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The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation

The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation 1IntroductionNonlinear partial differential equations (NLPDEs) are applied to describe a wide range of phenomena in biology, fluid mechanics, chemical physics, chemical kinematics, solid-state 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 Exp-function method [1,2], the G′G\left(\frac{{G}^{^{\prime} }}{G}\right)-expansion method [3,4], the tanh–sech method [5,6], the improved tanh-function method [7], the exp(−φ(η))\exp \left(-\varphi \left(\eta ))-expansion method [8], the perturbation method [9,10, 11,12], the extended tanh method [13,14], the sine-cosine method [15,16], the Adomian decomposition method [17,18, 19,20].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 Kuramoto-Sivashinsky (SKS) equation in one dimension with multiplicative noise in the Itô sense as follows: (1)du+[αu∂xu+p∂x2u+q∂x4u]dt=σudβ,{\rm{d}}u+\left[\alpha u{\partial }_{x}u+p{\partial }_{x}^{2}u+q{\partial }_{x}^{4}u]{\rm{d}}t=\sigma u{\rm{d}}\beta ,where α\alpha , pp, and qqare nonzero real constants, σ\sigma is a noise strength, and β(t)\beta \left(t)is the standard Wiener process and it depends only on tt.The Kuramoto-Sivashinsky (KS) equation (1) with σ=0\sigma =0was first proposed in the mid-1970s. Kuramoto was the first to derive the equations for the Belousov-Zabotinskii reaction using reaction-diffusion equations. Also, Sivashinsky used it to describe tiny thermal diffusive instabilities in laminar flamence Poiseuille flow of a film layer on an inclined surface in higher space dimensions. It may also be used to represent Benard convection in an elongated box in one space dimension, and it can be utilized to illustrate long waves at the interface between two viscous fluids and unstable drift waves in plasmas. The KS equation can be applied to control surface roughness in the growth of thin solid films by sputtering, step dynamics in epitaxy, amorphous film formation, and population dynamics models [21,22,23, 24,25].The deterministic Kuramoto-Sivashinsky equation (1) (i.e., σ=0\sigma =0) has been studied by a number of authors to attain its exact solutions by different methods such as the modified tanh–coth method [26], the tanh method and the extended tanh method [27], homotopy analysis method [28], the truncated expansion method [29], the G′G\left(\frac{{G}^{^{\prime} }}{G}\right)-expansion [30], the polynomial expansion method [31,32, 33,34], the perturbation method [35], the Painlevé expansions methods [36]. However, the analytical stochastic solutions of the stochastic Kuramoto-Sivashinsky have never been obtained till this moment.Our motivation of this article is to obtain the analytical stochastic solutions of the SKS (1) with multiplicative noise by using the G′G\left(\frac{{G}^{^{\prime} }}{G}\right)-expansion method. The results introduced here extend earlier studies, for instance, those reported in [27]. Also, we address the effects of multiplicative noise on these solutions.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 G′G\left(\frac{{G}^{^{\prime} }}{G}\right)-expansion method. In Section 4, we show several graphical representations to demonstrate the effect of multiplicative noise on the obtained solutions of SKS. Finally, the conclusions of this paper are shown.2Wave equation for SKS equationTo obtain the wave equation for SKS equation (1), we use the following wave transformation: (2)u(x,t)=φ(η)e(σβ(t)−12σ2t),η=x−ct,u\left(x,t)=\varphi \left(\eta ){e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)},\hspace{1em}\eta =x-ct,where ccis the wave speed and φ\varphi is the deterministic function. Substituting equation (2) into equation (1) and using (3)du=−cφ′+12σ2φ−12σ2φe(σβ(t)−12σ2t)dt+σφe(σβ(t)−12σ2t)dβ,ux=φ′e(σβ(t)−12σ2t),uxx=φ″e(σβ(t)−12σ2t)uxxx=φ‴e(σβ(t)−12σ2t),uxxxx=φ⁗e(σβ(t)−12σ2t),\begin{array}{rcl}{\rm{d}}u& =& \left(-c{\varphi }^{^{\prime} }+\frac{1}{2}{\sigma }^{2}\varphi -\frac{1}{2}{\sigma }^{2}\varphi \right){e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}{\rm{d}}t+\sigma \varphi {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}{\rm{d}}\beta ,\\ {u}_{x}& =& {\varphi }^{^{\prime} }{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)},\hspace{2.07em}{u}_{xx}={\varphi }^{^{\prime\prime} }{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\\ {u}_{xxx}& =& {\varphi }^{\prime\prime\prime }{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)},\hspace{1em}{u}_{xxxx}={\varphi }^{&#x2057;}{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)},\end{array}where +12σ2φ+\frac{1}{2}{\sigma }^{2}\varphi is the Itô correction term, we obtain (4)−cφ′+αφφ′e(σβ(t)−12σ2t)+pφ″+qφ⁗=0.