Introduction

A main procedure in mathematical modeling, partial differential equations (PDEs) offer a strong base for involving events that change not just over time but also across different geographical scales. PDEs are used in many different scientific and engineering fields, from modeling fluid flow and quantum phenomena to understanding heat transport and wave behavior1,2,3,4,5. This investigation calls for a combination of analytical and numerical methods, including finite differences and finite elements, variable separation, and Fourier analysis. Because of this, it is now feasible to analyze intricate systems for which analytical solutions are frequently unattainable6,7. So, professionals and academics working in a range of fields need to be well-versed in PDEs theory and its applications. It equips researchers with the knowledge and skills necessary to conduct real-world research and gain a deeper understanding of both artificial and natural systems8,9,10,11,12. The soliton phenomena has been studied frequently in nonlinear integrable PDEs. For instance, various kinds of solitons have been studied in modified nonlinear Shrödinger equation (SE)13,14, perturbed SE15, Chaffee–Infante equation16, bound-state soliton in fiber laser17 and many more18,19.

Stochastic differential equations, or SDEs, offer a solid mathematical framework for modeling dynamic systems impacted by random fluctuations. They are essential tools in fields like economics, physics, biology, and engineering where uncertainty plays a significant role20,21. Unlike traditional differential equations, stochastic processes are a feature of SDEs that make the development of the system more unpredictable22,23. This unpredictability can be caused by a variety of factors, such as ambient noise, shifts in the economy, and the inherent inefficiencies of biological systems. SDEs examine the effects of combining deterministic and random elements on the behavior of the system24,25,26,27. Key components of SDEs are the concepts of the Itô calculus, which provide a solid foundation for handling stochastic integrals and differentials. strong and weak solutions, which describe the behavior of SDEs in diverse scenarios28,29,30. We study the foundations of SDEs, including how they work, how to understand their mathematical features, and real-world applications. This provides the foundation for further investigation into this intriguing and practical branch of mathematics.

The Wiener process, also known as Brownian motion, is a mathematical model that helps us understand and predict variations in a variety of systems. It captures the concept of random fluctuations and helps us understand and predict variations in physics, engineering, finance, and other fields. Its properties, such as being Gaussian and self-similar, make it useful for modeling a wide range of situations. Researching the role of randomness in improving processes requires an understanding of the Wiener process fundamentals31,32.

Robert Chaffee and Ernesto Infante are the names of those who discovered the equation. This equation relates mostly to optical fibers and other wave-controlling devices; it is not the same as the nonlinear Schrödinger equation (NLSE). A mathematical relationship that is commonly found in physics is the Chaffee–Infante (CI) equation, especially when studying nonlinear optics, fluid flow, and wave propagation. It describes how a wave envelope changes in a medium that is nonlinear, often because the refractive index of the material depends on the wave’s strength33,34,35. In (1+1) dimension, the Chaffee–Infante (CI) equation is

$$\begin{aligned} H_{t}+-H_{xx}+\mu ( H^{3}- H) = 0. \end{aligned}$$
(1)

The (2+1) dimensional form of CI equation36 is:

$$\begin{aligned} H_{xt}+(-H_{xx}+P H^{3}-P H)_{x}+\sigma H_{yy} = 0, \end{aligned}$$
(2)

where \(\sigma\) and P are arbitrary constants.

In the study of sensations like soliton propagation in optical fibers, where the balance between diffusion and nonlinearity can result in the formation of solitary waves that can travel great distances without bending, the CI equation is crucial. It also discovers applications in other areas such as fluid dynamics, where related balance between dispersion and nonlinearity can lead to the creation of clear structures like solitons and rogue waves37,38.

The above equation represents the stochastic CI equation.

$$\begin{aligned} H_{xt}+(-H_{xx}+P H^{3}-P H)_{x}+\sigma H_{yy} = {\theta }H_{x}\frac{d W(t) }{dt}, \end{aligned}$$
(3)

where \(W(t)\) represents Wiener process and \(\theta\) gives white noise39.

  • Since it starts at zero, so \(W(0) = 0\).

  • There is statistical independence between the increments \(W(t_2) - W(t_1)\) for \(t_1 < t_2\).

