Research Article  Open Access
Hossam A. Ghany, M. Zakarya, "Exact Traveling Wave Solutions for WickType Stochastic Schamel KdV Equation", Physics Research International, vol. 2014, Article ID 937345, 9 pages, 2014. https://doi.org/10.1155/2014/937345
Exact Traveling Wave Solutions for WickType Stochastic Schamel KdV Equation
Abstract
Fexpansion method is proposed to seek exact solutions of nonlinear partial differential equations. By means of Hermite transform, inverse Hermite transform, and white noise analysis, the variable coefficients and Wicktype stochastic Schamel KdV equations are completely described. Abundant exact traveling wave solutions for variable coefficients Schamel KdV equations are given. These solutions include exact stochastic Jacobi elliptic functions, trigonometric functions, and hyperbolic functions solutions.
1. Introduction
In this paper, we investigate the variable coefficients Schamel KdV equations [1, 2]: where , , and are bounded measurable or integrable functions on . Random wave is an important subject of stochastic partial differential equations (SPDEs). Many authors have studied this subject. Wadati first introduced and studied the stochastic KdV equations and gave the diffusion of soliton for KdV equation under Gaussian noise in [3, 4] and others [5â€“9] also researched stochastic KdV equations. Xie first introduced Wicktype stochastic KdV equations on white noise space and showed the autoBacklund transformation and the exact white noise functional solutions in [10]. Furthermore, Xie [11â€“14] and Ghany et al. [15â€“21] researched some Wicktype stochastic wave equations using white noise analysis.
In this paper we use Fexpansion method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solutions for Schamel KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics. The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena, for instance, the nonlinear wave phenomena observed in the fluid dynamics, plasma, and optical fibers [1, 2]. Many effective methods have been presented such as homotopy analysis method [22], variational iteration method [23, 24], tanhfunction method [25â€“27], homotopy perturbation method [28â€“30], tanhcoth method [26, 31, 32], Expfunction method [33â€“38], Jacobi elliptic function expansion method [39â€“42], and Fexpansion method [43â€“46]. The main objective of this paper is using Fexpansion method to construct the exact traveling wave solutions for Wicktype stochastic Schamel KdV equations via the Wicktype product, Hermite transform, and white noise analysis. If (1) is considered in a random environment, we can get stochastic Schamel KdV equations. In order to give the exact solutions of stochastic Schamel KdV equations, we only consider this problem in white noise environment. We will study the following Wicktype stochastic Schamel KdV equations: where â€śâ€ť is the Wick product on the Kondratiev distribution space and , , and are valued functions [47].
2. Description of the FExpansion Method
In order to simultaneously obtain more periodic wave solutions expressed by various Jacobi elliptic functions to nonlinear wave equations, we introduce an Fexpansion method which can be thought of as a succinctly overall generalization of Jacobi elliptic function expansion. We briefly show what Fexpansion method is and how to use it to obtain various periodic wave solutions to nonlinear wave equations. Suppose a nonlinear wave equation for is given by where is an unknown function and is a polynomial in and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of a deformation Fexpansion method.
Step 1. Look for traveling wave solution of (3) by taking Hence, under the transformation in (4), then, (3) can be transformed into ordinary differential equation (ODE) as follows:
Step 2. Suppose that can be expressed by a finite power series of of the form where are constants to be determined later, while in (6) satisfies and hence holds for : where , , and are constants.
Step 3. The positive integer can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in (5). Therefore, we can get the value of in (6).
Step 4. Substituting (6) into (5) with condition (7), we obtain polynomial in â€‰â€‰. Setting each coefficient of this polynomial to be zero yields a set of algebraic equations for and .
Step 5. Solving the algebraic equations with the aid of Maple we have and can be expressed by , , and . Substituting these results into Fexpansion (6), then a general form of traveling wave solution of (3) can be obtained.
Step 6. Since the general solutions of (6) have been well known for us, choose properly , , and in ODE (7) such that the corresponding solution of it is one of Jacobi elliptic functions (see Appendices A, B, and C) [43â€“45].
3. Exact Traveling Wave Solutions of (2)
In this section, we apply Hermite transform, white noise theory, and Fexpansion method to explore soliton and periodic wave solutions for (2). Applying Hermite transform to (2), we get the deterministic equation where is a vector parameter. To look for the traveling wave solution of (3), we make the transformations , , , and , , , with where and are arbitrary constants which satisfy and is a nonzero function of the indicated variables to be determined later. Thus, (3) can be transformed into the following ODE: where . The balancing procedure implies that . Therefore, in view of Fexpansion method the solution of (3) can be expressed in the form where , are constants to be determined later. Substitute (12) with conditions (7) and (8) into (11) and collect all terms with the same power of as follows: Setting each coefficient of to be zero, we get a system of algebraic equations which can be expressed by with solving the above system to get the following coefficients: Substituting coefficient (15) into (12) yields general form solutions to (2): with From Appendix A, we give the special cases as follows.