-c{\varphi }^{^{\prime} }+\alpha \varphi {\varphi }^{^{\prime} }{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}+p{\varphi }^{^{\prime\prime} }+q{\varphi }^{&#x2057;}=0.Taking expectation on both sides and considering that φ\varphi is the deterministic function, we have (5)−cφ′+αφφ′e−12σ2tE(eσβ(t))+pφ″+qφ⁗=0,-c{\varphi }^{^{\prime} }+\alpha \varphi {\varphi }^{^{\prime} }{e}^{-\tfrac{1}{2}{\sigma }^{2}t}{\mathbb{E}}\left({e}^{\sigma \beta \left(t)})+p{\varphi }^{^{\prime\prime} }+q{\varphi }^{&#x2057;}=0,Since β(t)\beta \left(t)is the standard Gaussian random variable, then for any real constant γ\gamma , we have E(eγβ(t))=eγ22t{\mathbb{E}}\left({e}^{\gamma \beta \left(t)})={e}^{\tfrac{{\gamma }^{2}}{2}t}. Now equation (5) has the form (6)−cφ′+αφφ′+pφ″+qφ⁗=0,-c{\varphi }^{^{\prime} }+\alpha \varphi {\varphi }^{^{\prime} }+p{\varphi }^{^{\prime\prime} }+q{\varphi }^{&#x2057;}=0,Integrating equation (6) once in terms of η\eta yields (7)qφ‴+pφ′+α2φ2−cφ=0,q{\varphi }^{\prime\prime\prime }+p{\varphi }^{^{\prime} }+\frac{\alpha }{2}{\varphi }^{2}-c\varphi =0,where we put the constant of integration equal zero.3The stochastic exact solutions of SKS equationIn this section, we use the G′G\frac{{G}^{^{\prime} }}{G}-expansion method [3] to find the solutions of equation (7). As a result, we have the exact stochastic solutions of the SKS (1). First, we assume that the solution of equation (7) has the form: (8)φ=∑k=0MℏkG′Gk,\varphi =\mathop{\sum }\limits_{k=0}^{M}{\hslash }_{k}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{k},where ℏ0,ℏ1,…,ℏM{\hslash }_{0},{\hslash }_{1},\ldots ,{\hslash }_{M}are uncertain constants that must be calculated later, and GGsolves (9)G″+λG′+μG=0,{G}^{^{\prime\prime} }+\lambda {G}^{^{\prime} }+\mu G=0,where λ,μ\lambda ,\mu are unknown constants. Let us now calculate the parameter MMby balancing φ2{\varphi }^{2}with φ‴{\varphi }^{\prime\prime\prime }in equation (7) as follows: 2M=M+3,2M=M+3,and hence, (10)M=3.\hspace{-4.5em}M=3.From (10), we can rewrite equation (8) as follows: (11)φ=ℏ0+ℏ1G′G+ℏ2G′G2+ℏ3G′G3.\varphi ={\hslash }_{0}+{\hslash }_{1}\left[\frac{{G}^{^{\prime} }}{G}\right]+{\hslash }_{2}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{2}+{\hslash }_{3}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{3}.Substituting equation (11) into equation (7) and using equation (9) , we obtain a polynomial with degree 6 of G′G\frac{{G}^{^{\prime} }}{G}as follows: 12αℏ32−60qℏ3G′G6+(−24qℏ2+αℏ2ℏ3−144qλℏ3)G′G5+12αℏ22−3pℏ3−6qℏ1+αℏ1ℏ3−111qλ2ℏ3−114qμℏ3−54qλℏ2G′G4+(−cℏ3+2pℏ2+αℏ0ℏ3+αℏ1ℏ2−3pλℏ3−38qλ2ℏ2−40qμℏ2−27λ3ℏ3−12qλℏ1−168qλμℏ3)G′G3\begin{array}{l}\left(\frac{1}{2}\alpha {\hslash }_{3}^{2}-60q{\hslash }_{3}\right){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{6}+\left(-24q{\hslash }_{2}+\alpha {\hslash }_{2}{\hslash }_{3}-144q\lambda {\hslash }_{3}){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{5}\\ \hspace{1.0em}+\left(\frac{1}{2}\alpha {\hslash }_{2}^{2}-3p{\hslash }_{3}-6q{\hslash }_{1}+\alpha {\hslash }_{1}{\hslash }_{3}-111q{\lambda }^{2}{\hslash }_{3}-114q\mu {\hslash }_{3}-54q\lambda {\hslash }_{2}\right){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{4}\\ \hspace{1.0em}+\left(-c{\hslash }_{3}+2p{\hslash }_{2}+\alpha {\hslash }_{0}{\hslash }_{3}+\alpha {\hslash }_{1}{\hslash }_{2}-3p\lambda {\hslash }_{3}-38q{\lambda }^{2}{\hslash }_{2}-40q\mu {\hslash }_{2}-27{\lambda }^{3}{\hslash }_{3}-12q\lambda {\hslash }_{1}-168q\lambda \mu {\hslash }_{3}){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{3}\end{array}+(−cℏ2+12αℏ12−pℏ1+αℏ0ℏ2−2pλℏ2−3pμℏ3−7qλ2ℏ1−8qμℏ1−8qλ3ℏ2−52qλμℏ2−60qμ2ℏ3−57qλ2μℏ3)G′G2+(−cℏ1+αℏ0ℏ1−pλℏ1−2pμℏ2−qλ3ℏ1−16qμ2ℏ2−8qλμℏ1−14qλ2μℏ2−36qμ2λℏ3)G′G+−cℏ0+12αℏ02−pμℏ1−qλ2μℏ1−6qμ2λℏ2−2qμ2ℏ1−6qμ3ℏ3=0.\begin{array}{l}\hspace{1.0em}+\left(-c{\hslash }_{2}+\frac{1}{2}\alpha {\hslash }_{1}^{2}-p{\hslash }_{1}+\alpha {\hslash }_{0}{\hslash }_{2}-2p\lambda {\hslash }_{2}-3p\mu {\hslash }_{3}-7q{\lambda }^{2}{\hslash }_{1}-8q\mu {\hslash }_{1}-8q{\lambda }^{3}{\hslash }_{2}-52q\lambda \mu {\hslash }_{2}-60q{\mu }^{2}{\hslash }_{3}\hspace{1em}-57q{\lambda }^{2}\mu {\hslash }_{3}){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{2}+\left(-c{\hslash }_{1}+\alpha {\hslash }_{0}{\hslash }_{1}-p\lambda {\hslash }_{1}-2p\mu {\hslash }_{2}-q{\lambda }^{3}{\hslash }_{1}\hspace{1em}-16q{\mu }^{2}{\hslash }_{2}-8q\lambda \mu {\hslash }_{1}-14q{\lambda }^{2}\mu {\hslash }_{2}-36q{\mu }^{2}\lambda {\hslash }_{3})\left[\frac{{G}^{^{\prime} }}{G}\right]\\ \hspace{1.0em}+\left(-c{\hslash }_{0}+\frac{1}{2}\alpha {\hslash }_{0}^{2}-p\mu {\hslash }_{1}-q{\lambda }^{2}\mu {\hslash }_{1}-6q{\mu }^{2}\lambda {\hslash }_{2}-2q{\mu }^{2}{\hslash }_{1}-6q{\mu }^{3}{\hslash }_{3}\right)=0.\end{array}Assuming coefficient of G′Gi{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{i}(i=0,1,2,3,4,5,6i=0,1,2,3,4,5,6) to zero, we obtain a system of algebraic equations. Solving this system by using Maple, we obtain two cases:First case: (12)ℏ0=±30p19α−p19q,ℏ1=90p19α,ℏ2=0,ℏ3=120qα,c=±30p19−p19q,λ=0,μ=p76q,ifpq<0.