Investigation of closed form solitons for the stochastic Chavy-Waddy-Kolokolnikov equation in bacterial aggregation

Abstract

In this study, we will investigates the stochastic Chavy-Waddy-Kolokolnikov equation in the Stratonovich sense analytically. This model is applicable which is highly useful for simulating the collective development of bacteria attracted to light under the noise environment. New closed form solitary wave structures are achieved in different shapes like elliptic, hyperbolic, trigonometric, and rational stochastic solutions are obtained by applying the $\phi ^6$-model expansion approach. This approach is gives us the jaccobi elliptic function solutions. These jaccobi elliptic function are provided us the solitons and solitary wave solutions under the effects of noise. The dynamic performances of the various derived solutions are presented using 3-D and 2-D graphs to help explain the effects of multiplicative noise. We deduce that multiplicative noise affects and modifies the behavior of solutions for stochastic Chavy-Waddy-Kolokolnikov equation.

Introduction

Partial differential equations that are nonlinear have a wide range of applications in solving various physical phenomena. Numerous complex dynamical systems are modeled using it. Understanding the dynamics of these models is greatly aided by mathematical modeling, which is applied in many academic disciplines, including the social sciences, oceanology, and engineering^1,2. For instance, by taking into account the chemotactic population’s density, which directly affects its sensitive reaction to chemicals, the Keller-Segel model was expanded in the research. Presented to explain chemotaxis phenomena in the fluid environment was the Keller-Segel-Navier-Stokes model^3,4,5. The authors achieved a satisfactory agreement with experimental results by presenting quasi-random motion in papers to explain the formation of aggregates for low to medium bacteria density. They mimicked a collective movement that followed certain patterns but was constantly changing and didn’t appear to be biased toward light. A simple model based on stochastic agents with many aggregation types was introduced^5,6.

In this study, we discuss the Chavy-Waddy-Kolokolnikov equation, playing role in bacterial aggregation by using Stochastic approach. In order to adjust and to endure in their surroundings, bacteria have developed a number of adaptation strategies. A significant issue in biological research is the motion of bacteria that yields to various operators such as heat, light, etc^7,8. One of the well-known models in the field of biology is the Chavy-Waddy-Kokolnikov equation (CWKE), which is highly useful for simulating the collective development of bacteria attracted to light. For example, they may migrate to areas with more nutrient concentrations or superior environmental circumstances. Common to all bacteria is the ability to recognize and react to chemical gradients via chemotaxing, or migrating toward an area where a certain substance is more concentrated. Moreover, photosynthetic, motile creatures may migrate toward light through a process called phototaxis. In evolutionary and ecological processes, both phototaxis and chemotaxis are significant. In biology and mathematics, chemotaxis has been thoroughly examined; among the earliest and best-known models is the Keller-Segel equation^9,10.

In a model of cyanobacteria motion by Galante, Wisen, Bhaya, and Levy, Chavy-Waddy and Kolokolnikov explore the formation of pattern. Their model’s continuum limit allows them to build a unique fourth-order nonlinear parabolic PDE equation that controls the model’s behavior¹¹. Ramírez et al. provided the analytical solutions to the Chavy-Waddy–Kokolnikov equation, which is a continuum approximation used to explain phototaxis–the process by which bacteria move toward light and form aggregates. They found traveling wave-like solutions using three extremely effective methods, the $e^{-R(\xi )}$-expansion, the exponential function technique, and the modified Kudryashov method⁹. A paradigm for characterizing bacterial colonies called the Chavy–Waddy–Kokolnikov (CWK) model is thought about by Kudryashov et al. The nonlinear ordinary differential equation, which is equivalent to the fourth-order partial differential equation, is subjected to the Painlevé test to determine if the mathematical model is integrable¹². One of the well-known models in the field of biology is the Chavy-Waddy-Kokolnikov equation (CWKE), which is highly useful for simulating the collective development of bacteria attracted to light. In order to build the soliton behaviors of this model, Zahran et al examined this effective model in the following. They extracted the analytical solutions of this model using two of the most effective semi-analytical techniques: the extended simple equation method (ESEM) and the (G’/G)-expansion method. From these obtained solutions, they derive the soliton behaviors using the 2-kind and 3-kind graphs¹⁰. The PDE model for the mobility of cyanobacteria, newly suggested by Chavy-Waddy and Kolokolnikov, is studied in terms of stationary solutions and long-time dynamics by Taranets and Chugunova. They examine a variety of related stable states and their stability for various values of the parameter $\alpha$, which determines the aggregate’s extent, taking into account symmetric and non-symmetric situations separately¹³. Researchers employ the very effective Daftardar-Gejji and Jafari Method (DJM) to tackle a wide range of nonlinear phenomena. Nevertheless, delay differential equations were not solved in the literature using the DJM. In his work, Bhalekar and Patade applied DJM to estimate analytical solutions of functional-differential equations that have proportionate delays in a neutral setting. The outcomes of DJM were contrasted with precise answers and those from other iterative techniques, including the Adomian decomposition approach¹⁴. Complicating the development of resistance are uncertainties in the environmental processes. With the use of stochastic partial differential equations, Gothwal and Thatikonda aims to create a mathematical model for the movement of bacteria resistant to fluoroquinolone in riverine environments. In the stochastic partial differential equations (SPDE), Poisson’s process is utilized for the diffusion approach¹⁵. Sahu, and Jena are worked on the fractional Kawahara and modified Kawahara equations in Caputo sense on integral transforms¹⁶, time fractional Klein-Gordon equation based on modified laplace adomian decomposition technique¹⁷, used the telegraph equation in one and two space variables¹⁸, investigated the traveling wave solution of ion acoustic waves as a fractional nonlinear evolution equation¹⁹, explored the analytical solution of sine–Gordon equation based on Laplace Adomian decomposition method²⁰ and kink-antikink single waves in dispersion systems by generalized PHI-four equation²¹.

