Nonlinear Evolution Equations Research Articles

Exploring the physical characteristics and nonlinear wave dynamics of a (3+1)-dimensional integrable evolution system

This study comprehensively explores the [Formula: see text]-dimensional Mikhailov–Novikov–Wang [Formula: see text] integrable equation, with the primary objective of elucidating its physical manifestations and establishing connections with analogous nonlinear evolution equations. The investigated model holds significant physical meaning across various disciplines within mathematical physics. Primarily, it serves as a fundamental model for understanding nonlinear wave propagation phenomena, offering insights into wave behaviors in complex media. Moreover, its relevance extends to nonlinear optics, where it governs the dynamics of optical pulses and solitons crucial for optical communication and signal processing technologies. Employing analytical methodologies, namely the unified [Formula: see text], Khater II ([Formula: see text]hat.II) method, and He’s variational iteration [Formula: see text] method, both numerical and analytical solutions are meticulously examined. Through this investigation, the intricate behaviors of the equation are systematically unveiled, shedding illuminating insights on various physical phenomena, notably including wave propagation in complex media and nonlinear optics. The outcomes not only underscore the efficacy of the analytical techniques deployed but also accentuate the equation’s pivotal role in modeling a broad spectrum of nonlinear wave dynamics. Consequently, this research significantly advances our comprehension of complex physical systems governed by nonlinear dynamics, thereby contributing notably to interdisciplinary pursuits in mathematical physics.

Investigating wave solutions and impact of nonlinearity: Comprehensive study of the KP-BBM model with bifurcation analysis.

In this paper, we investigate the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona Mahony equation using two effective methods: the unified scheme and the advanced auxiliary equation scheme, aiming to derive precise wave solutions. These solutions are expressed as combinations of trigonometric, rational, hyperbolic, and exponential functions. Visual representations, including three-dimensional (3D) and two-dimensional (2D) combined charts, are provided for some of these solutions. The influence of the nonlinear parameter p on the wave type is thoroughly examined through diverse figures, illustrating the profound impact of nonlinearity. Additionally, we briefly investigate the Hamiltonian function and the stability of the model using a planar dynamical system approach. This involves examining trajectories, isoclines, and nullclines to illustrate stable solution paths for the wave variables. Numerical results demonstrate that these methods are reliable, straightforward, and potent tools for analyzing various nonlinear evolution equations found in physics, applied mathematics, and engineering.

Critical structures and regularity for nonlinear evolutional partial differential equations

Critical structures and regularity for nonlinear evolutional partial differential equations

Recent advances in employing the Laplace homotopy analysis method to nonlinear fractional models for evolution equations and heat-typed problems

This study stems from the growing need for effective mathematical tools to tackle nonlinear fractional evolution equations, which find wide applications in physics and engineering. Traditional methods often struggle to provide exact closed-form solutions for such equations, leading to the development of novel approaches. In this context, the combination of the homotopy analysis method with the Laplace transform presents a promising avenue for obtaining exact solutions to these complex equations. By exploring this hybrid approach, the study aims to address the existing challenges in solving nonlinear fractional models and provide a reliable method for obtaining closed-form solutions. In this way, this study focuses on rendering the coupling between the homotopy analysis approach and the Laplace transform in the search for exact closed-form solutions for the class of fractional evolution and heat-typed equations. In fact, this class of equations features some interesting nonlinear Caputo fractional partial differential equations that model dissimilar physical processes in physics and engineering. A generalized recurrent scheme is derived, which is applicable to all the members of the governing class, and further analyzed the application of this very scheme on some test fractional initial-value problems of much concern. Eventually, the method has been found to be advantageous, with various advantages, including convergence to available exact solutions. Thus, of the advantage of the current study is innovative combination of mathematical methods, the applicability of these methods to a special class of fractional equations, the development of a generalized scheme, the successful solution of prototype problems, and the practical advantages of the method.