\begin{array}{rcl}{\hslash }_{0}& =& \pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}},\hspace{1em}{\hslash }_{1}=\frac{90p}{19\alpha },\hspace{1em}{\hslash }_{2}=0,\hspace{1em}{\hslash }_{3}=\frac{120q}{\alpha },\\ c& =& \pm \frac{30p}{19}\sqrt{\frac{-p}{19q}},\hspace{1.28em}\lambda =0,\hspace{2.65em}\mu =\frac{p}{76q},\hspace{1em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}\frac{p}{q}\lt 0.\end{array}In this situation, the solution of equation (7) is (13)φ(η)=ℏ0+ℏ1G′G+ℏ3G′G3.\varphi \left(\eta )={\hslash }_{0}+{\hslash }_{1}\left[\frac{{G}^{^{\prime} }}{G}\right]+{\hslash }_{3}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{3}.Solving equation (9) with λ=0,μ=p76q\lambda =0,\mu =\frac{p}{76q}if pq<0\frac{p}{q}\lt 0, we obtain (14)G(η)=c1exp−p76qη+c2exp−−p76qη,G\left(\eta )={c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right),where c1{c}_{1}and c1{c}_{1}are arbitrary constants. Substituting equation (14) into equation (13), we have φ(η)=±30p19α−p19q+90p19α−p76qc1exp−p76qη−c2exp−−p76qηc1exp−p76qη+c2exp−−p76qη+120qα−p76q3c1exp−p76qη−c2exp−−p76qηc1exp−p76qη+c2exp−−p76qη3.\hspace{-40em}\begin{array}{rcl}\varphi \left(\eta )& =& \pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}}+\frac{90p}{19\alpha }\sqrt{\frac{-p}{76q}}\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)-{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right)}{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right)}\right]\\ & & +\frac{120q}{\alpha }{\left(\sqrt{\frac{-p}{76q}}\right)}^{3}{\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)-{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right)}{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right)}\right]}^{3}.\end{array}Hence, the exact stochastic solution in this case of the SKS (1), by using (2), has the following form: (15)u1(x,t)=e(σβ(t)−12σ2t)±30p19α−p19q+90p19α−p76qc1exp−p76q(x−ct)−c2exp−−p76q(x−ct)c1exp−p76q(x−ct)+c2exp−−p76q(x−ct)+120qα−p76q3c1exp−p76q(x−ct)−c2exp−−p76q(x−ct)c1exp−p76q(x−ct)+c2exp−−p76q(x−ct)3,\begin{array}{rcl}{u}_{1}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left\{\pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}}+\frac{90p}{19\alpha }\sqrt{\frac{-p}{76q}}\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)-{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\left(x-ct)\right)}{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\left(x-ct)\right)}\right]\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{-p}{76q}}\right)}^{3}{\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)-{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\left(x-ct)\right)}{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\left(x-ct)\right)}\right]}^{3}\right\},\end{array}where c=±30p19−p19qc=\pm \frac{30p}{19}\sqrt{\frac{-p}{19q}}and pq<0\frac{p}{q}\lt 0.Second case: (16)ℏ0=±30p19α1119q,ℏ1=−270p19α,ℏ2=0,ℏ3=120qα,c=±30p1911p19q,λ=0,μ=−11p76q,ifpq>0.\begin{array}{rcl}{\hslash }_{0}& =& \pm \frac{30p}{19\alpha }\sqrt{\frac{11}{19q}},\hspace{1em}{\hslash }_{1}=\frac{-270p}{19\alpha },\hspace{1em}{\hslash }_{2}=0,\hspace{1em}{\hslash }_{3}=\frac{120q}{\alpha },\\ c& =& \pm \frac{30p}{19}\sqrt{\frac{11p}{19q}},\hspace{1.29em}\lambda =0,\hspace{3.67em}\mu =\frac{-11p}{76q},\hspace{1em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}\frac{p}{q}\gt 0.\end{array}In this situation, the solution of equation (7) is expressed as follows: (17)φ(η)=ℏ0+ℏ1G′G+ℏ3G′G3.\varphi \left(\eta )={\hslash }_{0}+{\hslash }_{1}\left[\frac{{G}^{^{\prime} }}{G}\right]+{\hslash }_{3}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{3}.Solving equation (9) with λ=0\lambda =0, μ=−11p76q\mu =\frac{-11p}{76q}, if pq>0\frac{p}{q}\gt 0, we obtain (18)G(η)=c1exp11p76qη+c2exp−11p76qη.G\left(\eta )={c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right).Substituting equation (14) into equation (13), we have φ(η)=±30p19α11p19q−270p19α11p76qc1exp11p76qη−c2exp−11p76qηc1exp11p76qη+c2exp−11p76qη+120qα11p76q3c1exp11p76qη−c2exp−11p76qηc1exp11p76qη+c2exp−11p76qη3.\begin{array}{rcl}\varphi \left(\eta )& =& \pm \frac{30p}{19\alpha }\sqrt{\frac{11p}{19q}}-\frac{270p}{19\alpha }\sqrt{\frac{11p}{76q}}\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)-{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right)}{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right)}\right]\\ & & +\frac{120q}{\alpha }{\left(\sqrt{\frac{11p}{76q}}\right)}^{3}{\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)-{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right)}{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right)}\right]}^{3}.