Zayed et al., used a new $\phi ^{6}$-model expansion method that offers a broader use in managing partial differential equations that are not linear. This approach allows to obtain the solutions for the hyperbolic and trigonometric functions, respectively, when the modulus of the Jacobi elliptic functions goes to one or zero²².

Bacterial colonies and their behavior when noise is added, particularly when there are stochastic conditions, is of great importance in comprehending the growth and aggregation of microbes in the environment and their responses to environmental effects. Stochastic Chavy-Waddy-Kolokolnikov equation applied to the bacterial aggregation is a significant step towards the modeling of collective evolution of bacteria under a noisy environment. Biologically, this model may be applied to explain the formation of complex patterns by bacterial colonies in response to a wide range of factors, including light, temperature, and chemical gradients. Noise, or in this instance as multiplicative noise, can be introduced to explain every random variation in these environmental factors that are usually observed in the real-world setting. This noise may have an impact on the motion and aggregation of bacteria and have an effect in how they form complex structures or patterns with time. The solitary wave solutions obtained with closed form in the study provide information on the adaptation of bacteria colonies to varying conditions in order to enable the researchers to understand growth and behavior of bacterial clusters in various environmental stresses more easily.

Physically, the mechanisms of aggregation of bacteria in the presence of noise can be useful in shedding light on other processes in nature including the formation of patterns, self-organizing and even resistance development in bacterial populations. Noise has the potential to change the form and stability of bacterial aggregates creating waves, clusters, or other spatial features. The stochasticity of the Chavy-Waddy-Kolokolnikov equation is also representative of the unpredictability of biological systems whereby minor disturbances or random occurrences can cause a significant impact on the structure of the colony on a macro scale. The dynamics of solitons and solitary waves in the presence of noise aid in explaining how such perturbations are carried on in the wave-like structures in the bacterial population. The results may have more general uses, e.g., in the study of bacterial infections expansion, biofilm development, or the response of bacteria to varying nutrient levels. It is through a combination of stochastic models and biological observations that improved predictions of bacterial behavior can be developed by scientists, which is important in such areas as microbial ecology, medicine, and biotechnology. The expansion of solutions using the $\phi ^6$-model approach into Jacobi-like elliptic functions provides a significant contribution to the understanding of soliton solutions in the context of the stochastic Chavy-Waddy-Kolokolnikov equation. This approach goes beyond simple mathematical expansion by offering a powerful framework for deriving closed-form solutions that describe wave phenomena under the influence of noise. Unlike standard methods, the $\phi ^6$-model expansion facilitates the identification of multiple waveforms, including elliptic, hyperbolic, trigonometric, and rational solutions, which are crucial for understanding the complex dynamics of bacterial aggregation and other physical systems subjected to random disturbances. Therefore, the use of the $\phi ^6$-model expansion not only enriches the mathematical analysis but also has practical implications for modeling real-world phenomena, making it a valuable contribution.

In the underlaying model we use the $\phi ^{6}$-expansion method to obtain the soliton solution Chavy-Waddy and Kolokolnikov (CWK) model. These are the main contribution in this article,

• 1. Analytical study of the stochastic partial differential equations is an active area of the research for the young researchers.

• 2. The stochastic Chavy-Waddy-Kolokolnikov equation is under consideration in Stratonovich sense.

• 3. The underlying model has application related with the development of the bacteria in the light.

• 4. Analytical study is carried out with the help of $\phi ^6-$model expansion method.

• 5. A variety of the solutions are gained under the impact of the noise.

• 6. Mathematica11.1 software is used for the calculations and simulations.

• 7. The explanation of the obtained results are discussed in the context of the real world phenomena.

Mathematical model

The evolving of phototaxic bacterial aggregation was proposed by the Chavy-Waddy and Kolokolnikov (CWK) model²³ that is nonlinear 4th order partial differential equation. The CWK model is given as

$$\begin{aligned} z_t=-z_{xx}-z_{xxxx}+\alpha \bigg (\frac{z_x z_{xx}}{z}\bigg )_x. \end{aligned}$$

(1)

The term $-z_{xx}$ represent the reverse diffusion and $-z_{xxxx}$ is the long range term. The reverse diffusion process occurs towards the high concentration^24,25. The fourth order term is related with the impact of the distant neighbors on the concentration at a point^26,27. The reverse diffusion process destabilize the model and and fourth order term stabilize the system, that mean both phenomena balance the each other. The only parameter in Eq. (1) is the $\alpha$ that controls the size of aggregation and it is given as

$$\begin{aligned} \alpha =\frac{c(1+2d)(1+d)^2}{(c[1+d^3+2d^2+3d]-2a)}, \end{aligned}$$

(2)

here a,c, and d represent the rate at which bacterium moves by keeping its orientation, the rate at which bacterium moves with new orientation and sensing radius of the bacterium respectively. The stationary solution of Eq. (1) was obtained in²³ by reducing the order of the equation and studied the system with the transformation $z(x,t)=e^{w(x,t)}$ and Eq. (1) becomes

$$\begin{aligned} w_t+w_{x}^2+w_{xx}+w_{xxxx}-(4\alpha -6)w_x^2w_{xx}-(\alpha -4)w_x w_{xxx}-(\alpha -3)w_{xx}^2-(\alpha -1)w_x^4=0, \end{aligned}$$

(3)