  • For \(t \ge 0\), the function W(t) is continuous.

  • A Gaussian distribution with a mean of 0 and a variance of \(t_2 - t_1\) characterizes the distribution of increments \(W(t_2) - W(t_1)\).

In a mathematical setting, events with sharp and notable fluctuations are abstractly represented by white noise, which is defined as the Wiener process’s time derivative.

In the quest to get analytic solutions for nonlinear partial differential equations (PDEs), several methods have appeared, including the Darboux transformation40, the Sardar-sub-equation method41, sine-cosine method42, the modified Kudryashov technique43, and the generalized Kudryashov technique (KT)44. The motivation of this study was to find new analytic solutions for the CI equation through the modified extended tanh analytical method, within a stochastic environment. For this purpose, by using this approach, this model can provide some insights about complex nonlinear phenomenon as it explains dynamics behind CI model in stochastic. This research aims to enhance mathematical physics by presenting accurate traveling wave solutions for these equations and different them with previous ones. In this respect, the effectiveness of the modified extended tanh method for addressing certain types of nonlinear evolution equations in stochastic situations is demonstrated. This method can be applied to a wide range of nonlinear PDEs, making it a valuable tool for researchers in various fields where complex systems are studied. Especially, this model has not been previously studied using the proposed methods in the stochastic perspective, thus representing a novel contribution to the existing literature.

Modified extended tanh method

In this section, the methodology has been given. For this consider a nonlinear equation

$$\begin{aligned} R(h_,h_{x},h_{y},h_{t},h_{xx},h_{xy},h_{xt},...)= & {} 0, \end{aligned}$$
(4)

Step.1 By using following travelling wave transformation

$$\begin{aligned} h(x,y,t) = H(\varrho )e^{-\frac{\theta ^{2} t}{2}+\theta W(\beta )},\,\,\varrho =x+y-\lambda t, \end{aligned}$$
(5)

Equation (4) reduce to required ordinary differential equation

$$\begin{aligned} F(H, H^{'}, H^{''},H^{'''},...)= & {} 0. \end{aligned}$$
(6)

Step.2 Introduce a new variable namely \(h\,=\,h(\varrho )\), which is a solution of:

$$\begin{aligned} \frac{dh}{d \varrho }= & {} \kappa + h(\varrho )^{2}, \end{aligned}$$
(7)

The modified extended tanh technique admits the proper solution of Eq. (6)

$$\begin{aligned} h(x,y,t)=H(\varrho )= \sum _{j=0}^{m}\gamma _j h^{j}(\varrho )+\sum _{j=1}^{m}\psi _j h^{-j}(\varrho ), \end{aligned}$$
(8)

where m is the balancing number which can be obtained by balancing the highest non-linear term with highest order term in Eq. (6). Step.3 Now putting the values of Eqs. (5), (8) together with Eq. (7) into Eq. (6), and combining all terms of similar order of \(h^{j}(\varrho )\). Then setting all coefficients of \(h^{j}(\varrho )\) equal to zero to obtain a system of algebraic equations for \(\kappa , \lambda , \gamma _0,\gamma _1,...,\gamma _i,\psi _0,\psi _1,...,\) and \(\psi _j\). By solving algebraic equations, we can obtain the values of constants with the help of Mathematica 11.0. The Eq. (6) has general solution:

Case I if \(\kappa <0\) then

$$\begin{aligned} h(\varrho )= & {} - \sqrt{-\kappa }\,tanh( \sqrt{-\kappa }\varrho ),\nonumber \\ h(\varrho )= & {} - \sqrt{-\kappa }\,coth( \sqrt{-\kappa }\varrho ), \end{aligned}$$
(9)

Case II if \(\kappa =0\) then

$$\begin{aligned} h(\varrho )= & {} -\frac{1}{\varrho } \end{aligned}$$
(10)

Case III if \(\kappa >0\) then

$$\begin{aligned} h(\varrho )= & {} \sqrt{\kappa }\,tan( \sqrt{\kappa }\varrho ),\nonumber \\ h(\varrho )= & {} - \sqrt{\kappa }\,cot( \sqrt{\kappa }\varrho ), \end{aligned}$$
(11)

Substituting the values of \(\gamma _{0},\, \gamma _{1},\, \psi _{1},\,\lambda\) into Eq. (8) to obtain the solutions of Eq. (4).