\end{array}Therefore, by using (2), the exact stochastic solution in this case of the SKS (1) has the following form: (19)u2(x,t)=e(σβ(t)−12σ2t)±30p19α11p19q−270p19α11p76qc1exp11p76q(x−ct)−c2exp−11p76q(x−ct)c1exp11p76q(x−ct)+c2exp−11p76q(x−ct)+120qα11p76q3c1exp11p76q(x−ct)−c2exp−11p76q(x−ct)c1exp11p76q(x−ct)+c2exp−11p76q(x−ct)3,\begin{array}{rcl}{u}_{2}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left\{\pm \frac{30p}{19\alpha }\sqrt{\frac{11p}{19q}}-\frac{270p}{19\alpha }\sqrt{\frac{11p}{76q}}\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)-{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\left(x-ct)\right)}{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\left(x-ct)\right)}\right]\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{11p}{76q}}\right)}^{3}{\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)-{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\left(x-ct)\right)}{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\left(x-ct)\right)}\right]}^{3}\right\},\end{array}where c=±30p1911p19qc=\pm \frac{30p}{19}\sqrt{\frac{11p}{19q}}and pq>0\frac{p}{q}\gt 0.Special cases:Case 1: If we choose c1=c2=1,{c}_{1}={c}_{2}=1,then equations (15) and (19) become (20)u1(x,t)=e(σβ(t)−12σ2t)±30p19α−p19q+90p19α−p76qtanh−p76q(x−ct)+120qα−p76q3tanh3−p76q(x−ct),\begin{array}{rcl}{u}_{1}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left[\pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}}+\frac{90p}{19\alpha }\sqrt{\frac{-p}{76q}}\tanh \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{-p}{76q}}\right)}^{3}{\tanh }^{3}\left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)\right],\end{array}where c=±30p19−p19qc=\pm \frac{30p}{19}\sqrt{\frac{-p}{19q}}and pq<0,\frac{p}{q}\lt 0,and (21)u2(x,t)=e(σβ(t)−12σ2t)±30p19α11p19q−270p19α11p76qtanh11p76q(x−ct)+120qα11p76q3tanh311p76q(x−ct),\begin{array}{rcl}{u}_{2}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left[\pm \frac{30p}{19\alpha }\sqrt{\frac{11p}{19q}}-\frac{270p}{19\alpha }\sqrt{\frac{11p}{76q}}\tanh \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{11p}{76q}}\right)}^{3}{\tanh }^{3}\left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)\right],\end{array}where c=±30p1911p19qc=\pm \frac{30p}{19}\sqrt{\frac{11p}{19q}}and pq>0.\frac{p}{q}\gt 0.Case 2: If we choose c1=1{c}_{1}=1and c2=−1,{c}_{2}=-1,then equations (15) and (19) become (22)u1(x,t)=e(σβ(t)−12σ2t)±30p19α−p19q+90p19α−p76qcoth−p76q(x−ct)+120qα−p76q3coth3−p76q(x−ct),\begin{array}{rcl}{u}_{1}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left\{\pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}}+\frac{90p}{19\alpha }\sqrt{\frac{-p}{76q}}\coth \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{-p}{76q}}\right)}^{3}{\coth }^{3}\left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)\right\},\end{array}where c=±30p19−p19qc=\pm \frac{30p}{19}\sqrt{\frac{-p}{19q}}and pq<0,\frac{p}{q}\lt 0,and (23)u2(x,t)=e(σβ(t)−12σ2t)±30p19α11p19q−270p19α11p76qcoth11p76q(x−ct)+120qα11p76q3coth311p76q(x−ct),\begin{array}{rcl}{u}_{2}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left\{\pm \frac{30p}{19\alpha }\sqrt{\frac{11p}{19q}}-\frac{270p}{19\alpha }\sqrt{\frac{11p}{76q}}\coth \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{11p}{76q}}\right)}^{3}{\coth }^{3}\left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)\right\},\end{array}where c=±30p1911p19qc=\pm \frac{30p}{19}\sqrt{\frac{11p}{19q}}and pq>0\frac{p}{q}\gt 0.Remark 1If we put σ=0\sigma =0(i.e., equation (1) without noise) in equations (20)–(23), then we obtain the same results stated in [27].4The influence of noise on SKS solutionsHere, we discuss the influence of multiplicative noise on the exact solutions of the SKS equation (1). Fix the parameters α=p=q=1\alpha =p=q=1. We present a number of simulations for different values of σ\sigma (noise intensity). We utilize the MATLAB package to simulate our figures as follows:In Figure 1, we can see that there is a kink solution, which indicates that the solution is not planar when σ=0\sigma =0. But in Figure 2, when the noise appears and the intensity of the noise increases, we find that the surface becomes much more planar after small transit patterns. This means that the multiplicative noise affects and stabilizes the solutions.Figure 1Graph of solution u2{u}_{2}in equation (21) with σ=0\sigma =0.Figure 2Graph of solution u2{u}_{2}in equation (21) with σ=0.1,0.3,0.5,1,2,3\sigma =0.1,0.3,0.5,1,2,3.5ConclusionIn this paper, we presented a large variety of exact stochastic solutions of the Kuramoto-Sivashinsky 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation

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de Gruyter
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© 2022 Sahar Albosaily et al., published by De Gruyter
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2391-5455
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10.1515/math-2022-0012
