [24].There exists Ψ(x, , z) ∈ () −1 such that Ũ(x, , z) = ( Ũ(, x))(z) for all (, x, z) ∈ G × K  () and (x, ) solves (in the strong sense in () −1 ) (1) in () −1 .Hence, by applying the inverse Hermite transformation to solutions ( 22)-( 26), we can obtain the following Wick-type solutions of (1).Based on solutions ( 22) and ( 23), we get the following Wick-type solutions as follows: where () = (1/( − ) 2 ) ∫  0 {(∓(√ 1 /2)( ± ) − ( − )) 1 () + ( 1 /2) 2 ()}, and where From (24), we obtain the following Wick-type solution of (1): where Solutions ( 25) and ( 26) can convert into the following: where where More precisely, the obtained exact solutions and stochastic explicit solutions have rich structures which can be used to discuss the behaviors of solutions as a function of these arbitrary functions and also to provide enough freedom to build up solutions that may correspond to some particular physical situations.Moreover, it is observed that the obtained white noise functional solutions are of exponential type.It is noted that for different forms of  1 () and  2 (), which are dependent on group velocities and Brownian effect (), we can obtain different stochastic exact solutions of (1) from solutions ( 27)-(31) as in the following example.

Conclusion
In this work, we have obtained new various explicit solutions of the Wick-type stochastic reaction-Duffing equation with the Hermite transformation and the white noise theory.These results can be useful for explaining some physical problems.The system technique has been successfully applied to find more explicit solutions of the Wick-type stochastic reaction-Duffing equation.From the obtained solutions, it is noted that if we take particular values for physical parameters then these solutions provide us with more analysis of new exact solutions than other existing methods.For proper values of parameters and proper functions, we show that Fitzhugh-Nagumo equation and Chafee-Infante equation are similar to the obtained solutions.