In mathematics and physics Brownian motion stands for the phenomenon that refers to the seemingly disorderly movement of particles in a fluid. It is called after Robert Brown who, in 1827 observed this motion in pollen grains dropped in water. It should be noted that this phenomenon can be described by the mathematical apparatus of a stochastic process, namely it can be a Wiener process or a process of a more general type – a Markov process with continuous paths. The impact of time noise on CWK model in the It$\hat{o}$ sense is given as

$$\begin{aligned} w_t+w_{x}^2+w_{xx}+w_{xxxx}-(4\alpha -6)w_x^2w_{xx}-(\alpha -4)w_x w_{xxx}-(\alpha -3)w_{xx}^2-(\alpha -1)w_x^4= \mu V \circ \beta _t, \end{aligned}$$

(4)

where $\mu$ is a control parameter, $\beta _t$ is a noise factor, and the symbol $\circ$ indicates that the stochastic integral is to be interpreted in the Stratonovich sense.

Definition 1

The stochastic process $\beta (t)$, $t\ge 0$ is known to be brownian motion is the following properties are satisfies (for more detail see^28,29,30)

• $\beta (t)$ is continuous and has independent increments,

• $\beta (0)$=0,

• $\beta (t)-\beta (s)$ has Gaussian distribution.

The Stratonovich integral, so called after the Russian mathematician Ruslan Stratonovich, is an alternative form of the stochastic integral. An alternate version that handles the stochastic calculus differently than the Itô integral is the Stratonovich integral. To be more precise, the Stratonovich integral generates a distinct set of integration rules by applying a different method to determine the limit of Riemann sums. Because the Stratonovich integral is independent of the precise description of the underlying stochastic process, it finds frequent use in the fields of engineering and physics. Modeling concerns typically determine which version is appropriate, but once selected, a similar equation of the opposite kind can be generated with the same answers. Hence, the Itô (represented by $\int _0^t G d\beta$) and Stratonovich (represented by $\int _0^t \circ G d\beta$) can be switched by the subsequent relation:

$$\begin{aligned} \int _0^t G(s, Z_s)d\beta (s)=\int _0^tG(s, Z_s)\circ d\beta (s)-\frac{1}{2}G(s, Z_s)\frac{\partial G(s, Z_s)}{\partial Z} ds, \end{aligned}$$

(5)

here $\{Z_t, t\ge 0 \}$ is stochastic process and G is assumed to be sufficiently regular.

The methodology

This section involves a step-by-step description of the $\phi ^6$-model expansion approach. The standard form of NLEE is (for more detail see^31,32)

$$\begin{aligned} G(w, w_{x}, w_{t}, w_{tt}, w_{xx}, \cdots )=0, \end{aligned}$$

(6)

where w is satisfy the NLEE and G is a polynomial. The main steps of this method are given as below.

Step 1: The variable is applied to reduce the wave transformation.

$$\begin{aligned} w(x, t)=\psi (\eta ), \qquad \eta = x + m t, \end{aligned}$$

(7)

where m is a nonzero constants.

$$\begin{aligned} H(\psi , \psi ^{'}, \psi ^{''}, \psi ^{'''}, \psi ^{''''},\cdots )=0, \end{aligned}$$

(8)

The polynomial H in $\psi (\eta )$, is denoted by the total derivatives $\psi ^{'}(\eta ), \psi ^{''}(\eta )$ and so on.

$$\begin{aligned} \psi (\eta )= \sum _{k=0}^{2N}\delta _k \Lambda ^k (\eta ), \end{aligned}$$

(9)

where $\delta _k$(k = 0,1,...2N) are the constants that can be found later, here $\Lambda (\eta )$ is to modify the Riccati eq. as shown by.

$$\begin{aligned} \Lambda '^2(\eta )= & \theta _0+\theta _2 \Lambda ^2(\eta )+\theta _4 \Lambda ^4(\eta )+\theta _6 \Lambda ^6(\eta ),\\ \nonumber \Lambda ''(\eta )= & \theta _2 \Lambda (\eta )+2\theta _4 \Lambda ^3(\eta )+3\theta _6 \Lambda ^5(\eta ). \end{aligned}$$

(10)

Where $\theta _i$(i = 0,2,4,6) are real constants.

Step 3: The homogenous balancing principle can be applied to determine the positive number P.

Step 4: The solution to that equation is widely known.

$$\begin{aligned} \psi (\theta )=\frac{S(\eta )}{\sqrt{k_1 S^2(\eta )+k_2}}, \end{aligned}$$

(11)

here $(k_1 S^2(\eta )+k_2) > 0$ and $S(\eta )$ and it is the solution of Jacobian elliptic equation’s such as

$$\begin{aligned} \psi '(\eta )^2=c_0+c_2 S^2(\eta ) +c_4 S^4(\eta ), \end{aligned}$$

(12)

and $c_{k}(k=0,2,4)$ are constants that are found later, where as $k_1$ and $k_2$ are given by

$$\begin{aligned} k_1= & \frac{\theta _4 \left( c_2-\theta _2\right) }{\left( c_2-\theta _2\right) {}^2-2 c_2 \left( c_2-\theta _2\right) +3 c _0 c_4}, \end{aligned}$$

(13)

$$\begin{aligned} k_2= & \frac{3 \theta _4 \theta _0}{\left( c_2-\theta _2\right) {}^2-2 c_2 \left( c_2-\theta _2\right) +3 c_0 c_4}. \end{aligned}$$

(14)

Step 5: The Eq. (8) has the following Jacobian elliptic solutions that are mentioned in Table 1.

Table 1 JEF solutions.