A new analytic approach and its application to new generalized Korteweg‐de Vries and modified Korteweg‐de Vries equations

The current study is important from two perspectives. Firstly, in this article, we suggest a novel analytical technique for creating the exact solutions to nonlinear partial differential equations (NLPDEs). In order to study the dynamical behaviors of various wave phenomena, we can construct the several exact solutions in the form of Jacobi elliptic solutions, hyperbolic solutions, trigonometric solutions, and exponential solutions by using this method. Secondly, we consider a more generalized form of the (2+1)‐dimensional Korteweg–de Vries (KdV) and modified Korteweg‐de Vries (mKdV) equations that plays an important role in describing the shallow water. We successfully extract several soliton solutions for the examined equation. By choosing the appropriate parameters, some graphs of the discovered solutions have been represented in the figures. Every obtained solution has been demonstrated to satisfy the corresponding equation. The findings demonstrate that the method can be applied to solve a number of nonlinear evolution equations. The new solutions and the paper's findings could improve our understanding of how the waves move through shallow water and open up new research avenues.

Exploring novel optical soliton solutions for the stochastic chiral nonlinear Schrödinger equation: Stability analysis and impact of parameters

The primary aim of this investigation is to uncover novel optical soliton solutions (OSSs) and conduct a comprehensive stability analysis of a particular model. The extraction of OSSs is facilitated through two potent techniques: the advanced auxiliary equation and the [Formula: see text]-expansion approaches, both proven to be highly effective in revealing soliton solutions in various nonlinear evolution equations. In this paper, we provide an in-depth exposition of the obtained solutions, elucidating them both graphically and in their physical context. Furthermore, we investigate the influence of diverse parameters on these results by presenting several graphical representations. Additionally, we delve into the stability of equilibria and conduct a thorough phase-plane analysis of the model. To provide a broader context, we also offer a comparative discussion of our solutions in relation to previous works in the field. Considering the significance of the stochastic chiral nonlinear Schrödinger equation in soliton theory, the solutions obtained here hold the potential to illustrate intricate physical phenomena and find applications across diverse fields in the realm of communication systems.

The dynamical perspective of soliton solutions, bifurcation, chaotic and sensitivity analysis to the (3+1)-dimensional Boussinesq model

In this study, we examine multiple perspectives on soliton solutions to the (3+1)-dimensional Boussinesq model by applying the unified Riccati equation expansion (UREE) approach. The Boussinesq model examines wave propagation in shallow water, which is derived from the fluid dynamics of a dynamical system. The UREE approach allows us to derive a range of distinct solutions, such as single, periodic, dark, and rational wave solutions. Furthermore, we present the bifurcation, chaotic, and sensitivity analysis of the proposed model. We use planar dynamical system theory to analyze the structure and characteristics of the system’s phase portraits. The current study depends on a dynamic structure that has novel and unexplored results for this model. In addition, we display the behaviors of associated physical models in 3-dimensional, density, and 2-dimensional graphical structures. Our findings demonstrate that the UREE technique is a valuable mathematical tool in engineering and applied mathematics for studying wave propagation in nonlinear evolution equations.

Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

The stability characteristics of a thin film of viscoelastic (Walters’ B′ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number kc which include the viscoelastic parameter Γ, angle of inclination of the plane θ, Reynolds number Re and Weber number We. The weakly nonlinear stability analysis that is based on the second Landau constant J2, reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is 3(1+3Γ)Re−3cotθ−4ReWek2 = 0. It is found that all the four distinct flow zones of this problem exist in Re-k-, θ-k- and Γ-k-plane after the critical value of Rec,θc and Γc, respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of Γ irrespective of the values of Re and θ, as for example, a solution of polyisobutylene in cetane, compared with the viscous (Γ=0) film flow case. Finally, we scrutinize the effect of Γ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.

Utmost travelling wave phenomena to the fractional type nonlinear evolution equation in mathematical physics

In applied physics and engineering, non-linear fractional types of partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed-form traveling wave solution of the time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.

On some novel solitonic structures for the Zhiber–Shabat model in modern physics

Abstract In this article, the modified Kudryashov and extended simple equation methods are employed to obtain analytical solutions for the Zhiber–Shabat problem. The outcomes of this study clearly indicate that the provided methodologies are appropriate techniques for generating some new exact solutions for nonlinear evolution equations. Furthermore, the nature of the solutions would be presented in three dimensions for various parameters applying the most advanced scientific instruments. The physical behavior of the solutions are graphically displayed, and it is established that the acquired solutions are newly constructed in the form of bright, dark, optical, singular, and bell-shaped periodic soliton wave structures. The properties of the nonlinear model have been illustrated using 3D, 2D, and contour plots by selecting an appropriate set of parameters, which is demonstrated to visualize the physical structures more productively. Finally, it is concluded that similar strategies can also be implemented to study many contemporary models. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the governing model appearing in diverse disciplines. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.

