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Introduction
Lie symmetry method
Geometric integration
Meshfree methods
Publications
Publications (168)
This paper extensively studies the propagation of optical solitons within the framework of (2 + 1)-dimensional generalized coupled nonlinear Schrödinger equations. The investigation employs three worldly integration techniques: the enhanced direct algebraic method, the enhanced Kudryashov method, and the new projective Riccati equation method. Thro...
This study introduces some novel soliton solutions and other analytic wave solutions for the highly dispersive perturbed nonlinear Schrödinger equation with generalized nonlocal laws and sextic-power law refractive index. This equation models the behavior of complex wave propagation. The improved modified extended tanh function governs the transmis...
In this paper, we study the Schrödinger–Hirota equation in birefringent fibers incorporating cubic–quartic dispersion. The model studied comes with a cubic–quintic nonlinear structure. The governing model introduced in this study is novel and original, and the obtained solitons have not been reported before. The Schrödinger–Hirota equation is an im...
The Ito equation belongs to the Korteweg–de Vries (KdV) family and is commonly employed to predict how ships roll in regular seas. Additionally, it characterizes the interaction between two internal long waves. In the 1980s, Ito extended the bilinear KdV equation, resulting in the well-known (1+1)-dimensional and (2+1)-dimensional Ito equations. In...
This paper explores innovative solutions for the Stochastic Schrödinger-Hirota equation within the context of birefringent fibers with cubic-quintic nonlinearity, emphasizing incorporating multiplicative white noise in the Itô sense. Leveraging the Nucci reduction method, the study focuses on obtaining exact solutions, shedding light on the intrica...
This article explores a wide range of envelope soliton pulses and their behavior during propagation through a birefringent optical fiber. The propagation of light in such fibers is governed by two coupled non-linear Schrödinger equations (CNLSEs), which account for both coherent and incoherent non-linear couplings. To investigate these solitons, an...
In this paper, a nonlinear mechanical system of ordinary differential equations (ODEs) with multi-point boundary conditions is considered by a novel type of reproducing kernel Hilbert space method (RKHSM). To begin, we define the unknown variables in terms of the reproducing kernel function. The roots of the Shifted Chebyshev polynomials (SCPs) are...
This paper introduces the Nucci reduction method, a novel and efficient approach for deriving exact solutions to the perturbed Gerdjikov–Ivanov equation, offering a significant advancement in the field. The suggested technique involves transforming the equation into real and imaginary components prior to application. We successfully obtained four d...
This study investigates optical solitons and other traveling wave solutions in fiber Bragg gratings for coupled nonlinear Schrödinger equations with cubic quadratic nonlinearity using the improved modified extended tanh-function method (IMETFM). We discover a highly dispersive solitons in fiber Bragg such as bright solitons, dark solitons, combo br...
In this paper, we investigate the conformable derivative effect on the travelling wave solutions for extended \((3+1)\)-dimensional Kudryashov’s equation with generalized anti-cubic nonlinearity. To accomplish this, we employ the improved modified extended tanh-function method, which enables us to obtain diverse optical soliton solutions. A variety...
This study examines the analytic wave solutions of a highly dispersive perturbed complex Ginzburg–Landau equation (CGLE) with conformable fractional derivative and polynomial law of nonlinearity using the improved modified extended tanh-function method. The results show a wide range of solutions including (bright, dark, singular) solitons, Jacobi e...
This study addresses the modeling pulse propagation in optical fibers, focusing on a coupled system of nonlinear Schrödinger’s equation for the generalized Kudryashov’s equation in a magneto-optic waveguide. A magneto-optic waveguide is a waveguide that uses the magneto-optic phenomenon to manipulate the movement of light. This phenomenon entails t...
The presented research investigates the (2+1)-dimensional perturbed nonlinear Schrödinger model. This model takes into account various effects such as fourth order dispersion, intermodal dispersion, nonlinear dispersion, group velocity dispersion, Kerr nonlinearity and self steepening effects. This model simulates the estimation of optical solitons...
Studying the highly dispersive model with arbitrary refractive index initiated by Kudryashov is the subject of this work. This model depicts the behavior of soliton propagation via polarization-preserving fibers. To secure a dark soliton and like-solitons with singularities solution for the proposed model, the generalized Kudryashov’s integration a...
This study centers on examining the characteristics of the recently extended (3+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3+1)$$\end{document}-dimensional nonl...
In this paper, (1+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+1)$$\end{document}-dimensional Schrödinger–Hirota equation with Kerr law having inter-modal dispe...
This article investigates the dynamics of optical solitons in a magneto-optic waveguide with the nonlinear perturbed Schrödinger equation. The model incorporates two generalized nonlocal laws, a Kudryashov’s sextic-power law nonlinear structure, and high dispersion up to the sixth-order dispersion. The proposed innovative model ensures that all the...
In this study, the space-time fractional generalised reaction duffing model is investigated analytically, which is a generalization for a collection of prominent fractional models describing various key phenomenon in science and engineering. The governing equation is converted to a nonlinear ODE by the compatible travelling wave transform. The inve...
