Nonlinear Evolution Equations Research Articles

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.

Wave propagation analysis in the modified nonlinear time fractional Harry Dym equation: Insights from Khater II method and B-spline schemes

This study aims to investigate the modified nonlinear time fractional Harry Dym equation using analytical and numerical techniques. The modified nonlinear time fractional Harry Dym equation is a generalization of the classical Harry Dym equation, which describes the propagation of nonlinear waves in a variety of physical systems. The conformable fractional derivative is used to define the time fractional derivative in the equation, which provides a natural and straightforward approach. The Khater II method, a powerful analytical technique, is employed to obtain approximate solutions for the equation. Additionally, three numerical schemes, namely, Cubic-B-spline, Quantic-B-spline and Septic-B-spline schemes, are developed and implemented to solve the equation numerically. The numerical results are compared with other numerical solutions to assess the accuracy and efficiency of the proposed schemes. The physical meaning of the modified nonlinear time fractional Harry Dym equation is discussed in detail, and its relation to other nonlinear evolution equations is highlighted. The results of this study provide new insights into the behavior of nonlinear waves in physical systems and contribute to a better understanding of the physical characterizations of the modified nonlinear time fractional Harry Dym equation.

Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation

We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns using analytical or semi-analytical approach. The interaction of quasiparticles stems from nonlocal competitive losses in the FKPP model. We developed the formalism of our approach relying on ideas of the Maslov method. The construction of the asymptotic expansion of a solution to the original nonlinear evolution equation is based on solutions to an auxiliary dynamical system of ODEs. The asymptotic solutions for various specific cases corresponding to various spatial profiles of the reproduction rate and nonlocal competitive losses are studied within the framework of the approach proposed.

On new computations of the time-fractional nonlinear KdV-Burgers equation with exponential memory

This paper examines the Korteweg–de Vries-Burgers (KdV-Burgers) equation with nonlocal operators using the exponential decay and Mittag-Leffler kernels. The Caputo-Fabrizio and Atangana-Baleanu operators are used in the natural transform decomposition method (NTDM). By coupling a decomposition technique with the natural transform methodology, the method provides an effective analytical solution. When the fractional order is equal to unity, the proposed approach computes a series form solution that converges to the exact values. By comparing the approximate solution to the precise values, the efficacy and trustworthiness of the proposed method are confirmed. Graphs are also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both operators outcome is examined using diagrams and numerical data. These graphs show how the approximated solution’s graph and the precise solution’s graph eventually converge as the non-integer order gets closer to 1. The outcomes demonstrate the method’s high degree of accuracy and its wide applicability to fractional nonlinear evolution equations. In order to further explain these concepts, simulations are run using a computationally packed software that helps interpret the implications of solutions. NTDM is considered the best analytical method for solving fractional-order phenomena, especially KdV-Burgers equations.

Non‐singular complexiton, singular complexiton and complex multiple soliton solutions to the (3 + 1)‐dimensional nonlinear evolution equation

This research mainly concerned with some new solutions of the (3 + 1)‐dimensional nonlinear evolution equation (NEE). First, we extract the resonant multiple soliton solutions (RMSSs) by taking advantage of the linear superposition principle (LSP) and weight algorithm (WA). Then the non‐singular complexiton and singular complexiton solutions are developed by introducing pairs of the conjugate parameters. Besides, the complex multiple‐soliton solutions are also explored with the aid of the bilinear approach. The graph descriptions of the attained solutions are paraded to show the dynamical properties. The outcomes of this work are all new and are desirous to bring some new perspective to the investigation of the complexiton solutions to the other partial differential equations (PDEs).

Wave analysis in generalized fractional Tzitzéica-type nonlinear PDEs: Contributions to nonlinear sciences

In this paper, the extended direct algebraic approach with the general fractional derivative is employed to attain various novel wave structures of the non-linear fractional Tzitzéica type non-linear evolution equations in the form of kink, singular, periodic singular, dark, bright and dark-bright combine solitons. For the purpose to illustrate the physical behavior of the acquired solutions, some of the extracted results are sketched in the pattern of 3-D and 2-D plots which show the efficiency and authenticity of the proposed method. The under consideration method can also utilize to any other non-linear model appears in optics and engineering.

Mild solutions to a class of nonlinear second order evolution equations

The purpose of this paper is to study the existence of mild solutions to a class of second order nonlinear evolution equations of the form \begin{equation*} \begin{cases} u''(t)+A(u'(t))+B(u(t))\ni f(t), &t\in(0,T),\\ u(0)=u_0, \quad u'(0)=g(u') \end{cases} \end{equation*} where $A\colon D(A)\subseteq X\rightarrow 2^{X}$ is an $m$-accretive operator on a Banach space $X,$ $B: X\rightarrow X$ is a lipschitz mapping, $g\colon C([0,T];X)\to X$ is a function and $f\in L^1(0,T,X)$. We obtain sufficient conditions for this problem to have at least a mild solution.

Bifurcation analysis and exact wave solutions of the nano-ionic currents equation: Via two analytical techniques

In this work, numerous nonlinear evolution equations, namely the nano-ionic current equation, have wide applications in various fields of science and technology. To achieve this, we employ the generalized auxiliary equation and extended hyperbolic function techniques to investigate the exact wave solutions of the model under consideration. We obtained a range of exact wave solutions, including bright, dark, singular-periodic, periodic, and w-shape solutions, for the nonlinear model under examination, as well as rational and exponential function solutions with various structures, which are explored in our manuscript. In this paper, we compared the solutions obtained using the GAE and EHF approaches and also compared the obtained solutions using the GAE scheme with the solutions reported in Aljahdaly et al. [43]. Moreover, the three-dimensional and contour graphs are included to illustrate the structure of some obtained solutions by selecting specific parameter values. Bifurcation analysis is illustrated by phase plane analysis; phase plots are employed to depict the dynamic structure and are also analyzed for the existence of other solutions. These techniques serve as reliable, simple, and potent tools to analyze various nonlinear evolution equations found in physics, applied mathematics, and engineering.

Novel solitonic structure, Hamiltonian dynamics and lie symmetry algebra of biofilm

In this study, the Lie point symmetries and optimal system have been established. We discuss the biofilm model’s soliton solutions. To examine a nonlinear dynamical biofilm system, which is simply a bistable Allen–Cahn equation with quartic potential, to determine solitary wave profiles using the considered equation, a new auxiliary equation technique is used. A suitable variable transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The unified approach is utilized to evaluate periodic solutions, solitary and soliton solutions as well as several newly discovered exact solitary wave solutions, it can be accomplished via Mathematica (https://www.wolfram.com/mathematica/online/). The new auxiliary equation approach is an efficient method for creating unique wave profiles based on a variety of soliton families. Also, the results are graphically visualized, by using the appropriate parametric settings. The outcomes are shown graphically in two dimensions, three dimensions, and contour form. The Hamiltonian conditions are satisfied by the planer dynamical framework of equations to ensure as the system that was generated is a conservative Hamiltonian dynamical system which includes all traveling wave structures. A sensitivity evaluation is used to explore the governing model’s thoroughly dynamical properties. It is demonstrated that the model becomes more dependent on the beginning conditions than the variables. The strategy could be used to look for exact solutions to various nonlinear evolution equations. We believe that this study will have important applications in different areas of science.