Exploring chaos and stability: dynamic insights into the stochastic Davey-Stewartson system through advanced sensitivity analysis

This research study constructs and examines solitary wave solutions to the Katugampola time-fractional [Formula: see text]-dimensional generalized Painlev'e-type (also known as P-type) equation with the help of the generalized [Formula: see text]-expansion method and tanh-coth method. The targeted model is extensively used to understand the dynamics of the plasma waves and instabilities in soliton theory, plasma physics and nonlinear wave theory. By offering closed form traveling solutions, the proposed methods transform the desired model into a nonlinear algebraic system. A new set of solutions in terms of trigonometric, rational and hyperbolic functions is obtained by analytically solving the resulting algebraic system using the symbolic computation tool Maple. We provide a range of 3D, density, and 2D visual representations that show the presence of bright and dark solitary wave solutions in the framework of the resulting wave profiles that illustrate the strong dynamics of the model. Also, a more dynamic and practical framework proposed by Katugampola-derivative is used to enhance the validity of the results by investigating how the time-fractional derivative affects the given model's solutions. Being the initial accounts of developing solutions of the aimed model, our comparison brings out the uniqueness of the obtained solutions. The variegated array of findings also shows that the suggested approaches are useful mathematical tools for solving nonlinear Fractional Partial Differential Equations (FPDEs), with applications in mathematical sciences such as engineering, physics, and biology, as well as in other nonlinear evolution problems.

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

A novel integration approach to study the perturbed Biswas-Milovic equation with Kudryashov’s law of refractive index

The fractional mKDV–Zakharov–Kuznetsov (FmKDV-ZK) equation with M-truncated derivatives (MTDs) is considered. To obtain a novel rational, trigonometric, hyperbolic, and elliptic, solutions for FmKDV-ZK, we use two different methods including the modified mapping technique and generalized Riccati equation method. These methods are effective and simple. Also, they give us different kinds of solutions such as elliptic, trigonometric, hyperbolic, and rational solutions. The FmKDV-ZK equation is widely used in engineering, geophysics, meteorology, ocean dynamics, and plasma physics, therefore the obtained solutions may be applied to a wide range of important physical phenomena. Moreover, we display a wide range of solutions in two-dimensional and three-dimensional spaces to examine the effects of the MTDs operator. We conclude that the surface expands as the derivative order decreases.

On the basis of symbolic computation a generalized sub-equation expansion method is presented for constructing some exact analytical solutions of nonlinear partial differential equations. To illustrate the validity of the method, we investigate the exact analytical solutions of the inhomogeneous high-order nonlinear Schrödinger equation (IHNLSE) including not only the group velocity dispersion, self-phase-modulation, but also various high-order effects, such as the third-order dispersion, self-steepening and self-frequency shift. As a result, a broad class of exact analytical solutions of the IHNLSE are obtained. From our results, many previous solutions of some nonlinear Schrödinger-type equations can be recovered by means of suitable selections of the arbitrary functions and arbitrary constants. With the aid of computer simulation, the abundant structure of bright and dark solitary wave solutions, combined-type solitary wave solutions, dispersion-managed solitary wave solutions, Jacobi elliptic function solutions and Weierstrass elliptic function solutions are shown by some figures.

Novel exact double periodic Soliton solutions to strain wave equation in micro structured solids

New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp(-Φ(ξ))-expansion method

We describe the finite-gap solutions of different modifications of the Davey-Stewartson (DS) equations. The restrictions on the spectral data which give us solutions of the real forms DS1 and DS2+ of DS are the same as those in the case of KP1 and KP2 of the Kadomtsev-Petviashvily equation. But for DS2− the restrictions that we regard have no analogues in other integrable systems. We describe also the restrictions that provide regularity of those solutions for DS1 and DS2±. The finite-gap solutions include rational and soliton solutions. We give some classes of those solutions. The well-known dromions for DS1 are examples of that kind.

Analysis of the dynamical behaviors for the generalized Bogoyavlvensky–Konopelchenko equation and its analytical solutions occurring in mathematical physics

The focus of this study is to investigate novel and precise solitary wave solutions of the modified Kawahara (MK) equation using recent and accurate computational techniques. The MK equation, a nonlinear partial differential equation, has significant applications in various fields, including fluid dynamics and plasma physics The comprehension of nonlinear wave phenomena relies heavily on solitary wave solutions, making it necessary to develop accurate and efficient computational techniques for their identification. In this study, the Khater II method is employed as a computational technique, while the variational iteration method is used as a numerical scheme.The Khater II method is a powerful computational technique that has exhibited promising outcomes in solving nonlinear wave equations. It uses an analytical framework to transform the partial differential equation into a set of ordinary differential equations that are amenable to straightforward solutions, thereby enabling the construction of exact solutions. Conversely, the variational iteration method is a numerical scheme that enhances solution accuracy through iterative approximations.By using the Khater II method as a computational technique and the variational iteration method as a numerical scheme, this study identifies novel and precise solitary wave solutions of the MK equation. These solutions provide valuable insights into the dynamics and behavior of nonlinear waves across various applications. The effectiveness of the employed techniques is demonstrated by comparing them with other commonly utilized computational methods for solving the MK equation. The results indicate that the combined application of the Khater II method and the variational iteration method yields more accurate and efficient solutions. This heightened accuracy and efficiency are critical for comprehending and predicting the behavior of the studied model in real-world applications. Thus, this study underscores the significance of employing recent and accurate computational techniques, such as the Khater II method and the variational iteration method, to identify novel and precise solitary wave solutions of the MK equation. The obtained solutions contribute to an improved understanding of the model’s behavior and hold practical implications across diverse scientific and engineering domains.