The nonlinear Schrödinger equation (NLSE) is a fundamental equation in the field of nonlinear optics and plays an important role in the study of many physical phenomena. The present study introduces a new model that demonstrates the novelty of the paper and provides the advancement of knowledge in the area of nonlinear optics by solving a challengi...
Our goal in this paper is to obtain the travelling wave solutions of the (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili equation (DSKPE), which considered as a fundamental tool in studying a variety of phenomena appear in many applications of fluid mechanics and ocean engineering. This equation is used to describe the non-elastic and ela...
In this work, we investigate the optical solitons and other waves through magneto-optic waveguides with Kudryashov’s law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation factors using the modified extended mapping approach. Many classifications of solutions are established like bright solitons,...
This article extracts analytical solutions of the Fokas–Lenells equation describing pulse propagation in optical fibers and presents a comprehensive analysis of the analytical solutions. This paper focuses on deriving analytical, non‐perturbative solutions for the Fokas–Lenells equation by reducing it to a system of ordinary differential equations...
The newly developed (3+1)-dimensional nonlinear Kudryashov’s equation offers a framework for studying the behavior of propagating modulated envelope signals and is particularly useful in the fields, where the propagation of wave-like phenomena is essential
such as in the fled of fluid dynamics, it helps to analyze phenomena like water waves
and f...
The main purpose of this study is to reach the exact solutions of the p-forced nonlinear Klein–Gordon equation by using a novel method called the Nucci reduction method. The nonlinear Klein–Gordon equation finds applications in various real-world scenarios. One notable application is in the field of nonlinear optics, where the equation is used to s...
The Newell-Whitehead-Segel (NWS) model is a reaction-diffusion system that has been widely used to study pattern formation in biological and physical systems. In this paper, we present a powerful method for obtaining exact solutions of the stochastic NWS model by applying Nucci’s reduction method. The method involves transforming the original stoch...
In this work, a well-known non-homogeneous wave equation with temporal fractional derivative is approximately investigated. A recently defined generalized non-local fractional derivative is utilized as the fractional operator. A novel technique is proposed to approximate the solutions of wave equation with generalized fractional derivative. The pro...
In the present paper, a model of the gravitational waves of water known as the (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2 + 1)$$\end{document}-dimensional...
In this study, we investigated Schrödinger–Hirota equation Kerr law in the presence of spatio-temporal dispersion which describes pulse propagation in optical fibers provoked by nonlinear effect and reduced to multi-plane dispersion. We used the Lie symmetry method to reduce the model into the nonlinear ordinary differential equations. Moreover, th...
The study focuses on the combination of three mathematical models, namely nonlinear Schrödinger’s equation, Lakshmanan–Porsezian–Daniel model, and Sasa–Satsuma model, which is called the concatenation model. The study investigates optical solitons and other exact solutions using a modified extended mapping approach. The solutions derived include br...
The improved modified extended tanh‐function approach was used to study optical stochastic soliton solutions and other exact stochastic solutions for the nonlinear Schrödinger‐Hirota equation with multiplicative white noise. The derived solutions include stochastic bright solitons, stochastic singular solitons, stochastic periodic solutions, stocha...
The current research investigates the behavior of femtosecond solitary waves in an inhomogeneous optical fiber using the generalized derivative nonlinear Schrödinger equation with quintic nonlinearity. The extended F-expansion technique is utilized to obtain various exact solutions such as bright soliton solutions, dark soliton solutions, combo bri...
The present paper aims to investigate the chirped optical soliton solutions of nonlinear Schr¨odinger equation with nonlinear chromatic dispersion and quadratic-cubic law of refractive index. The exquisite balance between the chromatic dispersion and the nonlinearity
associated with the refractive index of a fiber gives rise to optical solitons, wh...
The current paper raises notice of a novel approach known as the “heir-equations method.” Nucci, for the first time in the literature, discovered that iterating the nonclassical symmetry method to generate new nonlinear equations secures the Lie point symmetry properties of the given equation, or, in other words, inherits them. The technique that c...
This paper investigates the (2+1)-dimensional Korteweg-de Vries (KdV) equation with a local M-derivative and beta derivative in the time direction. Investigation on the (2+1)-dimensional Korteweg-de Vries (KdV) equation with a local M-derivative and beta derivative in the time direction is important because it provides insights into the behavior of...
This paper discusses the existence of a diverse range of novel periodic nonlinear waves in the generalized (3+1)-dimensional Sasa–Satsuma equation. This equation models the transmission of femtosecond light pulses through optical fibers, taking into account third-order dispersion, self-frequency shift, and self-steepening effects in all three spati...
This paper focuses on the use of the Sardar sub-equation method to obtain optical solitons for a nonlinear model that incorporates Kudryashov’s quintuple power law and dual-form nonlocal nonlinearity. The model studies the propagation of pulses in optical fibers and the proposed technique is used to investigate various types of solitons including b...
This paper explores the interactions of capillary–gravity waves by presenting novel exact solutions to a coupled Schrödinger–KdV equation. The authors introduce a reduction method based on Nucci’s approach to derive these exact solutions. The results provide a new approach to solving the Schrödinger–KdV equation, offering potential applications in...
