ANALYSIS OF SOLITON PROFILES OF THE GENERALIZED P-TYPE EQUATION WITH CONFORMABLE DERIVATIVE VIA TWO DISTINCT APPROACHES

The paper analyzes the two-mode Sawada–Kotera equation with a conformable derivative, which describes nonlinear wave behaviour in fluid dynamics and optical systems. Using the extended direct mapping method, several soliton solutions are obtained. The significance of these solutions is highlighted through visual representations, including contour plots, as well as three-dimensional and two-dimensional graphs, depicting various soliton forms such as bell-shaped, bright, and dark solitons. Physically, the three-dimensional and two-dimensional plots provide a comprehensive visualization of the soliton's behaviour, capturing the amplitude and spatial-temporal evolution. The contour plot offers insights into the intensity distribution of the soliton across different spatial and temporal domains. This study analyzes the influence of the conformable fractional derivative and the temporal parameter in order to demonstrate their impact on the dynamical behaviour of soliton solutions. The Sawada-Kotera equation describes nonlinear wave phenomena in shallow water, ion-acoustic waves in plasma physics, and fluid dynamics.

New optical soliton solutions for time-fractional Kudryashov’s equation in optical fiber

The characteristics of fundamental and mutipole dark solitons in the nonlocal nonlinear couplers are studied through numerical simulation in this work. Firstly, the fundamental dark solitons with different parameters are obtained by the Newton iteration. It is found that the amplitude and beam width of the ground state dark soliton increase with the enhancement of the nonlocality degree. As the nonlinear parameters increase or the propagation constant decreases, the amplitude of the fundamental dark soliton increases and the beam width decreases. The power of the fundamental dark soliton increases with the nonlocality degree and nonlinear parameters increasing, and decreases with the propagation constant increasing. The refractive index induced by the light field decreases with the nonlocality degree increasing and the propagation constant decreasing. The amplitudes of the two components of the fundamental dark soliton can be identical by adjusting the coupling coefficient. These numerical results are also verified in the case of multipole dark solitons. Secondly, the transmission stability of fundamental and mutipole dark solitons are studied. The stability of dark soliton is verified by the linear stability analysis and fractional Fourier evolution. It is found that the fundamental dark solitons are stable in their existing regions, while the stable region of the multipolar dark solitons depends on the nonlocality degree and the propagation constant. Finally, these different types of dark dipole solitons and dark tripole solitons are obtained by changing different parameters, and their structures affect the stability of dark soliton. It is found that the multipole dark soliton with potential well is more stable than that with potential barrier. The refractive-index distribution dependent spacing between the adjacent multipole dark solitons favors their stability.

The present paper applies the exp (−φ(ξ))-expansion and the extended tanh-function methods to the (2+1)-dimensional Heisenberg Ferromagnetic Spin Chain (HFSC) equation. The applied methods acquire some new exact traveling wave solutions to the HFSC equation, which are representing the hyperbolic, trigonometric, exponential and rational function solutions. All solutions exhibit distinct physical configurations, such as the periodic, dark and singular soliton solutions. Three dimensional (3D) and two dimensional (2D) cross sectional graphics of some obtained solutions are confirmed the periodic, dark and singular behaviors. Furthermore, the conformable derivative is also considered and discussed for aforesaid methods to the HFSC equation. As outcomes, some new optical solutions are also attained in terms of fractionality. Obtained new solutions ensured that aforesaid methods are the reliable treatment for seeking nonlinear phenomena to HFSC equation as well as any NLEEs arising in mathematical physics and fiber optics.

In this paper, three eminent types of time-fractional nonlinear partial differential equations are considered, which are the fractional foam drainage equation, fractional Gardner equation, and fractional Fornberg–Whitham equation in the sense of conformable fractional derivative. The approximate solutions of these considered problems are constructed and discussed using the conformable fractional variational iteration method and conformable fractional reduced differential transform method. The conformable derivative is one of the admirable choices to handle nonlinear physical problems of different fields of interest. Comparisons of approximate solution obtained by two techniques, to each other and with the exact solutions are also presented and affirm that the considered methods are efficient and reliable techniques to study other nonlinear fractional equations and models in the sense of conformable derivative. To explain the effects of several parameters and variables on the movement, the approximate results are shown in tables and two-and three-dimensional surface graphs.

This paper presents a new study that incorporates the Stratonovich integral and conformal fractional derivative into the fractional stochastic Bogoyavlenskii equation with multiplicative noise. The study exposes the behavior of the system, including sensitivity, chaos and traveling wave solutions, by using the planar dynamical systems approach. Time series, periodic perturbation, phase portraits, and the Poincaré section are used to comprehensively study the dynamic properties. Notably, the research uses the planar dynamic systems method to build multiple traveling wave solutions, including kink wave, dark soliton, and double periodic solutions. Furthermore, a comparative study approach is applied to investigate the effects of fractional derivative and multiplicative noise on the traveling wave solutions, which demonstrate a significant influence of both variables. This work demonstrates the creative application of the planar dynamic system method to the analysis of nonlinear wave equations, offering insightful information that may be generalized to more complex wave phenomena.