Case 1. If we take , , and , then ; with In the limit case when , we have ; thus (18) becomes with In the limit case when , we have ; thus (18) becomes with
Case 2. If we take , , and , then and with In the limit case when , we have ; thus (24) becomes with In the limit case when , we have ; thus (24) becomes
Case 3. If we take , and , then and with In the limit case when , we have ; thus (29) becomes In the limit case when , we have ; thus (29) becomes with Remark that there are other solutions for (2). These solutions come from setting different values for the coefficients , , and (see Appendices A, B, and C). The abovementioned cases are just to clarify how far our technique is applicable.
4. White Noise Functional Solutions of (2)
In this section, we employ the results of Section 3 by using Hermite transform to obtain exact white noise functional solutions for Wicktype stochastic Schamel KdV equations (2). The properties of exponential and trigonometric functions yield the fact that there exists a bounded open set ,â€‰â€‰,â€‰â€‰ such that the solution of (9) and all its partial derivatives which are involved in (9) are uniformly bounded for , continuous with respect to for all , and analytic with respect to , for all . From Theorem in [47], there exists such that for all and solves (2) in . Hence, by applying the inverse Hermite transform to the results of Section 3, we get exact white noise functional solutions of (2) as follows.
(i) Exact stochastic Jacobi elliptic functions solutions: with
(ii) Exact stochastic trigonometric solutions: with
(iii) Exact stochastic hyperbolic solutions: with We observe that, for different forms of , , and , we can get different types of exact stochastic functional solutions of (2) from (34)â€“(38).
5. Example
It is well known that Wick version of function is usually difficult to evaluate. So, in this section, we give nonWick version of solutions of (2). Let be the Gaussian white noise, where is the Brownian motion. We have the Hermite transform [47]:
Since Suppose that where , , and are arbitrary constants and is integrable or bounded measurable function on . Therefore, for , exact white noise functional solutions of (2) are as follows: with with
6. Summary and Discussion
We have discussed the solutions of SPDEs driven by Gaussian white noise. There is a unitary mapping between the Gaussian white noise space and the Poisson white noise space. This connection was given by Benth and Gjerde [48]. From [47, section 4.9] and by the aid of the connection, we can derive some stochastic exact soliton solutions, which are Poisson white noise functions in (2). In this paper, using Hermite transformation, white noise theory, and Fexpansion method, we study the white noise functional solutions for Wicktype stochastic Schamel KdV equations. This paper shows that Fexpansion method is sufficient to solve the stochastic nonlinear equations in mathematical physics. The method which we have proposed in this paper is standard, direct, and computerized method, which allows us to do complicated and tedious algebraic calculation. It is shown that the algorithm can be also applied to other nonlinear SPDEs in mathematical physics such as modified HirotaSatsuma coupled KdV, KdVBurgers, modified KdV Burgers, SawadaKotera, and ZhiberShabat equations and BenjaminBonaMahony (BBM) equations. Since (2) has other solutions of Jacobi elliptic functions, trigonometric functions, and hyperbolic functions if we select other values of , , and (see Appendices A, B, and C), there are many other exact traveling wave solutions for Wicktype stochastic Schamel KdV equations.
Appendices
A.
The Jacobi elliptic functions degenerate into trigonometric functions when :
B.
The Jacobi elliptic functions degenerate into hyperbolic functions when :
C.
The ODE and Jacobi elliptic functions: for relation between values of , , and and corresponding in ODE, see Table 1.

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
References
 A. H. Khater, M. M. Hassan, and R. S. Temsah, â€śExact solutions with Jacobi elliptic functions of two nonlinear models for ionacoustic plasma waves,â€ť Journal of the Physical Society of Japan, vol. 74, no. 5, pp. 1431â€“1435, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. Schamel, â€śA modified Korteweg de Vries equation for ion acoustic waves due to resonant electrons,â€ť Journal of Plasma Physics, vol. 9, pp. 377â€“387, 1973. View at: Google Scholar
 M. Wadati, â€śStochastic Kortewegde Vries equation,â€ť Journal of the Physical Society of Japan, vol. 52, no. 8, pp. 2642â€“2648, 1983. View at: Publisher Site  Google Scholar  MathSciNet
 M. Wadati and Y. Akutsu, â€śStochastic Kortewegde Vries equation with and without damping,â€ť Journal of the Physical Society of Japan, vol. 53, no. 10, pp. 3342â€“3350, 1984. View at: Publisher Site  Google Scholar  MathSciNet
 A. de Bouard and A. Debussche, â€śOn the stochastic Kortewegde Vries equation,â€ť Journal of Functional Analysis, vol. 154, no. 1, pp. 215â€“251, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. de Bouard, A. Debussche, and Y. Tsutsumi, â€śWhite noise driven Kortewegde Vries equation,â€ť Journal of Functional Analysis, vol. 169, no. 2, pp. 532â€“558, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 A. Debussche and J. Printems, â€śNumerical simulation of the stochastic Kortewegde Vries equation,â€ť Physica D, vol. 134, no. 2, pp. 200â€“226, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 A. Debussche and J. Printems, â€śEffect of a localized random forcing term on the Kortewegde Vries equation,â€ť Journal of Computational Analysis and Applications, vol. 3, no. 3, pp. 183â€“206, 2001. View at: Publisher Site  Google Scholar  MathSciNet