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Abstract

1IntroductionNonlinear partial differential equations (NLPDEs) are applied to describe a wide range of phenomena in biology, fluid mechanics, chemical physics, chemical kinematics, solid-state 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 Exp-function method [1,2], the G′G\left(\frac{{G}^{^{\prime} }}{G}\right)-expansion method [3,4], the tanh–sech method [5,6], the improved tanh-function method [7], the exp(−φ(η))\exp \left(-\varphi \left(\eta ))-expansion method [8], the perturbation method [9,10, 11,12], the extended tanh method [13,14], the sine-cosine method [15,16], the Adomian decomposition method [17,18, 19,20].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 Kuramoto-Sivashinsky (SKS) equation in one dimension with multiplicative noise in the Itô sense as follows: (1)du+[αu∂xu+p∂x2u+q∂x4u]dt=σudβ,{\rm{d}}u+\left[\alpha u{\partial }_{x}u+p{\partial }_{x}^{2}u+q{\partial }_{x}^{4}u]{\rm{d}}t=\sigma u{\rm{d}}\beta ,where α\alpha , pp, and qqare nonzero real constants, σ\sigma is a noise strength, and β(t)\beta \left(t)is the standard Wiener process and it depends only on tt.The Kuramoto-Sivashinsky (KS) equation (1) with σ=0\sigma =0was first proposed in the mid-1970s. Kuramoto was the first to derive the equations for the Belousov-Zabotinskii reaction using reaction-diffusion equations. Also, Sivashinsky used it to describe tiny thermal diffusive instabilities in laminar flamence Poiseuille flow of a film layer on an inclined surface in higher space dimensions. It may also be used to represent Benard convection in an elongated box in one space dimension, and it can be utilized to illustrate long waves at the interface between two viscous fluids and unstable drift waves in plasmas. The KS equation can be applied to control surface roughness in the growth of thin solid films by sputtering, step dynamics in epitaxy, amorphous film formation, and population dynamics models [21,22,23, 24,25].The deterministic Kuramoto-Sivashinsky equation (1) (i.e., σ=0\sigma =0) has been studied by a number of authors to attain its exact solutions by different methods such as the modified tanh–coth method [26], the tanh method and the extended tanh method [27], homotopy analysis method [28], the truncated expansion method [29], the G′G\left(\frac{{G}^{^{\prime} }}{G}\right)-expansion [30], the polynomial expansion method [31,32, 33,34], the perturbation method [35], the Painlevé expansions methods [36]. However, the analytical stochastic solutions of the stochastic Kuramoto-Sivashinsky have never been obtained till this moment.Our motivation of this article is to obtain the analytical stochastic solutions of the SKS (1) with multiplicative noise by using the G′G\left(\frac{{G}^{^{\prime} }}{G}\right)-expansion method. The results introduced here extend earlier studies, for instance, those reported in [27]. Also, we address the effects of multiplicative noise on these solutions.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 G′G\left(\frac{{G}^{^{\prime} }}{G}\right)-expansion method. In Section 4, we show several graphical representations to demonstrate the effect of multiplicative noise on the obtained solutions of SKS. Finally, the conclusions of this paper are shown.2Wave equation for SKS equationTo obtain the wave equation for SKS equation (1), we use the following wave transformation: (2)u(x,t)=φ(η)e(σβ(t)−12σ2t),η=x−ct,u\left(x,t)=\varphi \left(\eta ){e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)},\hspace{1em}\eta =x-ct,where ccis the wave speed and φ\varphi is the deterministic function. Substituting equation (2) into equation (1) and using (3)du=−cφ′+12σ2φ−12σ2φe(σβ(t)−12σ2t)dt+σφe(σβ(t)−12σ2t)dβ,ux=φ′e(σβ(t)−12σ2t),uxx=φ″e(σβ(t)−12σ2t)uxxx=φ‴e(σβ(t)−12σ2t),uxxxx=φ⁗e(σβ(t)−12σ2t),\begin{array}{rcl}{\rm{d}}u& =& \left(-c{\varphi }^{^{\prime} }+\frac{1}{2}{\sigma }^{2}\varphi -\frac{1}{2}{\sigma }^{2}\varphi \right){e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}{\rm{d}}t+\sigma \varphi {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}{\rm{d}}\beta ,\\ {u}_{x}& =& {\varphi }^{^{\prime} }{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)},\hspace{2.07em}{u}_{xx}={\varphi }^{^{\prime\prime} }{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\\ {u}_{xxx}& =& {\varphi }^{\prime\prime\prime }{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)},\hspace{1em}{u}_{xxxx}={\varphi }^{&#x2057;}{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)},\end{array}where +12σ2φ+\frac{1}{2}{\sigma }^{2}\varphi is the Itô correction term, we obtain (4)−cφ′+αφφ′e(σβ(t)−12σ2t)+pφ″+qφ⁗=0.