This paper investigates the time-fractional extended (3+1)-dimensional nonlinear conformable Schrödinger equation in a dispersion and nonlinearity managed fiber laser. By utilizing the new direct mapping method and the unified Riccati equation technique, we derive various optical soliton solutions for the nonlinear Schrödinger equation with conformable derivative. The importance of these newly constructed soliton solutions is demonstrated through contour, three-dimensional, and two-dimensional graphs, demonstrating dark, bell-shaped, bright, and wave soliton formation. Further, the effect of the fractional order parameter and the temporal parameter on these solutions is analyzed, offering valuable insights into the conformable nonlinear Schrödinger model. The algorithms developed in this study hold promise for broader application across various nonlinear Schrödinger equations in fields such as nonlinear optics and applied mathematics. Ultimately, this study deepens our understanding of nonlinear optics by uncovering new soliton dynamics and their governing mechanisms within dispersion and nonlinearity-managed fiber laser systems.

This study employs the unified method, a powerful approach, to address the intricate challenges posed by fractional differential equations in mathematical physics. The principal objective of this study is to derive novel exact solutions for the time-fractional thin-film ferroelectric material equation. Fractional derivatives in this study are defined using the conformable fractional derivative, ensuring a robust mathematical foundation. Through the unified method, we derive solitary wave solutions for the governing equation, which models wave dynamics in these materials and holds significance in various fields of physics and hydrodynamics. The behavior of these solutions is analyzed using the conformable derivative, shedding light on their dynamic properties. Analytical solutions, formulated in hyperbolic, periodic, and trigonometric forms, illustrating the impact of fractional derivatives on these physical phenomena. This paper highlights the capability of the unified method in tackling complex issues associated with fractional differential equations, expanding both mathematical techniques and our understanding of nonlinear physical phenomena.

This paper presents travelling wave solutions for the nonlinear time-fractional Gardner and Benjamin–Ono equations via the exp( $$- \Phi ( \varepsilon ))$$ -expansion approach. Specifically, both the models are studied in the sense of conformable fractional derivative. The obtained travelling wave solutions are structured in rational, trigonometric (periodic solutions) and hyperbolic functions. Further, the investigation of symmetry analysis and nonlinear self-adjointness for the governing equations are discussed. The exact derived solutions could be very significant in elaborating physical aspects of real-world phenomena. We have 2D and 3D illustrations for free choices of the physical parameter to understand the physical explanation of the problems. Moreover, the underlying equations with conformable derivative have been investigated using the new conservation theorem.

Analytic solutions for a modified fractional three wave interaction equations with conformable derivative by unified method

Temporal behavior of bright and dark spatial solitons in photorefractive crystals having both the linear and quadratic electro-optic effects based on low amplitude approximations

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.

This article suggests solving the traveling wave solutions of the time.fractional Kaup-Kupershmidt (KK) equation via 1/ G -expansion and sub-equation methods. Non-local fractional derivatives have some advantages over local fractional derivatives. The most important of these advantages are the chain rule and the Leibniz rule. The conformable derivative, which has a local fractional derivative feature, is taken into account in this study. Different types of traveling wave solutions of the time-fractional KK equation have been produced by using the important benefits of the time-dependent conformable derivative operator. These wave types are dark, singular, rational, trigonometric and hyperbolic type solitons. 2D, 3D and contour graphics are presented by giving arbitrary values to the constants in the solutions produced by analytical methods. These presented graphs represent the shape of the standing wave at any given moment. Besides, the advantages and disadvantages of the two analytical methods are discussed and presented in the result and discussion section. In addition, wave behavior analysis for different velocity values of the dark soliton produced by the analytical method is analyzed by simulation. The conditional convergence and asymptotic stability of the dark soliton discussed are analyzed. Computer software is also used in operations such as drawing graphs, complex operations, and solving algebraic equation systems.

Modified extended mapping method is further modified to discover traveling wave solutions of non- linear complex physical models, arising in various fields of applied sciences. The method is applied to three-dimensional ZakharovKuznetsov-Burgers equation in magnetized dusty plasma. Consequently different kinds of families of exact traveling wave solutions that represent electric field potential, electric and magnetic fields are fruitfully surveyed, with the help of Mathematica. The obtained novel exact traveling wave solutions are in different forms such as bright and dark solitary wave, periodic solitary wave, dark and bright soliton, etc., that are represented in the forms of trigonometric, hyperbolic, exponential and rational functions. The properties of some of the novel traveling wave solutions are shown by figures. The obtained results exhibit the effectiveness, power and exactness of the method that can be used for many other nonlinear problems.

The aim of this work is to investigate the Wick-type stochastic nonlinear evolution equations with conformable derivatives. The general Kudryashov method is improved by a new auxiliary equation. So, a new technique, which we call “the general improved Kudryashov method (GIKM)”, is introduced to produce exact solutions for the nonlinear evolution equations with conformable derivatives. By means of GIKM, white noise theory, Hermite transform, and computerized symbolic computation, a novel technique is presented to solve the Wick-type stochastic nonlinear evolution equations with conformable derivatives. This technique is applied to construct exact traveling wave solutions for Wick-type stochastic combined KdV–mKdV equation with conformable derivatives. Moreover, numerical simulations with 3D profiles are shown for the obtained results.