On lump and solitonic wave structures for the (3+1)-dimensional nonlinear evolution model

In order to understand many complex situations in wave propagation, such as heat transfer, fluid dynamics, optical fibers, electrodynamics, physics, chemistry, biology, condensed matter physics, ocean engineering, and many other branches of nonlinear science, the majority of natural processes are routinely modelled and analysed using nonlinear evolution equations. In this study, the (3+1)-dimensional nonlinear evolution equation is investigated analytically. Initially, the Hirota bilinear approach is used to develop the bilinear version of the higher dimensional nonlinear model. Consequently, we are able to design periodic wave soliton solutions, lump wave and single-kink soliton solutions, and collisions between lumps and periodic waves. Later on, the unified method is applied to develop several new travelling wave solutions for the governing model substantially. Furthermore, numerous exact solutions are analyzed graphically to explore many fascinating nonlinear dynamical structures with the aid of 3D, contour, and 2D visualizations. A variety of higher dimensional nonlinear evolution models can also be investigated by employing present approaches arising in many fields of contemporary science and technology.

Exact solutions and modulation instability analysis of a generalized Kundu-Eckhaus equation with extra-dispersion in optical fibers

This study aims to examine the nonlinear partial differential equation known as the (1+1)-dimensional generalized Kundu-Eckhaus equation with extra-dispersion, which is used to model the transmission of ultra-short femtosecond pulses in an optical fiber. Two versatile techniques, namely the extended G′G2 -expansion as well as the extended exp(−ϕ(ξ)) -expansion techniques, are utilized to generate numerous precise answers. Diverse novel collections of exact traveling wave solutions, such as bright solitons, dark solitons, singular solitons, W-shape solutions, M-shape solutions, and rational solutions, are identified as a result. Several of the acquired solutions are interpreted physically through the use of figures. In addition, the modulation instability analysis of the considered equation is performed and presented via 3D and 2D graphs. In the field of nonlinear sciences, the proposed methods have great value and can be applied to other nonlinear evolutionary equations that are used to represent nonlinear physical models.

Dynamic nature of analytical soliton solutions of the nonlinear ZKBBM and GZKBBM equations

A solitary wave, characterized as a localized perturbation in a medium, emerges as a result of a delicate equilibrium between nonlinear and dispersive phenomena. Solitons, a subtype of solitary waves, exhibit persistent shape and velocity during propagation, representing a fundamental phenomenon observed widely in natural systems and possessing various applications in nonlinear dynamics. This investigation focuses on two nonlinear evolution equations (NEEs), specifically the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation and the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony (GZKBBM) equation, which find relevance in domains such as fluid dynamics and ocean engineering. Utilizing an ansatz-based methodology, soliton solutions of both bright and dark characteristics are derived, alongside exploration of rogue wave-type solutions. Notably, the manifestation of dark, bright, and rogue waves aligns with the physical interpretation of the generated solitons. Computational simulations conducted using Wolfram Mathematica aim to provide a comprehensive description of the physical phenomena. The novelty of this study lies in its unreported investigation, contributing new insights into the solitonic dynamics within the considered models.

New Conformable P-Type $(3+1)$-Dimensional Evolution Equation and its Analytical and Numerical Solutions

The paper examines the conformable nonlinear evolution equation in $(3+1)$-dimensions. First, basic definitions and characteristics for the conformable derivative are given. Then, the modified extended tanh-function and $\exp(–\phi(\xi))$-expansion techniques are utilized to determine the exact solutions to this problem. The consequences of some of the acquired data's physical 3D and 2D contour surfaces are used to demonstrate the findings, providing insight into how geometric patterns are physically interpreted. These solutions help illustrate how the studied model and other nonlinear representations in physical sciences might be used in real-world scenarios. It is clear that these methods have the capacity to solve a large number of fractional differential equations with beneficial outcomes.