Mathematical investigation of Chaffee–Infante equation

Using travelling wave transformation.

$$\begin{aligned} h(x,y,t)=H(\varrho ) e^{-\frac{\theta ^{2} t}{2}+\theta W(\beta )},\,\,\varrho =x+y-\lambda t \end{aligned}$$
(12)

The Eq. (3) reduce to following ordinary partial differential equation (ODE).

$$\begin{aligned} -\lambda H^{''}-H^{'''}+3 P H^{2} H^{'}-Pk^{'}+\sigma H^{''}= 0, \end{aligned}$$
(13)

under the balancing principle we have \(m=1\), therefore,

$$\begin{aligned} H(\varrho )=\gamma _{0}+\gamma _{1} h(\varrho )+\frac{\psi _{1}}{h(\varrho )}, \end{aligned}$$
(14)

Substitute the value of Eq. (14) with Eq. (7) into Eq. (13), to get algebraic equations:

$$\begin{aligned}{} & {} -2 \gamma _{1} \kappa ^2+2 \kappa \psi _{1}+3 \gamma _{0}^2 \gamma _{1} \kappa P-3 \gamma _{0}^2 P \psi _{1}+3 \gamma _{1}^2 \kappa P \psi 1-\gamma _{1} \kappa P-3 \gamma _{1} P \psi _{1}^2+P \psi _{1} =0,\\{} & {} \quad 6 \kappa ^3 \psi _{1}-3 \kappa P \psi _{1}^3=0,\\{} & {} -2 c \kappa ^2 \psi _{1}+2 \kappa ^2 \sigma \psi _{1}-6 \gamma _{0} \kappa P \psi _{1}^2=0,\\{} & {} \quad 8 \kappa ^2 \psi _{1}-3 \gamma _{0}^2 \kappa P \psi _{1}-3 \gamma _{1} \kappa P \psi _{1}^2+\kappa P \psi _{1}-3 P \psi _{1}^3=0,\\{} & {} -2 c \kappa \psi _{1}+2 \kappa \sigma \psi _{1}-6 \gamma _{0} P \psi _{1}^2=0,\\{} & {} -2 c \gamma _{1} \kappa +2 \gamma _{1} \kappa \sigma +6 \gamma _{0} \gamma _{1}^2 \kappa P=0,\\{} & {} -8 \gamma _{1} \kappa +3 \gamma _{0}^2 \gamma _{1} P+3 \gamma _{1}^3 \kappa P+3 \gamma _{1}^2 P \psi _{1}-\gamma _{1} P=0,\\{} & {} -2 c \gamma _{1}+2 \gamma _{1} \sigma +6 \gamma _{0} \gamma _{1}^2 P=0, \\{} & {} \quad 3 \gamma _{1}^3 P-6 \gamma _{1}=0, \end{aligned}$$

By setting all polynomials equal to zero and solving by Mathematica, we have:

$$\begin{aligned} \gamma _{0}=-\frac{\sqrt{8 \kappa +P}}{\sqrt{3} \sqrt{P}};\,\, \gamma _{1}=\frac{\sqrt{2}}{\sqrt{P}};\,\, \psi _{1}=-\frac{\sqrt{2} \kappa }{\sqrt{P}};\,\,\lambda = -\sqrt{6} \sqrt{8 \kappa +P}+\sigma ; \end{aligned}$$
(15)

Case I if \(\kappa <0\), then substituting the values of Eq. (15) into Eq. (14) with Eq. (9) to get the solution of Eq. (3) under stochastic behaviour, we have

$$\begin{aligned} h_{1}(x,y,t)= & {} \left( -\frac{\sqrt{8 \kappa +P}}{\sqrt{3} \sqrt{P}}+\frac{\sqrt{2} \kappa \coth \left( \sqrt{-\kappa } (x+y-\lambda t)\right) }{\sqrt{-\kappa } \sqrt{P}}-\frac{\sqrt{2} \sqrt{-\kappa } \tanh \left( \sqrt{-\kappa } (x+y-\lambda t)\right) }{\sqrt{P}}\right) e^{-\frac{\theta ^{2}t}{2}+\theta W(\beta )},\nonumber \\ {}{} & {} \end{aligned}$$
(16)
Figure 1
figure 1

This figure represent the outcome of \(h_{1}(x,y,t)\).