In this table, $sn(\eta )=sn(\eta ,\sigma ),$ $cd(\eta )=cd(\eta ,\sigma ),$ $dn(\eta )=dn (\eta ,\sigma ),$ $ns(\eta )= ns(\eta ,\sigma ),$ $ns(\eta )= ns(\eta ,\sigma ),$ $cs(\eta )= cs(\eta ,\sigma ),$ $ds(\eta )= ds(\eta ,\sigma ),$ $sc(\eta )= sc(\eta ,\sigma ),$ $sd(\eta )= sd(\eta ,\sigma )$ are the Jacobi elliptic functions with the modulus $0<\sigma <1.$ These functions degenerate into hyperbolic functions, when $\sigma \rightarrow 1$ as follows: $sn(\eta ,1)=\tanh (\eta ),$ $cn(\eta ,1)=sech(\eta ),$ $dn(\eta ,1)=sech(\eta ),$ $ns(\eta ,1)=\coth (\eta ),$ $cs(\eta ,1)=csch(\eta ),$ $ds(\eta ,1)=csch(\eta ),$ $sc(\eta ,1)=\sinh (\eta ),$ $nc(\eta ,1)=\cosh (\eta ),$ $cd(\eta ,1)=1.$

and into trigonometric functions, when $\sigma \rightarrow 0$ as follows:

$sn(\eta ,0)=\sin (\eta ),$ $cd(\eta ,0)=\cos (\eta ),$ $cn(\eta ,0)=\cos (\eta ),$ $ns(\eta ,0)=\csc (\eta ),$ $cs(\eta ,1)=\cot (\eta ),$ $ds(\eta ,0)=\csc (\eta ),$ $sc(\eta ,0)=\tan (\eta ),$ $sd(\eta ,0)=\sin (\eta ),$ $nc(\eta ,0)= \sec (\eta ),$ $dn(\eta ,0)=1.$

Step 6: Inserting Eq. (9) with the help of Eqs. (10) and (11) into Eq. (8), we obtained the Jacobi elliptic function (JEF) solutions of Eq. (8).

Stochastic wave transformation

The stochastic wave transformation for the given model is consider as^33,34

$$\begin{aligned} w(x,t)=V(\eta ) e^{\mu \beta (t)-\mu ^2t}, \qquad \text{ where }\qquad \eta = x+ m t, \end{aligned}$$

(15)

Taking the derivatives of Eq. (15) and substitute into the Eq. (4) and get the following form as

$$\begin{aligned} m V′+V'^{2} e^{(\mu \beta (t)-\mu ^2t)}+V''+V^{''''}-(4 \alpha -6)V'^{2} e^{(\mu \beta (t)-\mu ^2t)} V^{'''}-(\alpha -4)V' V^{'''} \alpha -3)V''^{2} e^{(\mu \beta (t)-\mu ^2t)}-(\alpha -1)V'^{4} e^{3(\mu \beta (t)-\mu ^2t)}=0, \end{aligned}$$

(16)

now, we take the Expectation both side of above equation

$$\begin{aligned} m V′+V'^{2} e^{-\mu ^2 t} \mathbb {E}( e^{\mu \beta (t)}) +V''+V^{''''}-(4 \alpha -6)V'^{2}e^{-\mu ^2 t} \mathbb {E}( e^{\mu \beta (t)}) V^{'''}-(\alpha -4)V' V^{'''}-\nonumber \\(\alpha -3)V''^{2} e^{-\mu ^2 t} \mathbb {E}( e^{\mu \beta (t)})-(\alpha -1)V'^{4} e^{-3\mu ^2 t} \mathbb {E}( e^{3\mu \beta (t)}) =0, \end{aligned}$$

(17)

as we know $\mathbb {E}( e^{2 \mu \beta (t)})=e^{\mu ^2 t}$, then we get the following form

$$\begin{aligned} m V′+V'^{2}+V''+V^{''''}-(4 \alpha -6)V'^{2} V^{'''}\nonumber \alpha -4)V' V^{'''}-(\alpha -3)V''^{2}-(\alpha -1)V'^{4}=0, \end{aligned}$$

(18)

Also, substituting $V'=\psi$

$$\begin{aligned} m\psi +\psi ^2+\psi '+\psi '''-(4\alpha -6)\psi ^2\psi '-(\alpha -4)\psi \psi ''-(\alpha -3)\psi '^2-(\alpha -1)\psi ^4=0, \end{aligned}$$

(19)

Application to CWK model

Now, we suppose the general solution of Eq. (19) in the form of polynomial such as

$$\begin{aligned} \psi (\eta )=\sum _{k=0}^{2N}\delta _k\Lambda ^k(\eta ), \end{aligned}$$

(20)

where constant $\delta _k( k=0,1,2,3,\cdots ,2N)$ are calculated later, while $\Lambda (\eta )$ satisfies the following nonlinear ODE

$$\begin{aligned} \Lambda '^2(\eta )=\, & \theta _0+\theta _2 \Lambda ^2(\eta )+\theta _4 \Lambda ^4(\eta )+\theta _6 \Lambda ^6(\eta ),\qquad \\\nonumber \Lambda ''(\eta )= \,& \theta _2 \Lambda (\eta )+2\theta _4 \Lambda ^3(\eta )+3\theta _6 \Lambda ^5(\eta ), \end{aligned}$$

(21)

where $(\theta _i, i=0,2,4,6.)$ are real constants. By using the homogeneous balancing principle, we get the N = 1,

$$\begin{aligned} \psi (\eta )=\delta _0+\delta _1\Lambda (\eta )+\delta _2\Lambda ^2(\eta ), \end{aligned}$$

