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It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions. [ABSTRACT FROM AUTHOR]

This manuscript presents some new exact solutions for the dispersive Schrödinger Hirota's (SH) equation considering the effects of spatio-temporal dispersion with parabolic law nonlinearity. A wave transformation technique is implemented to transform the SH equation into a nonlinear ordinary differential equation. The solutions for the dispersive SH equation are derived using the modified extended tanh approach. These solutions exhibits various sorts of solitons, including bright, dark, periodic, and singular solitons. The findings of this research hold potential significance in the domain of fiber-optic communication, as they contribute to a deeper understanding of soliton dynamics. [ABSTRACT FROM AUTHOR]

New solitary wave solutions are developed for various shapes of three-dimensional nonlinear partial differential equations using the Sardar-Subequation method. Some periodic solitary wave solutions, including bright, dark, dark-bright, and singular periodic soliton solutions, are obtained. These solutions have broad applications in a particular category of applied sciences and physics. Graphical representations of the obtained results have been created using Maple 18 software to illustrate the accuracy and simplicity of the proposed method. Two-dimensional (2-D), three-dimensional (3-D), and 2-D contour plots have been used to depict the physical behavior of the observed patterns by incorporating relevant parameters. To demonstrate the robustness, sustainability, capability, and effectiveness of the present method, we have applied it to solve the nonlinear (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation. [ABSTRACT FROM AUTHOR]

In this article, the Korteweg-de Vries-Caudrey-Dodd-Gibbon (KdV-CDG) equation is explored via a fractional operator. A nonlocal differential operator with a nonsingular kernel is used to study the KdV-CDG equation. Some theoretical features concerned with the existence and uniqueness of the solution, convergence, and Picard-stability of the solution by using the concepts of fixed point theory are discussed. Analytical solutions of the KdV-CDG equation by using the Laplace transformation (LT) associated with the Adomian decomposition method (ADM) are retrieved. The solutions are presented using 3D and surface graphics. [ABSTRACT FROM AUTHOR]

In this manuscript, we study new solutions of generalized version of geophysical KdV equation which is called generalized perturbed KdV (gpKdV) under time–space conformable operator. We implement two methods to get some novel waves solution of the gpKdV equation. First, we use extended Tanh-method to extract new solutions of considered equations in the form of trigonometric hyperbolic functions. To achieve Sine and Cosine hyperbolic solutions, we use generalized Kudryashov (GK) technique with Riccati equation. We show the behaviour of solutions via 2D and 3D figures. Also, we analyze the Corioles effect on the evolution of waves solutions of the considered equation. [ABSTRACT FROM AUTHOR]

KdV equations have a lot of applications of in fluid mechanics. The exact solutions of the KdV equations play a vital role in the wave dynamics of fluids. In this paper, some new exact solutions of a generalized geophysical KdV equation are computed with the aid of tanh-coth method. To implement the tanh-coth procedure, we first convert the PDEs to ODEs with the help of wave transformation. Then, using a system of algebraic equations, we obtain several soliton solutions. To verify and clearly illustrate the exact solutions, several graphic presentations are developed by giving the parameter values, which are then thoroughly discussed in the relevant components. [ABSTRACT FROM AUTHOR]

The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop and analyze modified and generalized forms of the standard KdV equation. In this paper, we inspect the KdV-mKdV equation, which is a modified and generalized form of the ordinary KdV equation. We use the fractional operator in the Caputo sense to analyze the equation. We examine some theoretical results concerned with the solution's existence, uniqueness, and stability. We employ a modified Laplace method to extract the numerical results of the considered equation. We use MATLAB-2020 to simulate the results in a few fractional orders. We report the effects of the fractional order on the wave dynamics of the proposed equation. [ABSTRACT FROM AUTHOR]

There are some confusion and complexity in our everyday lives, as we live in an uncertain environment. In such type of environment, an accurate calculation of the data and finding a solution to a problem is not an easy job. So, fuzzy differential equations are the better tools to model problems in the fuzzy domain. Modeling the real-world phenomenon more accurately requires such operators. Therefore, we investigate the fractional-order Swift–Hohenberg equation in the fuzzy concept. We study this equation under the fuzzy Caputo fractional derivative. We use the fuzzy Sumudu transform to find out the semianalytical solution of the considered equation. To deal with the nonlinear term of the problem, we also use the Adomian decomposition method. To confirm the accuracy of the proposed procedure, we give two test problems. Lastly, we plot the numerical results for various fractional orders, and uncertainty belongs to [0, 1]. [ABSTRACT FROM AUTHOR]

The purpose of this article is to extend the fractional third order dispersive PDE under singular and non-singular fractional operators via the notion of fuzziness. We investigate the fuzzy dispersive PDE in one and higher dimension under Caputo, Caputo-Fabrizio, and Atangana-Baleanu fractional operators and provide two examples to each derivative. We derive the general algorithm and numerical results in series of the models and test problems with the help of fuzzy Laplace transform. The numerical results confirm that solutions obtained in the fuzzy sense are more generalized than the fractional-order solution. We mention in remarks following each example that we recover the solutions of the fractional-order equations by putting the lower and upper functions of the fuzzy number g ̃ equals to 1 in the fuzzy solutions of the proposed dispersive PDEs. We demonstrate the numerical results through 2D and 3D plots for different fractional-order and uncertainty k ∈ [ 0 , 1 ]. We provide a comparison between Caputo, Caputo-Fabrizio and Atangana-Baleanu fuzzy fractional dispersive PDE. In the end, we give the conclusion of the article and future work. [ABSTRACT FROM AUTHOR]

Using the Hirota bilinear method (HBM) and Cole-Hopf transformation, we extract the interaction between the lump and kink-soliton solutions for the generalized perturbed Korteweg–de Vries (KdV) equation in this manuscript. In the literature [15] , the authors have applied the Cole-Hopf transformation with a value of R to study the particular case of the generalized perturbed KdV equation, which is found to be an inappropriate value. Alqarni et al., computed an appropriate value of R in the literature [29]. We use the correct value of R and get more clearly the outcomes for a lump-single kink soliton, a lump-two kink soliton, and a lump-periodic soliton solution. We display these solutions in 2D and 3D space using Mathematica by selecting the proper values for the parameters. • The generalized perturbed Korteweg–de Vries (KdV) equation is considered. • The Hirota bilinear method and Cole-Hopf transformation is used. • The lump and kink-soliton solutions are presented. • The obtained results are given by figures. [ABSTRACT FROM AUTHOR]

Advanced mathematics has analyzed complex systems, a key area of research in nonlinear sciences, including fluid mechanics, the theory of solitons, hydrodynamics, optical fibers, and chaos theory. Nonlocal integrable mKdV equations could help understand dispersive waves in nonlinear and complex media. In this manuscript, we derive bright one and two soliton solutions for a nonlocal nonlinear integrable KdV equation using an improved Hirota bilinear method (HBM). We use MATLAB to visualize the obtained results in 3D space by choosing the appropriate parameter values. It is demonstrated that these solutions have novel characteristics that differ from those of the mKdV equation. • The novel nonlocal nonlinear integrable KdV equation is considered. • The bright one and two soliton solutions for the proposed problem are presented using an improved Hirota bilinear method. • The obtained results are given in 3D figures by choosing the appropriate parameter values. [ABSTRACT FROM AUTHOR]