 V. V. Konotop and L. Vazquez, Nonlinear Random Waves, World Scientific Publishing, Singapore, 1994.
 Y. Xie, â€śExact solutions for stochastic KdV equations,â€ť Physics Letters A, vol. 310, no. 23, pp. 161â€“167, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 Y. C. Xie, â€śExact solutions of the Wicktype stochastic KadomtsevPetviashvili equations,â€ť Chaos, Solitons and Fractals, vol. 21, no. 2, pp. 473â€“480, 2004. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Xie, â€śPositonic solutions for Wicktype stochastic KdV equations,â€ť Chaos, Solitons and Fractals, vol. 20, no. 2, pp. 337â€“342, 2004. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Xie, â€śAn autoBäcklund transformation and exact solutions for Wicktype stochastic generalized KdV equations,â€ť Journal of Physics A: Mathematical and General, vol. 37, no. 19, pp. 5229â€“5236, 2004. View at: Publisher Site  Google Scholar  MathSciNet
 Y. C. Xie, â€śExact solutions for stochastic KdV equations,â€ť Physics Letters A, vol. 310, no. 23, pp. 161â€“167, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 H. A. Ghany, â€śExact solutions for stochastic generalized HirotaSatsuma coupled KdV equations,â€ť Chinese Journal of Physics, vol. 49, no. 4, pp. 926â€“940, 2011. View at: Google Scholar
 H. A. Ghany and A. Fathallah, â€śExact solutions for KDVburger equations with an application of whitenoise analysis,â€ť International Journal of Pure and Applied Mathematics, vol. 78, no. 1, pp. 17â€“27, 2012. View at: Google Scholar  Zentralblatt MATH
 H. A. Ghany and A.A. Hyder, â€śWhite noise functional solutions for the Wicktype twodimensional stochastic ZakharovKuznetsov equations,â€ť International Review of Physics, vol. 6, no. 2, pp. 153â€“157, 2012. View at: Google Scholar
 H. A. Ghany and A.A. Hyder, â€śExact solutions for the wicktype stochastic timefractional KdV equations,â€ť Kuwait Journal of Science, vol. 41, no. 1, pp. 75â€“84, 2014. View at: Google Scholar  MathSciNet
 H. A. Ghany and A. Hyder, â€śAbundant solutions of Wicktype stochastic fractional 2D KdV equations,â€ť Chinese Physics B, vol. 23, no. 6, Article ID 060503, 2014. View at: Publisher Site  Google Scholar
 H. A. Ghany and M. S. Mohammed, â€śWhite noise functional solutions for Wicktype stochastic fractional KdVBurgersKuramoto equations,â€ť Chinese Journal of Physics, vol. 50, no. 4, pp. 619â€“627, 2012. View at: Google Scholar  MathSciNet
 H. A. Ghany, A. S. Okb El Bab, A. M. Zabel, and A.A. Hyder, â€śThe fractional coupled KdV equations: exact solutions and white noise functional approach,â€ť Chinese Physics B, vol. 22, no. 8, Article ID 080501, 2013. View at: Publisher Site  Google Scholar
 S. Abbasbandy, â€śApproximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method,â€ť Chemical Engineering Journal, vol. 136, no. 23, pp. 144â€“150, 2008. View at: Publisher Site  Google Scholar
 M. Dehghan, J. M. Heris, and A. Saadatmandi, â€śApplication of semianalytic methods for the FitzhughNagumo equation, which models the transmission of nerve impulses,â€ť Mathematical Methods in the Applied Sciences, vol. 33, no. 11, pp. 1384â€“1398, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. Dehghan and M. Tatari, â€śIdentifying an unknown function in a parabolic equation with overspecified data via He's variational iteration method,â€ť Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 157â€“166, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 E. Fan, â€śExtended tanhfunction method and its applications to nonlinear equations,â€ť Physics Letters A, vol. 277, no. 45, pp. 212â€“218, 2000. View at: Publisher Site  Google Scholar  MathSciNet
 A. M. Wazwaz, â€śThe tanh method for generalized forms of nonlinear heat conduction and BurgersFisher equations,â€ť Applied Mathematics and Computation, vol. 169, no. 1, pp. 321â€“338, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 S. Zhang and T.C. Xia, â€śSymbolic computation and new families of exact nontravelling wave solutions of (2 + 1)dimensional BroerKaup equations,â€ť Communications in Theoretical Physics, vol. 45, no. 6, pp. 985â€“990, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 D. D. Ganji and A. Sadighi, â€śApplication of He's homotopyperturbation method to nonlinear coupled systems of reactiondiffusion equations,â€ť International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 4, pp. 411â€“418, 2006. View at: Google Scholar