(22)

here $\delta _i, i=0,1,2$ are constants and will be calculated later. Putting Eq. (22) and all its related derivatives in Eq. (19), a system of polynomials is obtained and collecting the coefficients by using the MATHEMATICA software. We get the following solution set

$$\begin{aligned} \delta _0=-\frac{1}{\sqrt{6 \alpha -10}},~~\delta _1=0,~~\delta _2=-\frac{3 \sqrt{6 \alpha -10} \theta _4}{2 \alpha -3},~~m=\frac{(21-2 \alpha ) \alpha -30}{4 \sqrt{2} (3 \alpha -5)^{3/2}},~~\theta _0=\frac{(3-2 \alpha )^2}{48 (5-3 \alpha )^2 \theta _4},~~\\ \theta _2=-\frac{\alpha -2}{4 (3 \alpha -5)},~~\theta _6=-\frac{3 (3 \alpha -5) (5 \alpha (2 \alpha -9)+48) \theta _4^2}{2 (3-2 \alpha )^2 (\alpha -4)}. \end{aligned}$$

The following solution are generated for the family and JEFs from table and solution of Eq. (4) are given as Type-I $c_0=1,~~ c_2=-\left( \sigma ^2+1\right) ,~~ c_4=\sigma ^2,~~ 0< \sigma < 1$, then $S(\eta )=sn(\eta ,\sigma ) or \delta (\eta )=cd(\eta ,\sigma )$, and obtained JEFs as

$$\begin{aligned} \psi _{1,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 sn(\eta )^4}{(2 \alpha -3) \left( A_{1} sn(\eta )^2+A_{2}\right) }, \end{aligned}$$

(23)

or

$$\begin{aligned} \psi _{2}(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 cd(\eta )^4}{(2 \alpha -3) \left( A_{1} cd(\eta )^2+A_{2}\right) }, \end{aligned}$$

(24)

where

$$\begin{aligned} A_{1}= & \frac{4 (3 \alpha -5) \theta _4 \left( 4 (3 \alpha -5) \sigma ^2+11 \alpha -18\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396},\end{aligned}$$

(25)

$$\begin{aligned} A_{2}= & -\frac{48 (5-3 \alpha )^2 \theta _4}{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}. \end{aligned}$$

(26)

$\bullet$ For $\sigma \rightarrow 1$, the kink type solution is obtained

$$\begin{aligned} w_{1,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \tanh ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \tanh ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(27)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{1,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \sin ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \sin ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(28)

or

$$\begin{aligned} w_{2,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \cos ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \cos ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(29)

Type-II $c_0=1-\sigma ^2,~~ c_2=-\left( 2\sigma ^2-1\right) ,~~ c_4=-\sigma ^2,~~ 0< \sigma < 1$, then $S(\eta )=cn(\eta ,\sigma )$, and obtained JEFs as

$$\begin{aligned} \psi _{3,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 cn(\eta )^4}{(2 \alpha -3) \left( A_{1} cn(\eta )^2+A_{2}\right) }. \end{aligned}$$

(30)

where

$$\begin{aligned} A_{1}= & -\frac{4 (3 \alpha -5) \theta _4 \left( 8 (3 \alpha -5) \sigma ^2-11 \alpha +18\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}\\ A_{2}= & \frac{48 (5-3 \alpha )^2 \theta _4 \left( \sigma ^2-1\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the bright solution is obtained

$$\begin{aligned} w_{3,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {sech}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {sech}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(31)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{3,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {cos}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {cos}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(32)

Type-III $c_0=\sigma ^2-1,~~ c_2=2-\sigma ^2,~~ c_4=-1,~~ 0< \sigma < 1$, then $S(\eta )=dn(\eta ,\sigma )$, and obtained JEFs as

$$\begin{aligned} \psi _{4,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 dn(\eta )^4}{(2 \alpha -3) \left( A_{1} dn(\eta )^2+A_{2}\right) }, \end{aligned}$$

(33)

where

$$\begin{aligned} A_{1}= & \frac{4 (3 \alpha -5) \theta _4 \left( 4 (3 \alpha -5) \sigma ^2-25 \alpha +42\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396},\\ A_{2}= & -\frac{48 (5-3 \alpha )^2 \theta _4 \left( \sigma ^2-1\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the bright solution is obtained

$$\begin{aligned} w_{4,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {sech}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {sech}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(34)

Type-IV $c_0=\sigma ^2,~~ c_2=-(1-\sigma ^2),~~ c_4=1,~~ 0< \sigma < 1$, then $S(\eta )=ns(\eta ,\sigma )$ or $S(\eta )=cd(\eta ,\sigma )$, and obtained JEFs as

$$\begin{aligned} \psi _{5,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 ns(\eta )^4}{(2 \alpha -3) \left( A_{1} ns(\eta )^2+A_{2}\right) }, \end{aligned}$$

(35)

or

$$\begin{aligned} \psi _{6,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 cd(\eta )^4}{(2 \alpha -3) \left( A_{1} cd(\eta )^2+A_{2}\right) }. \end{aligned}$$

(36)

where

$$\begin{aligned} A_{1}= & -\frac{4 (3 \alpha -5) \theta _4 \left( 4 (3 \alpha -5) \sigma ^2-11 \alpha +18\right) }{\left( 4 (3 \alpha -5) \sigma ^2-13 \alpha +22\right) \left( 4 (3 \alpha -5) \sigma ^2-11 \alpha +18\right) -48 (5-3 \alpha )^2 c_4 \sigma ^2},\\ A_{2}= & \frac{3 \theta _4 \sigma ^2}{-\frac{(11 \alpha -18) (13 \alpha -22)}{16 (5-3 \alpha )^2}+3 c_4 \sigma ^2-\sigma ^4+2 \sigma ^2}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the bright solution is obtained

$$\begin{aligned} w_{5,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \coth ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \coth ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(37)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{5,0}(x,t)=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \csc ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \csc ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }, \end{aligned}$$