Rogue waves for the (2+1)-dimensional Myrzakulov–Lakshmanan-IV equation on a periodic background

In this paper, the rogue wave solutions of the (2+1)-dimensional Myrzakulov–Lakshmanan (ML)-IV equation, which is described by five component nonlinear evolution equations, are studied on a periodic background. By using the Jacobian elliptic function expansion method, the Darboux transformation (DT) method and the nonlinearization of the Lax pair, two kinds of rogue wave solutions which are expressed by Jacobian elliptic functions dn and cn, are obtained. The relationship between these five kinds of potential is summarized systematically. Firstly, the periodic rogue wave solution of one potential is obtained, and then the periodic rogue wave solutions of the other four potentials are obtained directly. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u=be−ax2, a<0, a,b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u=be−ax22, respectively, for the third-order PDE, u=AeiΦ, where the amplitude b and the phase function a are complex-valued functions, A>0, and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.

Lump, periodic, travelling, semi-analytical solutions and stability analysis for the Ito integro-differential equation arising in shallow water waves

Nonlinear evolution equations are typically used to model and analyse the majority of natural processes. These equations are extremely important for comprehending many intricate situations involving wave propagation, including heat transfer, fluid dynamics, optical fibres, electrodynamics, physics, chemistry, biology, condensed matter physics, ocean engineering, and many other nonlinear science fields. In this study, the anticipation of a bilinear Kdv model known as the (1+1)-dimensional integro-differential Ito equation is considered which depicts the oceanic shallow water wave dynamics. Firstly, we determine the bilinear form of the model and then we develop lump waves and analyse their interactions with periodic waves using symbolic computation approaches. Later, the interactions between single, double-kink and lump waves solutions, as well as interactions between periodic, lump and soliton solutions is considered. Furthermore, the (G′/G2) expansion approach and the Adomian technique are implemented to acquire a variety of novel configurations for the governing dynamical model. Later, the corresponding absolute accuracy is evaluated through a comparison of several types of observed solutions. Additionally, the stability of the solitary travelling wave solutions to the specified model is also established. The 3D, 2D, and contour plots have been employed to demonstrate the characteristics of the nonlinear model and aid in determine the proper set of parameters. As a result, a collection of options for bright, dark, periodic, rational, and elliptic functions is established. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the integro-differential equation appearing in shallow water waves. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.

Similarity Analysis of Lax Pairs for a Class of Nonlinear Evolution Equations.

Similarity Analysis of Lax Pairs for a Class of Nonlinear Evolution Equations.

Optical soliton solutions to the coupled Kaup-Newell equation in birefringent fibers

One of the fastest growing fields of study in nonlinear optics is optical solitons. This article investigates the optical soliton solutions of the coupled Kaup-Newell equation in birefringent fibers without four wave mixing. This model helps explain how modified structures, particularly Alfven waves, propagate in optical fiber and plasma physics. The results presented in this research clearly show how well the new mapping method handle nonlinear Kaup-Newell equation in coupled vector form, in birefringent fiber without four-wave mixing, to extract the analytical soliton solutions. Four-wave mixing expresses a nonlinear optical effect in which four waves interact each other as a result of the third order nonlinearity. As a valuable outcome of this article, a variety of solitons are discovered including bright, dark, periodic, singular bell-shaped, gap, flat-kink, dark-singular, and w-shaped. The existence criteria for these solitons are also obtained that are presented as constraint conditions. The constraint relations guarantee the existence of such solitons. Furthermore, graphs in 2D, 3D and contour are created to show the physical characteristics of the solutions that are produced. The new mapping method is clear and effective, and the solutions provide a wealth of avenues for additional research. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. The outcomes presented in this work are new versions which have never been observed before in existing literature.

Exclusive soliton solutions arise in mono-mode optical fibre connecting to nonlinear Fokas system

Nonlinear partial evolution equations are mostly significant to illustrate critical phenomena in wave theory concerning real-world problems. The current study deals with the (2 + 1)-dimensional nonlinear Fokas model depicting the nonlinear pulse through the mono-mode optical fibers. Improved auxiliary equation and improved tanh schemes are executed on the considering governing system. Subsequently, a variety of optical soliton solutions with the nature of dynamic nonlinear waves are made accessible throughout the present exploration. Some of constructed solutions are figured out in 3D, 2D and contour sense for the visualization to the readers for making them understand of the characteristics of dynamic waves. The solitons are visible to be bright, dark, kink, anti-kink, singular kink, periodic, compacton, anti-compacton etc. in the current exploration. Involved free parameters are assigned with various numerical values and brought out the effects of nonlinear pulses in wave propagation along mono-mode optical fibers. The entire work might claim to be recorded in the literature as new aspects of research.