-c{\varphi }^{^{\prime} }+\alpha \varphi {\varphi }^{^{\prime} }{e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}+p{\varphi }^{^{\prime\prime} }+q{\varphi }^{&#x2057;}=0.Taking expectation on both sides and considering that φ\varphi is the deterministic function, we have (5)−cφ′+αφφ′e−12σ2tE(eσβ(t))+pφ″+qφ⁗=0,-c{\varphi }^{^{\prime} }+\alpha \varphi {\varphi }^{^{\prime} }{e}^{-\tfrac{1}{2}{\sigma }^{2}t}{\mathbb{E}}\left({e}^{\sigma \beta \left(t)})+p{\varphi }^{^{\prime\prime} }+q{\varphi }^{&#x2057;}=0,Since β(t)\beta \left(t)is the standard Gaussian random variable, then for any real constant γ\gamma , we have E(eγβ(t))=eγ22t{\mathbb{E}}\left({e}^{\gamma \beta \left(t)})={e}^{\tfrac{{\gamma }^{2}}{2}t}. Now equation (5) has the form (6)−cφ′+αφφ′+pφ″+qφ⁗=0,-c{\varphi }^{^{\prime} }+\alpha \varphi {\varphi }^{^{\prime} }+p{\varphi }^{^{\prime\prime} }+q{\varphi }^{&#x2057;}=0,Integrating equation (6) once in terms of η\eta yields (7)qφ‴+pφ′+α2φ2−cφ=0,q{\varphi }^{\prime\prime\prime }+p{\varphi }^{^{\prime} }+\frac{\alpha }{2}{\varphi }^{2}-c\varphi =0,where we put the constant of integration equal zero.3The stochastic exact solutions of SKS equationIn this section, we use the G′G\frac{{G}^{^{\prime} }}{G}-expansion method [3] to find the solutions of equation (7). As a result, we have the exact stochastic solutions of the SKS (1). First, we assume that the solution of equation (7) has the form: (8)φ=∑k=0MℏkG′Gk,\varphi =\mathop{\sum }\limits_{k=0}^{M}{\hslash }_{k}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{k},where ℏ0,ℏ1,…,ℏM{\hslash }_{0},{\hslash }_{1},\ldots ,{\hslash }_{M}are uncertain constants that must be calculated later, and GGsolves (9)G″+λG′+μG=0,{G}^{^{\prime\prime} }+\lambda {G}^{^{\prime} }+\mu G=0,where λ,μ\lambda ,\mu are unknown constants. Let us now calculate the parameter MMby balancing φ2{\varphi }^{2}with φ‴{\varphi }^{\prime\prime\prime }in equation (7) as follows: 2M=M+3,2M=M+3,and hence, (10)M=3.\hspace{-4.5em}M=3.From (10), we can rewrite equation (8) as follows: (11)φ=ℏ0+ℏ1G′G+ℏ2G′G2+ℏ3G′G3.\varphi ={\hslash }_{0}+{\hslash }_{1}\left[\frac{{G}^{^{\prime} }}{G}\right]+{\hslash }_{2}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{2}+{\hslash }_{3}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{3}.Substituting equation (11) into equation (7) and using equation (9) , we obtain a polynomial with degree 6 of G′G\frac{{G}^{^{\prime} }}{G}as follows: 12αℏ32−60qℏ3G′G6+(−24qℏ2+αℏ2ℏ3−144qλℏ3)G′G5+12αℏ22−3pℏ3−6qℏ1+αℏ1ℏ3−111qλ2ℏ3−114qμℏ3−54qλℏ2G′G4+(−cℏ3+2pℏ2+αℏ0ℏ3+αℏ1ℏ2−3pλℏ3−38qλ2ℏ2−40qμℏ2−27λ3ℏ3−12qλℏ1−168qλμℏ3)G′G3\begin{array}{l}\left(\frac{1}{2}\alpha {\hslash }_{3}^{2}-60q{\hslash }_{3}\right){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{6}+\left(-24q{\hslash }_{2}+\alpha {\hslash }_{2}{\hslash }_{3}-144q\lambda {\hslash }_{3}){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{5}\\ \hspace{1.0em}+\left(\frac{1}{2}\alpha {\hslash }_{2}^{2}-3p{\hslash }_{3}-6q{\hslash }_{1}+\alpha {\hslash }_{1}{\hslash }_{3}-111q{\lambda }^{2}{\hslash }_{3}-114q\mu {\hslash }_{3}-54q\lambda {\hslash }_{2}\right){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{4}\\ \hspace{1.0em}+\left(-c{\hslash }_{3}+2p{\hslash }_{2}+\alpha {\hslash }_{0}{\hslash }_{3}+\alpha {\hslash }_{1}{\hslash }_{2}-3p\lambda {\hslash }_{3}-38q{\lambda }^{2}{\hslash }_{2}-40q\mu {\hslash }_{2}-27{\lambda }^{3}{\hslash }_{3}-12q\lambda {\hslash }_{1}-168q\lambda \mu {\hslash }_{3}){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{3}\end{array}+(−cℏ2+12αℏ12−pℏ1+αℏ0ℏ2−2pλℏ2−3pμℏ3−7qλ2ℏ1−8qμℏ1−8qλ3ℏ2−52qλμℏ2−60qμ2ℏ3−57qλ2μℏ3)G′G2+(−cℏ1+αℏ0ℏ1−pλℏ1−2pμℏ2−qλ3ℏ1−16qμ2ℏ2−8qλμℏ1−14qλ2μℏ2−36qμ2λℏ3)G′G+−cℏ0+12αℏ02−pμℏ1−qλ2μℏ1−6qμ2λℏ2−2qμ2ℏ1−6qμ3ℏ3=0.\begin{array}{l}\hspace{1.0em}+\left(-c{\hslash }_{2}+\frac{1}{2}\alpha {\hslash }_{1}^{2}-p{\hslash }_{1}+\alpha {\hslash }_{0}{\hslash }_{2}-2p\lambda {\hslash }_{2}-3p\mu {\hslash }_{3}-7q{\lambda }^{2}{\hslash }_{1}-8q\mu {\hslash }_{1}-8q{\lambda }^{3}{\hslash }_{2}-52q\lambda \mu {\hslash }_{2}-60q{\mu }^{2}{\hslash }_{3}\hspace{1em}-57q{\lambda }^{2}\mu {\hslash }_{3}){\left[\frac{{G}^{^{\prime} }}{G}\right]}^{2}+\left(-c{\hslash }_{1}+\alpha {\hslash }_{0}{\hslash }_{1}-p\lambda {\hslash }_{1}-2p\mu {\hslash }_{2}-q{\lambda }^{3}{\hslash }_{1}\hspace{1em}-16q{\mu }^{2}{\hslash }_{2}-8q\lambda \mu {\hslash }_{1}-14q{\lambda }^{2}\mu {\hslash }_{2}-36q{\mu }^{2}\lambda {\hslash }_{3})\left[\frac{{G}^{^{\prime} }}{G}\right]\\ \hspace{1.0em}+\left(-c{\hslash }_{0}+\frac{1}{2}\alpha {\hslash }_{0}^{2}-p\mu {\hslash }_{1}-q{\lambda }^{2}\mu {\hslash }_{1}-6q{\mu }^{2}\lambda {\hslash }_{2}-2q{\mu }^{2}{\hslash }_{1}-6q{\mu }^{3}{\hslash }_{3}\right)=0.\end{array}Assuming coefficient of G′Gi{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{i}(i=0,1,2,3,4,5,6i=0,1,2,3,4,5,6) to zero, we obtain a system of algebraic equations. Solving this system by using Maple, we obtain two cases:First case: (12)ℏ0=±30p19α−p19q,ℏ1=90p19α,ℏ2=0,ℏ3=120qα,c=±30p19−p19q,λ=0,μ=p76q,ifpq<0.