(38)

or

$$\begin{aligned} w_{6,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \cos ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \cos ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(39)

Type-V $c_0=-\sigma ^2,~~ c_2=2\sigma ^2-1,~~ c_4=1-\sigma ^2,~~ 0< \sigma < 1$, then $S(\eta )=nc(\eta ,\sigma )$ and obtained JEFs as

$$\begin{aligned} \psi _{7,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 nc(\eta )^4}{(2 \alpha -3) \left( A_{1} nc(\eta )^2+A_{2}\right) }. \end{aligned}$$

(40)

where

$$\begin{aligned} A_{1}= & -\frac{4 (3 \alpha -5) \theta _4 \left( 8 (3 \alpha -5) \sigma ^2-11 \alpha +18\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396},\\ A_{2}= & \frac{48 (5-3 \alpha )^2 \theta _4 \sigma ^2}{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the solution is obtained

$$\begin{aligned} w_{7,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \cosh ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \cosh ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(41)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{7,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \sec ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \sec ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(42)

Type-VI $c_0=-1,~~ c_2=2-\sigma ^2,~~ c_4=-(1-\sigma ^2),~~ 0< \sigma < 1$, then $S(\eta )=nd(\eta ,\sigma )$ and obtained JEFs as

$$\begin{aligned} \psi _{8,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{2 i \sqrt{3} \sqrt{\alpha -4} \sqrt{\theta _6} nd(\eta )^4}{\sqrt{10 \alpha ^2-45 \alpha +48} \left( A_{1} nd(\eta )^2+A_{2}\right) }, \end{aligned}$$

(43)

where

$$\begin{aligned} A_{1}= & \frac{4 (3 \alpha -5) \theta _4 \left( 4 (3 \alpha -5) \sigma ^2-25 \alpha +42\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396},\\ A_{2}= & \frac{48 (5-3 \alpha )^2 \theta _4}{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the solution is obtained

$$\begin{aligned} w_{8,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {sech}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {sech}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(44)

Type-VII $c_0=1,~~ c_2=2-\sigma ^2,~~ c_4=1-\sigma ^2,~~ 0< \sigma < 1$, then $S(\eta )=sc(\eta ,\sigma )$ and obtained JEFs as

$$\begin{aligned} \psi _{9,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{2 i \sqrt{3} \sqrt{\alpha -4} \sqrt{\theta _6} sc(\eta )^4}{\sqrt{10 \alpha ^2-45 \alpha +48} \left( A_{1} sc(\eta )^2+A_{2}\right) }, \end{aligned}$$

(45)

where

$$\begin{aligned} A_{1}= & \frac{4 (3 \alpha -5) \theta _4 \left( 4 (3 \alpha -5) \sigma ^2-25 \alpha +42\right) }{\left( 4 (3 \alpha -5) \sigma ^2-25 \alpha +42\right) \left( 4 (3 \alpha -5) \sigma ^2-23 \alpha +38\right) -48 (5-3 \alpha )^2 c_4},\\ A_{2}= & \frac{3 \theta _4}{-\frac{(23 \alpha -38) (25 \alpha -42)}{16 (5-3 \alpha )^2}+3 c_4-\sigma ^4+4 \sigma ^2}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the solution is obtained

$$\begin{aligned} w_{8,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {sinh}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {sinh}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(46)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{8,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {tan}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {tan}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(47)

Type-VIII $c_0=1,~~ c_2=(2\sigma ^2-1),~~ c_4=-\sigma ^2(1-\sigma ^2),~~ 0< \sigma < 1$, then $S(\eta )=sd(\eta ,\sigma )$ and obtained JEFs as

$$\begin{aligned} \psi _{10,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{2 i \sqrt{3} \sqrt{\alpha -4} \sqrt{\theta _6} sd(\eta )^4}{\sqrt{10 \alpha ^2-45 \alpha +48} \left( A_{1} sd(\eta )^2+A_{2}\right) }, \end{aligned}$$

(48)

where

$$\begin{aligned} A_{1}= & -\frac{4 (3 \alpha -5) \theta _4 \left( 8 (3 \alpha -5) \sigma ^2-11 \alpha +18\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396},\\ A_{2}= & -\frac{48 (5-3 \alpha )^2 \theta _4}{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the solution is obtained

$$\begin{aligned} w_{10,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {sinh}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {sinh}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(49)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{10,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {sin}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {sin}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(50)

Type-IX $c_0=1-\sigma ^2,~~ c_2=(2-\sigma ^2),~~ c_4=1,~~ 0< \sigma < 1$, then $S(\eta )=cs(\eta ,\sigma )$ and obtained JEFs as

$$\begin{aligned} \psi _{11,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{2 i \sqrt{3} \sqrt{\alpha -4} \sqrt{\theta _6} sc(\eta )^4}{\sqrt{10 \alpha ^2-45 \alpha +48} \left( A_{1} sc(\eta )^2+A_{2}\right) }, \end{aligned}$$

(51)

where

$$\begin{aligned} A_{1}= & \frac{4 (3 \alpha -5) \theta _4 \left( 4 (3 \alpha -5) \sigma ^2-25 \alpha +42\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396},\\ A_{2}= & \frac{48 (5-3 \alpha )^2 \theta _4 \left( \sigma ^2-1\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the solution is obtained

$$\begin{aligned} w_{11,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {csch}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {csch}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(52)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{11,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {cot}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {cot}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(53)