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Figure 2
figure 2

This figure represent the outcome of \(h_{1}(x,y,t)\) under stochastic behaviour.

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$$\begin{aligned} h_{2}(x,y,t)= & {} \left( -\frac{\sqrt{8 \kappa +P}}{\sqrt{3} \sqrt{P}}-\frac{\sqrt{2} \sqrt{-\kappa } \coth \left( \sqrt{-\kappa } (x+y-\lambda t)\right) }{\sqrt{P}}+\frac{\sqrt{2} \kappa \tanh \left( \sqrt{-\kappa } (x+y-\lambda t)\right) }{\sqrt{-\kappa } \sqrt{P}}\right) e^{-\frac{\theta ^{2} t}{2}+\theta W(\beta )}, \end{aligned}$$
(17)
Figure 3
figure 3

This figure represent the outcome of \(h_{2}(x,y,t)\).

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Figure 4
figure 4

This figure represent the outcome of \(h_{2}(x,y,t)\) under stochastic behaviour.

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Case II if \(\kappa =0\), then substituting the values of Eq. (15) into Eq. (14) with Eq. (10) to get the solution of Eq.(3) under stochastic behaviour, we have

$$\begin{aligned} h_{3}(x,y,t)= & {} \left( -\frac{\sqrt{8 \kappa +P}}{\sqrt{3} \sqrt{P}}-\frac{\sqrt{2}}{\sqrt{P} (x+y-\lambda t)}-\frac{\sqrt{2} \kappa (-x-y+\lambda t)}{\sqrt{P}}\right) e^{-\frac{\theta ^{2} t}{2}+\theta W(\beta )}, \end{aligned}$$
(18)
Figure 5
figure 5

This figure represent the outcome of \(h_{3}(x,y,t)\).

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Figure 6
figure 6

This figure represent the outcome of \(h_{3}(x,y,t)\) under stochastic behaviour.

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Case III if \(\kappa >0\), then substituting the values of Eq. (15) into Eq. (14) with Eq. (11) to get the solution of Eq. (3) under stochastic behaviour, we have

$$\begin{aligned} h_{4}(x,y,t)= & {} \left( -\frac{\sqrt{8 \kappa +P}}{\sqrt{3} \sqrt{P}}-\frac{\sqrt{2} \sqrt{\kappa } \cot \left( \sqrt{\kappa } (x+y-\lambda t)\right) }{\sqrt{P}}+\frac{\sqrt{2} \sqrt{\kappa } \tan \left( \sqrt{\kappa } (x+y-\lambda t)\right) }{\sqrt{P}}\right) e^{-\frac{\theta ^{2} t}{2}+\theta W(\beta )},\nonumber \\ {}{} & {} \end{aligned}$$
(19)
Figure 7
figure 7

This figure represent the outcome of \(h_{4}(x,y,t)\).

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Figure 8
figure 8

This figure represent the outcome of \(h_{4}(x,y,t)\) under stochastic behaviour.

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$$\begin{aligned} h_{5}(x,y,t)= & {} \left( -\frac{\sqrt{8 \kappa +P}}{\sqrt{3} \sqrt{P}}-\frac{\sqrt{2} \sqrt{\kappa } \cot \left( \sqrt{\kappa } (x+y-\lambda t)\right) }{\sqrt{P}}+\frac{\sqrt{2} \sqrt{\kappa } \tan \left( \sqrt{\kappa } (x+y-\lambda t)\right) }{\sqrt{P}}\right) e^{-\frac{\theta ^{2} t}{2}+\theta W(\beta )}, \end{aligned}$$
(20)
Figure 9
figure 9

This figure represent the outcome of \(h_{2}(x,y,t)\).

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Figure 10
figure 10

This figure represent the outcome of \(h_{2}(x,y,t)\) under stochastic behaviour.