 F. Shakeri and M. Dehghan, â€śSolution of delay differential equations via a homotopy perturbation method,â€ť Mathematical and Computer Modelling, vol. 48, no. 34, pp. 486â€“498, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. Ma, L. Wei, and Z. Guo, â€śHe's homotopy perturbation method to periodic solutions of nonlinear Jerk equations,â€ť Journal of Sound and Vibration, vol. 314, no. 12, pp. 217â€“227, 2008. View at: Publisher Site  Google Scholar
 A.M. Wazwaz, â€śThe tanhcoth method for new compactons and solitons solutions for the K($n;n$) and the K($n+1$, $n+1$) equations,â€ť Applied Mathematics and Computation, vol. 188, no. 2, pp. 1930â€“1940, 2007. View at: Publisher Site  Google Scholar
 A.M. Wazwaz, â€śTravelling wave solutions for combined and double combined sinecosineGordon equations by the variable separated ODE method,â€ť Applied Mathematics and Computation, vol. 177, no. 2, pp. 755â€“760, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 H. A. Ghany and M. Zakarya, â€śGeneralized solutions of Wicktype stochastic KdVburgers equations using expfunction method,â€ť International Review of Physics, vol. 8, no. 2, 2014. View at: Google Scholar
 J.H. He and X.H. Wu, â€śExpfunction method for nonlinear wave equations,â€ť Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700â€“708, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 J.H. He and M. A. Abdou, â€śNew periodic solutions for nonlinear evolution equations using Expfunction method,â€ť Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1421â€“1429, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 X.H. Wu and J.H. He, â€śEXPfunction method and its application to nonlinear equations,â€ť Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 903â€“910, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 S.D. Zhu, â€śExpfunction method for the HybridLattice system,â€ť International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 461â€“464, 2007. View at: Google Scholar
 S.D. Zhu, â€śExpfunction method for the discrete mKdV lattice,â€ť International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 465â€“468, 2007. View at: Google Scholar
 Z. Fu, S. Liu, and Q. Zhao, â€śNew Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations,â€ť Physics Letters A, vol. 290, no. 12, pp. 72â€“76, 2001. View at: Publisher Site  Google Scholar  MathSciNet
 S. K. Liu, Z. T. Fu, S. D. Liu, and Q. Zhao, â€śJacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,â€ť Physics Letters A, vol. 289, no. 12, pp. 69â€“74, 2001. View at: Publisher Site  Google Scholar  MathSciNet
 J. Liu, L. Yang, and K. Yang, â€śNonlinear transform and Jacobi elliptic function solutions of nonlinear equations,â€ť Chaos, Solitons and Fractals, vol. 20, no. 5, pp. 1157â€“1164, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. J. Parkes, B. R. Duffy, and P. C. Abbott, â€śThe Jacobi ellipticfunction method for finding periodicwave solutions to nonlinear evolution equations,â€ť Physics Letters A, vol. 295, no. 56, pp. 280â€“286, 2002. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Zhou, M. Wang, and Y. Wang, â€śPeriodic wave solutions to a coupled KdV equations with variable coefficients,â€ť Physics Letters A, vol. 308, no. 1, pp. 31â€“36, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 S. Zhang and T. Xia, â€śA generalized $F$expansion method with symbolic computation exactly solving BroerKaup equations,â€ť Applied Mathematics and Computation, vol. 189, no. 1, pp. 836â€“843, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 S. Zhang and T. Xia, â€śA generalized Fexpansion method and new exact solutions of KonopelchenkoDubrovsky equations,â€ť Applied Mathematics and Computation, vol. 183, no. 2, pp. 1190â€“1200, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 S. Zhang and T. Xia, â€śAn improved generalized Fexpansion method and its application to the $(2+1)$dimensional KdV equations,â€ť Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 7, pp. 1294â€“1301, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 H. Holden, B. Øksendal, J. Ubøe, and T. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Basel, Switzerland, 1996.
 F. E. Benth and J. Gjerde, â€śA remark on the equivalence between Poisson and Gaussian stochastic partial differential equations,â€ť Potential Analysis, vol. 8, no. 2, pp. 179â€“193, 1998. View at: Publisher Site  Google Scholar  MathSciNet
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Copyright © 2014 Hossam A. Ghany and M. Zakarya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.