Type-X $c_0=-\sigma ^2(1-\sigma ^2),~~ c_2=(2\sigma ^2-1),~~ c_4=1,~~ 0< \sigma < 1$, then $S(\eta )=ds(\eta ,\sigma )$ and obtained JEFs as

$$\begin{aligned} \psi _{12,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{2 i \sqrt{3} \sqrt{\alpha -4} \sqrt{\theta _6} ds(\eta )^4}{\sqrt{10 \alpha ^2-45 \alpha +48} \left( A_{1} ds(\eta )^2+A_{2}\right) }, \end{aligned}$$

(54)

where

$$\begin{aligned} A_{1}= & -\frac{4 (3 \alpha -5) \theta _4 \left( 8 (3 \alpha -5) \sigma ^2-11 \alpha +18\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396},\\ A_{2}= & -\frac{48 (5-3 \alpha )^2 \theta _4 \sigma ^2 \left( \sigma ^2-1\right) }{143 \alpha ^2+16 (5-3 \alpha )^2 \sigma ^4-16 (5-3 \alpha )^2 \sigma ^2-476 \alpha +396}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the solution is obtained

$$\begin{aligned} w_{12,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {csch}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {csch}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(55)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{12,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}-\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {csc}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( A_{1} \text {csc}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +A_{2}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(56)

Type-XI $c_0=\frac{1-\sigma ^2}{4},~~ c_2=\frac{1+\sigma ^2}{2},~~ c_4=\frac{1-\sigma ^2}{4},~~ 0< \sigma < 1$, then $S(\eta )=ns(\eta ,\sigma )\mp sc(\eta ,\sigma )$ or $S(\eta )=\frac{cn(\eta ,\sigma )}{1\pm sn(\eta ,\sigma )}$ and obtained JEFs as

$$\begin{aligned} \psi _{13,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{2 i \sqrt{3} \sqrt{\alpha -4} \sqrt{\theta _6} (ns(\eta )\mp sc(\eta ) )^4}{\sqrt{10 \alpha ^2-45 \alpha +48} \left( A_{1} (ns(\eta )\mp sc(\eta ) )^2+A_{2}\right) }, \end{aligned}$$

(57)

or

$$\begin{aligned} \psi _{14,\sigma }(\eta )=-\frac{1}{\sqrt{6 \alpha -10}}-\frac{2 i \sqrt{3} \sqrt{\alpha -4} \sqrt{\theta _6} ( \frac{cn(\eta )}{1\pm sn(\eta )} )^4}{\sqrt{10 \alpha ^2-45 \alpha +48} \left( A_{1} ( \frac{cn(\eta )}{1\pm sn(\eta )} )^2+A_{2}\right) }, \end{aligned}$$

(58)

where

$$\begin{aligned} A_{1}= & -\frac{4 (3 \alpha -5) \theta _4 \left( 2 (3 \alpha -5) \sigma ^2+7 \alpha -12\right) }{8 \alpha ^2+(5-3 \alpha )^2 \sigma ^4+14 (5-3 \alpha )^2 \sigma ^2-26 \alpha +21},\\ A_{2}= & \frac{12 (5-3 \alpha )^2 \theta _4 \left( \sigma ^2-1\right) }{8 \alpha ^2+(5-3 \alpha )^2 \sigma ^4+14 (5-3 \alpha )^2 \sigma ^2-26 \alpha +21}. \end{aligned}$$

$\bullet$ For $\sigma \rightarrow 1$, the solution is obtained

$$\begin{aligned} w_{13,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}\nonumber \hspace{9cm}\right. \\\left. -\frac{3 \sqrt{6 \alpha -10} \theta _4 \text {sech}^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( \tanh \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +1\right) ^4 \left( \frac{A_{11} \text {sech}^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{\left( \tanh \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +1\right) ^2}+A_{22}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(59)

or

$$\begin{aligned} w_{14,1}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}\nonumber \hspace{9cm}\right. \\\left. -\frac{3 \sqrt{6 \alpha -10} \theta _4 \left( \sinh \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +\cosh \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) \right) ^4}{(2 \alpha -3) \left( A_{11} \left( \sinh \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +\cosh \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) \right) ^2+A_{22}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(60)

For $\sigma \rightarrow 0$, the solitary wave solution is obtained

$$\begin{aligned} w_{13,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}\nonumber \hspace{9cm}\right. \\\left. -\frac{3 \sqrt{6 \alpha -10} \theta _4 \left( \tan \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +\sec \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) \right) ^4}{(2 \alpha -3) \left( A_{11} \left( \tan \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +\sec \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) \right) ^2+A_{22}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(61)

or

$$\begin{aligned} w_{14,0}(x,t)=\left( -\frac{1}{\sqrt{6 \alpha -10}}\nonumber \hspace{9cm}\right. \\\left. -\frac{3 \sqrt{6 \alpha -10} \theta _4 \cos ^4\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{(2 \alpha -3) \left( \sin \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +1\right) ^4 \left( \frac{A_{11} \cos ^2\left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) }{\left( \sin \left( \frac{((21-2 \alpha ) \alpha -30) t}{4 \sqrt{2} (3 \alpha -5)^{3/2}}+x\right) +1\right) ^2}+A_{22}\right) }\right) e^{\mu \beta (t)-\mu ^2t}. \end{aligned}$$

(62)