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Results and discussion

The physical aspects of numerous solutions which we have constructed are extensively analyzed in this section. The obtained wave figures in 3D, 2D and contour are highly localized and demonstrate an individual frequency shift under the behavior of sharp peak, periodic wave, and in singular soliton. Singular soliton solutions are important in nonlinear physics, offering stable, self-reinforcing waves that resist dispersion and maintain their shape over long distances. Their significance spans various fields, from optical communications and fiber optics to fluid dynamics and plasma physics, enabling efficient signal transmission and exact control in nonlinear systems. The section describes the new analytic solutions obtained using the Modified Extended Tanh analytical technique and their implications for understanding the behavior of the system. The results are significant because they offer a more comprehensive understanding of the Chaffee–Infante equation in stochastic environments and contribute to the advancement of mathematical physics. We illustrate through these figures that how to solitary waves respond and demonstrate the significant changes to varying level of \(\kappa\). Many physical systems explain the stochastic behavior which is reliable for the implementation of the stochastic technique. In our analysis, the localized wave packets become very important because they indicate the continued existence of isolated wave packets.

In Figs. 1 and 2: we used \(y=1.5, \,\sigma =0.3,\,\lambda =1.4,\,P=1.7,\,\kappa =-0.2,\) for 3D and contour plotes while in 2D the value of \(t=1.\) The range of \(-5\le x \le 5\) and \(-2\le t \le 2\).

In Figs. 3 and 4: we used \(y=0.5, \,\sigma =0.5,\,\lambda =0.4,\,P=0.5,\,\kappa =-0.2,\) for 3D and contour plotes while in 2D the value of \(t=1.\) The range of \(-5\le x \le 5\) and \(-1\le t \le 1\).

In Figs. 5 and 6: we used \(y=0.5, \,\sigma =0.5,\,\lambda =0.4,\,P=0.5,\,\kappa =0,\) for 3D and contour plotes while in 2D the value of \(t=1.\) The range of \(-5\le x \le 5\) and \(-2\le t \le 2\).

In Figs. 7 and 8: we used \(y=1.5, \,\sigma =1,\,\lambda =1.4,\,P=1.7,\,\kappa =1.2,\) for 3D and contour plotes while in 2D the value of \(t=1.\) The range of \(-5\le x \le 5\) and \(-1\le t \le 1\).

In Figs. 9 and 10: we used \(y=0.5, \,\sigma =0.5,\,\lambda =0.4,\,P=0.5,\,\kappa =1.2,\) for 3D and contour plotes while in 2D the value of \(t=1.\) The range of \(-5\le x \le 5\) and \(-1\le t \le 1\).

This research emphasis on the numerous ways in which stochastic aspects interact with the localized wave outcomes, and reveal significant newly acquired knowledge regarding the manner in which overall system interact to raising level of noise around it.

Conclusion

We have finally clarified the soliton solutions of the Chaffee–Infante (CI) equation in the Wiener process base of stochastic differential equations (SDEs). The Modified Extended Tanh Method has allowed us to get precise soliton solutions, which has motivated us to explore the dynamics governed by this fundamental mathematical model. The CI equation has implications in several scientific and technical fields due to its significance in diffusion processes and wave propagation phenomena.

For the purpose of analyzing nonlinear wave phenomena, the Modified Extended Tanh Method has shown to be an effective analytical tool that provides valuable insights into the behavior of soliton solutions. We obtained different types of singular soliton solutions, including bright solitons, dark solitons, periodic solitons and singular solitons, which provide valuable insights into the behavior of the system.Through numerical simulations in both two and three dimensions, we have demonstrated the three-dimensional and time-dependent features of these solutions. A comprehensive understanding of the origins and propagation of soliton solutions is obtained via the study of these solutions in three dimensions in both space and time.

Our results open the way for future research and applications in the domains of nonlinear dynamics and stochastic processes. A wide range of scientific fields, including biology, engineering, physics, and finance, are covered by the potential ideas. Our study combines practical suggestions with theoretical insights, opening up new avenues for understanding and linking the dynamics of complex systems governed by nonlinear equations.