The investigation centers on a generalized Toda lattice equation featuring four potentials. Initially, the continuous counterpart is first established using the continuous limit approach. Subsequently, leveraging its 3 × 3 matrix spectrum problem, the discrete generalized (m, 3N − m)-fold Darboux transformation (DT) is formulated for this discrete system. Finally, the derived generalized DT is employed to generate precise rational solutions and hybrid rational-exponential solutions, which are thoroughly analyzed and graphically illustrated. These novel findings may offer insights into the lattice dynamical phenomena described by Toda lattice equation.

In the present work, the exp(−φ(ξ))-expansion method is applied for solving the deterministic and stochastic Phi-4 equation. Namely, we introduce hyperbolic, trigonometric, and rational function solutions. The computational study shows that the offered method is pretentious, robust, and influential in applications of interesting analysis, observations of particle physics, plasma physics, quantum field theory, and fluid dynamics. The control on the randomness input (the coefficients are random variables) is studied in order to obtain stability stochastic process solution with beta distribution. In this work, we will deal with stability moment method and then we apply the mean square calculus for the stability concept.

The study of nonlinear partial differential equations through analytical methods is arguably the most important and fascinating to grasp various complex physical and engineering phenomena. The positive Gardner–Kadomtsev–Petviashvili equation, which combines the dimensions effects of the KP equation with the dispersion properties of the Gardner equation, is investigated in this study. Non-linear wave systems, fluid dynamics, plasma physics, and wave interactions are all explained by this equation. A Hamiltonian formulation is presented, followed by chaos analysis, which visualizes fractal structures and reveals underlying deterministic patterns in seemingly random systems. A new extended algebraic method is introduced to solve the positive–Gardner-KP equation, yielding rational, exponential, and trigonometric solutions. The graphical propagation of soliton solutions is examined, illustrating their behavior under various parametric conditions. The results highlight the efficiency of the proposed method in understanding the dynamics of shallow water waves, particularly in the absence of surface tension and viscosity, with implications for atmospheric and oceanic wave phenomena.

In this article, we investigate the generalized (3+1)-dimensional KdV-Benjamin-Bona-Mahony equation governed with constant coefficients. It applies the Painlevé analysis to test the complete integrability of the concerned KdV-BBM equation. The symbolic computational approach provides first-order, second-order rogue wave and lump solutions with center-controlled parameters. The rogue waves localized in space and time have a significant amplitude, and lumps are of rational form solution, localized decaying solutions in all space directions rationally. Utilizing a symbolic computation approach, we get the bilinear equation of the KdV-Benjamin-Bona-Mahony equation and show the center-controlled rogue waves and lumps. We employ the symbolic system software Mathematica to do the symbolic computations, form the first and second-order rogue waves, and lump solutions with appropriate values of constant coefficients. The KdV-Benjamin-Bona-Mahony equation analyses the evolution of long waves with modest amplitudes propagating in plasma physics and the motion of waves in fluids and other weakly dispersive mediums. Moreover, rogue waves and lumps occur in several scientific areas, such as fluid dynamics, optical fibers, dusty plasma, oceanography, water engineering, and other nonlinear sciences.

In this research article, we investigate the coupled breaking soliton (cBS) model using two distinct analytical methods, namely, the Lie symmetry approach and the Unified method. We start by applying the Lie group technique to the cBS model, allowing us to establish infinitesimals, vector fields, commutative and adjoint tables, and an adjoint transformation matrix. Through the utilization of the adjoint transformation matrix, we identify a one-dimensional optimal system of subalgebras. This essential stage allows the cBS model to be reduced into several collections of ordinary differential equations (ODEs) relating to similarity variables resulting from symmetry reduction. By solving these ODE systems under specific parametric constraints, we successfully obtain exact solutions in terms of closed form. Furthermore, the Unified method is employed to address the governing equation, leading us to deduce polynomial and rational function solutions. The dynamic behaviours and characteristics of these such solutions are comprehensively explored through 3-dimensional (3D) plots and contour plots. The graphics show breather solitons, cone-shaped solitons, lump solitons, and patterns of flower petals, periodic solitons, and solitary waves. Additionally, we have connected our mathematical findings with real-world phenomena, which enrich our research work. Furthermore, breathers and lumps arise in many fields of mathematical physics, including fluid dynamics, plasma physics, ocean engineering, nonlinear optics, and physical sciences, as well as several other areas of nonlinear dynamics.