\begin{array}{rcl}{\hslash }_{0}& =& \pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}},\hspace{1em}{\hslash }_{1}=\frac{90p}{19\alpha },\hspace{1em}{\hslash }_{2}=0,\hspace{1em}{\hslash }_{3}=\frac{120q}{\alpha },\\ c& =& \pm \frac{30p}{19}\sqrt{\frac{-p}{19q}},\hspace{1.28em}\lambda =0,\hspace{2.65em}\mu =\frac{p}{76q},\hspace{1em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}\frac{p}{q}\lt 0.\end{array}In this situation, the solution of equation (7) is (13)φ(η)=ℏ0+ℏ1G′G+ℏ3G′G3.\varphi \left(\eta )={\hslash }_{0}+{\hslash }_{1}\left[\frac{{G}^{^{\prime} }}{G}\right]+{\hslash }_{3}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{3}.Solving equation (9) with λ=0,μ=p76q\lambda =0,\mu =\frac{p}{76q}if pq<0\frac{p}{q}\lt 0, we obtain (14)G(η)=c1exp−p76qη+c2exp−−p76qη,G\left(\eta )={c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right),where c1{c}_{1}and c1{c}_{1}are arbitrary constants. Substituting equation (14) into equation (13), we have φ(η)=±30p19α−p19q+90p19α−p76qc1exp−p76qη−c2exp−−p76qηc1exp−p76qη+c2exp−−p76qη+120qα−p76q3c1exp−p76qη−c2exp−−p76qηc1exp−p76qη+c2exp−−p76qη3.\hspace{-40em}\begin{array}{rcl}\varphi \left(\eta )& =& \pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}}+\frac{90p}{19\alpha }\sqrt{\frac{-p}{76q}}\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)-{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right)}{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right)}\right]\\ & & +\frac{120q}{\alpha }{\left(\sqrt{\frac{-p}{76q}}\right)}^{3}{\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)-{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right)}{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\eta \right)}\right]}^{3}.\end{array}Hence, the exact stochastic solution in this case of the SKS (1), by using (2), has the following form: (15)u1(x,t)=e(σβ(t)−12σ2t)±30p19α−p19q+90p19α−p76qc1exp−p76q(x−ct)−c2exp−−p76q(x−ct)c1exp−p76q(x−ct)+c2exp−−p76q(x−ct)+120qα−p76q3c1exp−p76q(x−ct)−c2exp−−p76q(x−ct)c1exp−p76q(x−ct)+c2exp−−p76q(x−ct)3,\begin{array}{rcl}{u}_{1}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left\{\pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}}+\frac{90p}{19\alpha }\sqrt{\frac{-p}{76q}}\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)-{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\left(x-ct)\right)}{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\left(x-ct)\right)}\right]\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{-p}{76q}}\right)}^{3}{\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)-{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\left(x-ct)\right)}{{c}_{1}\exp \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)+{c}_{2}\exp \left(-\sqrt{\frac{-p}{76q}}\left(x-ct)\right)}\right]}^{3}\right\},\end{array}where c=±30p19−p19qc=\pm \frac{30p}{19}\sqrt{\frac{-p}{19q}}and pq<0\frac{p}{q}\lt 0.Second case: (16)ℏ0=±30p19α1119q,ℏ1=−270p19α,ℏ2=0,ℏ3=120qα,c=±30p1911p19q,λ=0,μ=−11p76q,ifpq>0.\begin{array}{rcl}{\hslash }_{0}& =& \pm \frac{30p}{19\alpha }\sqrt{\frac{11}{19q}},\hspace{1em}{\hslash }_{1}=\frac{-270p}{19\alpha },\hspace{1em}{\hslash }_{2}=0,\hspace{1em}{\hslash }_{3}=\frac{120q}{\alpha },\\ c& =& \pm \frac{30p}{19}\sqrt{\frac{11p}{19q}},\hspace{1.29em}\lambda =0,\hspace{3.67em}\mu =\frac{-11p}{76q},\hspace{1em}\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}\frac{p}{q}\gt 0.\end{array}In this situation, the solution of equation (7) is expressed as follows: (17)φ(η)=ℏ0+ℏ1G′G+ℏ3G′G3.\varphi \left(\eta )={\hslash }_{0}+{\hslash }_{1}\left[\frac{{G}^{^{\prime} }}{G}\right]+{\hslash }_{3}{\left[\frac{{G}^{^{\prime} }}{G}\right]}^{3}.Solving equation (9) with λ=0\lambda =0, μ=−11p76q\mu =\frac{-11p}{76q}, if pq>0\frac{p}{q}\gt 0, we obtain (18)G(η)=c1exp11p76qη+c2exp−11p76qη.G\left(\eta )={c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right).Substituting equation (14) into equation (13), we have φ(η)=±30p19α11p19q−270p19α11p76qc1exp11p76qη−c2exp−11p76qηc1exp11p76qη+c2exp−11p76qη+120qα11p76q3c1exp11p76qη−c2exp−11p76qηc1exp11p76qη+c2exp−11p76qη3.\begin{array}{rcl}\varphi \left(\eta )& =& \pm \frac{30p}{19\alpha }\sqrt{\frac{11p}{19q}}-\frac{270p}{19\alpha }\sqrt{\frac{11p}{76q}}\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)-{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right)}{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right)}\right]\\ & & +\frac{120q}{\alpha }{\left(\sqrt{\frac{11p}{76q}}\right)}^{3}{\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)-{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right)}{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\eta \right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\eta \right)}\right]}^{3}.