Abstract
We construct new explicit solutions of the Wick-type stochastic reaction-Duffing equation arising from mathematical 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 , , and Brownian motion which are the solitary wave group velocities.
1. 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 such as the white noise functional space and and [1, 2]. Here and are constants, is the Gaussian white noise, and 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 [3–6].
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 [19–22]. 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.
2. Descriptions of the Wick-Type Product and Algorithm for Finding Solutions
The space consists of all formal expansions with and . Here is the Hida distribution space on . The Wick product of two elements , with is defined by
For , with , the Hermite transformation of , denoted by , is defined by where (the set of all sequences of complex numbers) and for , where and .
For , by this definition, we have for all such that and exist. The product on the right hand side of the above formula is the complex bilinear product between two elements of defined by , where .
Let . Then the vector is called the generalized expectation of and is denoted by . Suppose that is an analytic function, where is a neighborhood of . Assume that the Taylor series of around have coefficients in . Then the Wick version is an element of . In other words, if has the power series expansion with , then .
The Wick exponent of is defined by . With the Hermite transformation, the Wick exponent shows the same algebraic properties as the usual one. For example, .
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 is a polynomial, in which the highest order derivatives and nonlinear terms are involved, and is the unknown (generalized) stochastic process and where the operators , when .
With the aid of the Hermite transformation, we transform the Wick-type equation into an ordinary product equation with variable coefficients where is the Hermite transform of and are complex numbers.
Suppose that we can get exact solutions of (7) for each , where for some , . For a start, we would like to introduce the process for obtaining exact solutions of (7) as follows.
Combining the variables and into one traveling wave variable , we suppose that traveling wave transformation, permits us to reduce (7) to an ordinary differential equation, for , where , ,
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 as follows: where and are satisfying the system where and are arbitrary constants, and we will call it the system, and and will be determined later with .
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 together, the left-hand side of (9) is converted into another polynomial in using derivatives in Remark 2. Equating each coefficient of this polynomial to zero, we have a set of algebraic equations for , and .
Assuming that the coefficients 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 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 . Moreover, under certain conditions, we take the inverse Hermite transformation 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 and without loss of generality. From the first part of (11), the solution 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 to zero in (9): , , , and so on.
3. Wick-Type Explicit Solutions of (1)
Taking the Hermit transformation in (1), we obtain the following equation: where is a vector parameter.
To simplify (18), we take traveling wave transformation and and . Then we have where , , 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 is given by (12) and and satisfy system (11).
By taking expression (21) and using Remark 2, we can obtain the derivatives and expressed via the function . By substituting , , and into (20) and equating to zero the expressions with the same degrees of , we can obtain the set of algebraic equations with respect to the unknowns , , , and 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).
Let and 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 − .
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 , , such that the exact solutions of (18) are uniformly bounded for , continuous with respect to and analytic with respect to , for all [24]. There exists such that for all and solves (in the strong sense in ) (1) in . 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 , andwhere .
From (24), we obtain the following Wick-type solution of (1): where − .
Solutions (25) and (26) can convert into the following: where , andwhere .
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 and , 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.
Example 3. Suppose that and are bounded or integrable functions on and put and , where and are real constants and is Gaussian white noise; that is, . is a Brownian motion. Further, taking the Hermite transformation of and , we have and , where and is a parameter vector and is defined in [25]. In this case, the stochastic type solutions of (18) can express the following for (27)–(31):where − ,where − ,where − ,where + , and where + .
Next, by using for nonrandom and , we can obtain new versions of exact solutions for solutions (27)–(31) as follows:where + , where + , where + ,where − , andwhere − .
Solutions (37)–(41) are obtained by the system technique and these solutions contain various pulse widths and group velocities for explaining the physical phenomena. In particular, we discuss the behaviors of solutions (37) and (40) with various parameter values and arbitrary functions and Brownian motions in Figures 1–4. Solution (37) indicates that pulse width is constant as , which are dependent on arbitrary values of parameters , , , and with some relations and , and group velocity is dependent on arbitrary functions , , and Brownian motion [26, 27].
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In Figure 1, solution (37) shows graphically the behaviors of kink traveling wave solution (37): (a) kink traveling wave solution with white noise term and (b) kink traveling wave solitons-like solution with under two functions , , and proper values of physical parameters , , , , , , , and with and is similar to the Chafee-Infante equation. In particular, the behaviors of (37) with have soft traveling waves as a big real constant is chosen. Also we represent two contours of motions (a) and (b), respectively.
In Figure 2, solution (37) shows graphically the behaviors of soliton solution (37): (a) soliton solution with white noise term and (b) solitons solution with under two functions , and proper values of physical parameters , , , , , , , and with , .
In Figure 3, assuming two functions are and and proper parameters , , , , , , , and with , , we can derive the motions of solution (37) that is like a solution of Fitzhugh-Nagumo equation; Figure 3(a) represents the process of the well-known dark soliton into the flat-bottom dark soliton with white noise term and Figure 3(b) represents the stochastic effect with deep concavity as multiple kinks solution with white noise term .
Solution (40) indicates that pulse width is constant , which are dependent on arbitrary parameter values , , , , and , and group velocity is dependent on arbitrary functions , , and Brownian motion . In Figure 4, assuming two functions are and and proper parameters , , , , , , , and with , , we can derive the soliton-like motions of solution (40): Figure 4(a) represents the symmetric motion of two solitons solutions with white noise term and Figure 4(b) represents the stochastic effect as nonsymmetric solitons solution with white noise term .
4. 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.
Competing Interests
The authors declare that they have no competing interests.