Abstract
In this article, we take into account the stochastic KuramotoSivashinsky equation forced by multiplicative noise in the Itô sense. To obtain the exact stochastic solutions of the stochastic KuramotoSivashinsky equation, we apply the
1 Introduction
Nonlinear partial differential equations (NLPDEs) are applied to describe a wide range of phenomena in biology, fluid mechanics, chemical physics, chemical kinematics, solidstate physics, optical fibers, plasma physics, geochemistry, and a lot of other fields. The research of analytical solutions for NLPDEs is important in the investigation of nonlinear physical phenomena. Throughout the past several decades, the discovery of new phenomena has been aided by new exact solutions. Thus, the seeking of exact solutions to those equations of NLPDEs has long been a feature of mathematics and science. To obtain exact solutions of NLPDEs, a variety of effective techniques have been applied, for instance, the Expfunction method [1,2], the
Until the 1950s, deterministic models of differential equations were commonly used to describe the dynamics of the system in implementations. However, it is evident that the phenomena that exist in today’s world are not always deterministic.
Noise has now been shown to be important in many phenomena, also called randomness or fluctuations. Therefore, random effects have become significant when modeling different physical phenomena that take place in oceanography, physics, biology, meteorology, environmental sciences, and so on. Equations that consider random fluctuations in time are referred to as stochastic differential equations.
Here, we treat the stochastic KuramotoSivashinsky (SKS) equation in one dimension with multiplicative noise in the Itô sense as follows:
where
The KuramotoSivashinsky (KS) equation (1) with
The deterministic KuramotoSivashinsky equation (1) (i.e.,
Our motivation of this article is to obtain the analytical stochastic solutions of the SKS (1) with multiplicative noise by using the
The format of this paper is as follows: In Section 2, we obtain the wave equation for SKS equation (1), while in Section 3, we have the exact stochastic solutions of the SKS (1) by applying the
2 Wave equation for SKS equation
To obtain the wave equation for SKS equation (1), we use the following wave transformation:
where
where
Taking expectation on both sides and considering that
Since
Integrating equation (6) once in terms of
where we put the constant of integration equal zero.
3 The stochastic exact solutions of SKS equation
In this section, we use the
where
where
and hence,
From (10), we can rewrite equation (8) as follows:
Substituting equation (11) into equation (7) and using equation (9) , we obtain a polynomial with degree 6 of
Assuming coefficient of
First case:
In this situation, the solution of equation (7) is
Solving equation (9) with
where
Hence, the exact stochastic solution in this case of the SKS (1), by using (2), has the following form:
where
Second case:
In this situation, the solution of equation (7) is expressed as follows:
Solving equation (9) with
Substituting equation (14) into equation (13), we have
Therefore, by using (2), the exact stochastic solution in this case of the SKS (1) has the following form:
where
Special cases:
Case 1: If we choose
where
where
Case 2: If we choose
where
where
4 The influence of noise on SKS solutions
Here, we discuss the influence of multiplicative noise on the exact solutions of the SKS equation (1). Fix the parameters
In Figure 1, we can see that there is a kink solution, which indicates that the solution is not planar when
5 Conclusion
In this paper, we presented a large variety of exact stochastic solutions of the KuramotoSivashinsky equation (1) forced by multiplicative noise. Moreover, several results were extended such as those described in [27]. These types of solutions can be utilized to explain a variety of fascinating and complex physical phenomena. Finally, we used the MATLAB program to generate some graphical representations to show the impact of multiplicative noise on the solutions of the SKS (1). In the future work, we can consider the multiplicative noise with more dimensions or we can take this equation with additive noise.
Acknowledgments
This research has been funded by Scientific Research Deanship at University of Ha’il  Saudi Arabia through project number RG21001.

Funding information: This research has been funded by Scientific Research Deanship at University of Ha’il  Saudi Arabia through project number RG21001.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors declare no coflict of interest.

Data availability statement All data are available in this paper.
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