\end{array}Therefore, by using (2), the exact stochastic solution in this case of the SKS (1) has the following form: (19)u2(x,t)=e(σβ(t)−12σ2t)±30p19α11p19q−270p19α11p76qc1exp11p76q(x−ct)−c2exp−11p76q(x−ct)c1exp11p76q(x−ct)+c2exp−11p76q(x−ct)+120qα11p76q3c1exp11p76q(x−ct)−c2exp−11p76q(x−ct)c1exp11p76q(x−ct)+c2exp−11p76q(x−ct)3,\begin{array}{rcl}{u}_{2}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left\{\pm \frac{30p}{19\alpha }\sqrt{\frac{11p}{19q}}-\frac{270p}{19\alpha }\sqrt{\frac{11p}{76q}}\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)-{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\left(x-ct)\right)}{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\left(x-ct)\right)}\right]\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{11p}{76q}}\right)}^{3}{\left[\frac{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)-{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\left(x-ct)\right)}{{c}_{1}\exp \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)+{c}_{2}\exp \left(-\sqrt{\frac{11p}{76q}}\left(x-ct)\right)}\right]}^{3}\right\},\end{array}where c=±30p1911p19qc=\pm \frac{30p}{19}\sqrt{\frac{11p}{19q}}and pq>0\frac{p}{q}\gt 0.Special cases:Case 1: If we choose c1=c2=1,{c}_{1}={c}_{2}=1,then equations (15) and (19) become (20)u1(x,t)=e(σβ(t)−12σ2t)±30p19α−p19q+90p19α−p76qtanh−p76q(x−ct)+120qα−p76q3tanh3−p76q(x−ct),\begin{array}{rcl}{u}_{1}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left[\pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}}+\frac{90p}{19\alpha }\sqrt{\frac{-p}{76q}}\tanh \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{-p}{76q}}\right)}^{3}{\tanh }^{3}\left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)\right],\end{array}where c=±30p19−p19qc=\pm \frac{30p}{19}\sqrt{\frac{-p}{19q}}and pq<0,\frac{p}{q}\lt 0,and (21)u2(x,t)=e(σβ(t)−12σ2t)±30p19α11p19q−270p19α11p76qtanh11p76q(x−ct)+120qα11p76q3tanh311p76q(x−ct),\begin{array}{rcl}{u}_{2}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left[\pm \frac{30p}{19\alpha }\sqrt{\frac{11p}{19q}}-\frac{270p}{19\alpha }\sqrt{\frac{11p}{76q}}\tanh \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{11p}{76q}}\right)}^{3}{\tanh }^{3}\left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)\right],\end{array}where c=±30p1911p19qc=\pm \frac{30p}{19}\sqrt{\frac{11p}{19q}}and pq>0.\frac{p}{q}\gt 0.Case 2: If we choose c1=1{c}_{1}=1and c2=−1,{c}_{2}=-1,then equations (15) and (19) become (22)u1(x,t)=e(σβ(t)−12σ2t)±30p19α−p19q+90p19α−p76qcoth−p76q(x−ct)+120qα−p76q3coth3−p76q(x−ct),\begin{array}{rcl}{u}_{1}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left\{\pm \frac{30p}{19\alpha }\sqrt{\frac{-p}{19q}}+\frac{90p}{19\alpha }\sqrt{\frac{-p}{76q}}\coth \left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{-p}{76q}}\right)}^{3}{\coth }^{3}\left(\sqrt{\frac{-p}{76q}}\left(x-ct)\right)\right\},\end{array}where c=±30p19−p19qc=\pm \frac{30p}{19}\sqrt{\frac{-p}{19q}}and pq<0,\frac{p}{q}\lt 0,and (23)u2(x,t)=e(σβ(t)−12σ2t)±30p19α11p19q−270p19α11p76qcoth11p76q(x−ct)+120qα11p76q3coth311p76q(x−ct),\begin{array}{rcl}{u}_{2}\left(x,t)& =& {e}^{\left(\sigma \beta \left(t)-\tfrac{1}{2}{\sigma }^{2}t)}\left\{\pm \frac{30p}{19\alpha }\sqrt{\frac{11p}{19q}}-\frac{270p}{19\alpha }\sqrt{\frac{11p}{76q}}\coth \left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)\right.\\ & & \left.+\frac{120q}{\alpha }{\left(\sqrt{\frac{11p}{76q}}\right)}^{3}{\coth }^{3}\left(\sqrt{\frac{11p}{76q}}\left(x-ct)\right)\right\},\end{array}where c=±30p1911p19qc=\pm \frac{30p}{19}\sqrt{\frac{11p}{19q}}and pq>0\frac{p}{q}\gt 0.Remark 1If we put σ=0\sigma =0(i.e., equation (1) without noise) in equations (20)–(23), then we obtain the same results stated in [27].4The influence of noise on SKS solutionsHere, we discuss the influence of multiplicative noise on the exact solutions of the SKS equation (1). Fix the parameters α=p=q=1\alpha =p=q=1. We present a number of simulations for different values of σ\sigma (noise intensity). We utilize the MATLAB package to simulate our figures as follows:In Figure 1, we can see that there is a kink solution, which indicates that the solution is not planar when σ=0\sigma =0. But in Figure 2, when the noise appears and the intensity of the noise increases, we find that the surface becomes much more planar after small transit patterns. This means that the multiplicative noise affects and stabilizes the solutions.Figure 1Graph of solution u2{u}_{2}in equation (21) with σ=0\sigma =0.Figure 2Graph of solution u2{u}_{2}in equation (21) with σ=0.1,0.3,0.5,1,2,3\sigma =0.1,0.3,0.5,1,2,3.5ConclusionIn this paper, we presented a large variety of exact stochastic solutions of the Kuramoto-Sivashinsky 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.

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: stochastic Kuramoto-Sivashinsky; multiplicative noise; stochastic exact solutions; (G′/G)-expansion method; 35Q51; 35A20; 60H10; 60H15; 83C15

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