Graphical effect of noise

In this section, we will see the effect of noise of our obtained soliton solutions. The different form of solitons and solitary wave solutions are obtained via $\phi ^6$-model expansion method. These results are new and very fruitful for the Chavy-Waddy-Kolokolnikov equation. Moreover, we are mainly focused on the impact of noise on these solutions. Depending on the noise level, the amplitude and form of solitons are constant in the conditions of soliton existence. Amplitude modulation is observed to shift up and down due to noise, and the soliton profiles are altered. The amplitude and contour of the wave can change with the noise for a brief period of time. Solitons are long-lived and live for an extremely long time on non- noisy environments. Solitons themselves could become unstable in the presence of noise and in such a situation they might either continue to spread or appear to split into several smaller solitons. However, solitons can sometimes reveal stochastically stability and therefore even if fluctuations occur in the wave, the main structure of solitons will remain largely intact. In the absence of noise, the interfacing between fermions is amicable; these particles tend to merge or simply go through each other, taking on no forms of the other on the outcome of the process. The complexity of the transition processes of solitons and their chaotic behavior grows with noise. It has been observed that due to the presence of noise, there can be unsymmetrical soliton mixing, partition, or deviations. Some solutions are drawn for the various values of parameters in the 3D, and 2D and plots as well. With the use of 3D, and 2D plots the development of the influenced waveforms is illustrated as a result of the stochastic term. In these solutions $\mu$ is the controlling factor of Brownian motion. In Fig. 1, there is the graphical representation of the solution $w_{1,1}(x,t)$ which demonstrates the various noise characteristics, demonstrating that noise in the physical system grows as $\mu$ values rise. The Fig. 4a, d are drawn for $\mu = 0$ which shows classical solution with no randomness and zero noise giving dark soliton solution, while in Fig. 4b, e and 4c, f the value of noise is increased as $\mu = 0.3$ and $\mu = 0.5$ respectively. It can be observed from the graphs that when we increase noise strength from 0 to 0.3 and then to 0.5, the abrupt spikes are emerged which shows randomness. The graphs of the solution $w_{1,0}(x,t)$ is presented in Fig. 2 that will provides the solitary wave solutions. Figures 3 and 4 clearly gives us the bright solitons at noise zero. So, these plots clearly shows that how the noise is effected out soliton solutions. The physical systems impact on noise is observed in the real world through a variety of natural and artificial phenomena, especially the biological systems, such as bacterial colonies. The stochastic Chavy-Waddy-Kolokolnikov equation incorporates noise to model the random environmental variations that impact microbial phenomena. An example of this is the environmental conditions, such as light, temperature, nutrient availability, etc, are not fixed and may vary at random in a case of bacterial aggregation. These are the fluctuations that are predicted by the stochastic term, and cause the changes in the ways of aggregation, movement or formation of biofilms by bacteria. This type of noise modeling has been used to explain microbial behavior in uncontrollable settings, including both natural ecosystems and industrial fermentation processes. An illustration of the increase in the intensity of noise with the parameter. Using $\mu$ exhibits the change in the structure formation and behaviour of the colonies over randomness change, providing an idea of the way bacterial populations evolve in response to the change in noise.

The extension of the idea of solitons and solitary waves to noise, in more material sense, has a wide-ranging application to the study of systems that are characterized by non-linear wave propagation, including fluid mechanics, optics and even modeling of traffic flow. Noise can cause the behaviour of waves in such systems to become random or chaotic in some cases in the same way as it does in the appearance of the sudden spikes in the soliton solutions in response to an increase in noise. As an example, solitons have been used in nonlinear optical fibers to ensure pulses are preserved in their shape as they pass through the medium, although the noise present can alter such pulses, causing loss of coherence or energy. On the same note, within the materials science framework, noise effects on the propagation of waves may play an important role in designing materials that are robust to changes in the environment. The findings in the figures, including the classical to noisy solution transitions, can be directly applied to such areas as communication technology and materials engineering where stability and predictable waveform behavior is essential. Therefore, the stochastic models give important information about the control and perception of the unpredictable behavior of real-life physical systems. The stochastic Chavy-Waddy-Kolokolnikov equation requires numerical simulation and experimental validation for real-world application. Simulations Discretization of the equation and running the equation at different noise levels ($\mu$) can be used to simulate the formation of solitons and how they respond to noise. Experimental validation is the observation of bacterial colonies under controlled conditions with changing experimental conditions e.g. light or nutrient changes in order to compare with simulation results. These experiments can verify whether noise causes the emergent appearance of randomness and soliton like structures which is anticipated. The model’s simplified assumptions, however, limit its applicability. It assumes a specific noise type and a constant environment, whereas real biological systems feature more complex noise dynamics and interactions.

Fig. 1

Effects of noise on dark soliton solution in the form of 3D and 2D for the solutions $w_{1,1}(x,t)$.

Fig. 2

Effects of noise on solitary wave solution in the form of 3D and 2D for the solutions $w_{1,0}(x,t)$.

Fig. 3

Effects of noise on bright soliton solution in the form of 3D and 2D for the solutions $w_{3,1}(x,t)$.

Fig. 4

Effects of noise on bright soliton solution in the form of 3D and 2D for the solutions $w_{8,1}(x,t)$.

Conclusion

In this study, the Chavy-Waddy-Kolokolnikov equation is considered under the impact of temporal noise. The underlying model has numerous real life applications such as it helps to understand the formation of bacterial colonies and their response to chemical signals. It also explains the behavior of bacteria in the human body particulary in understanding the spread of disease. As the governing model has important uses for different scenario of life, so we considered the stochastic Chavy-Waddy-Kolokolnikov equation in Ito sense. For the soliton and solitary wave solution, the $\phi ^6$-model expansion method is used on the underlying model. This method provides the jaccobi elliptic functions which includes the stochastic elliptic, hyperbolic, rational,trigonometric solutions. 3D, 2D and their corresponding counters are plotted for various choices of the parameters. The impact of noise is also analyzed and plots for various choices of noise strength is also drawn. Plots are also explained in the sense of real life problems.