In this paper, we study the extended (3+1)-dimensional nonlinear conformable Schrödinger equation with cubic–quintic nonlinearity. We use three different methods to obtain exact solutions of this equation: the G′/G expansion method, the extended hyperbolic method, and Nucci’s reduction method. We show that these methods are effective in finding sol...
In this study, the Nucci's reduction approach and the method of generalized projective Riccati equations (GPREs) were utilized to derive novel analytical solutions for the (1+1)-dimensional classical Boussinesq equations, the generalized reaction Duffing model, and the nonlinear Pochhammer-Chree equation. The nonlinear systems mentioned earlier hav...
The objective of this study is to investigate a few solutions to the nonlinear Schrödinger problem with parabolic law. The first integral and exact solutions for the reduced ODE of the model under consideration are extracted using Nucci’s reduction approach. Finally, using the efficient and effective solutions technique, we display density plots an...
In mathematics, physics, and engineering, establishing numerical or analytical solutions for fractional mathematical models for specific phenomena and developing fractional mathematical models for specific phenomena are important topics. In this work the (G′G)-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfo...
In this study, our focus is on the construction of novel soliton solutions of non-linear Schrödinger equation with parabolic law and non-local law non-linearities via new extended hyperbolic function method. The acquired solutions are original and could be helpful in the field of nonlinear optics, fluid dynamics, and plasma physics. From these outc...
This study presents a novel reduction method to obtain exact solutions for the behavior of a porous fin under a uniform magnetic field, considering the effects of convection, radiation, and internal heat generation. The study applies the reduction method to simplify the governing equations and reduce the complexity of the mathematical model, making...
The existence of solutions of coupled system of bi-harmonic Schrödinger equations as fixed point of an operator has been given in this article. The corresponding estimates for the length of continuity of solutions have been constructed. By applying new ϕ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \...
The current work deals with the classical and non-classical Lie group analysis for the fractional type system of nonlinear Chen–Lee–Liu (CLL) equations. The invariance surface condition is imposed to this system of fractional differential equations to find non-classical generators. Non-classical and classical Lie symmetry analysis are used to simil...
The main purpose of this study is to introduce symmetry analysis and Nucci’s reduction method for revealing the exact solutions of the periodic Hunter–Suxon equation, which is parameterized by the speed of the Galilean frame. Under one-point similarity transformations, we obtained a set of generators that transformed the given equation into an ordi...
This paper presents an analytical treatment of the perturbed nonlinear Schrödinger equation (P-NLSE) using the Lie symmetry method. The NLSE is a fundamental equation in nonlinear optics and describes the propagation of optical pulses in a nonlinear medium. In the presence of perturbations, the solution to the NLSE becomes more complicated and requ...
In this paper, we provide the Noether symmetries with gauge fields of the area-minimizing hypersurface Lagrangian according to some vacuum classes of pp-waves. Also we show that they are elements of the Killing algebra of vacuum classes of pp-waves. Finally, we construct the conserved fields for the area-minimizing Lagrangian according to Noether’s...
In this study, a general second-order evolution equation of the Fisher-type, with time-dependent variable coefficients, is considered. This equation contains many well-known equations, and obtained results may be applicable in investigating other evolution equations. Lie symmetries and corresponding invariant solutions of the considered problem are...
The purpose of this work is to seek various innovative exact solutions using the new Kudryashov approach to the nonlinear partial differential equations (NLPDEs). This technique obtains novel exact solutions of soliton types. Moreover, several 3D and 2D plots of the higher dimensional Klein-Gordon, Kadomtsev-Petviashvili, and Boussinesq equations a...
In this paper, the symmetry reduction method and Nucci’s reduction method are used to obtain exact solutions to the Triki–Biswas equation. Furthermore, the new conservation theorem is utilized for finding the conservation laws of the given model. The conservation laws are derived for each admitted symmetry of the Triki–Biswas equation and satisfy t...
This work is devoted to the time-fractional differential equations with the regularized Prabhakar derivative and their analytical solutions. We generalize the invariant subspace method to find the exact solutions of such equations. Then, we apply this method to obtain the exact solutions of different time-fractional nonlinear differential equations...
In this research, we study the numerical solution of the singular Abel’s equation of the second kind. Solving this equation is challengeable, because of the nonlinear and singularity. For this purpose, we present an efficient algorithm based on the Galerkin method using biorthogonal Hermite cubic spline multiwavelets (BHCSMWs). Because of the spars...
In this study, we handled the solitary waves of the generalized nonlinear wave equation in (3+1) dimensions with gas bubbles because liquids with gas bubbles are widespread in engineering, science, life, nature, and physics. The considered model is solved by the enhanced and modified Kudryashov’s methods for obtaining the solitary waves. Nucci’s di...
In this work, we use the symmetry of the Lie group analysis as one of the powerful tools that deal with the wide class of fractional order differential equations in the Riemann-Liouville concept. We employ the classical Lie symmetries to obtain similarity reductions of nonlinear time-fractional Benjamin-Ono equation and then, we find the related ex...
Fractional derivatives are significant mathematical instruments that have been practiced to model real phenomena in different areas of science. Through the current investigation study, we develop the Human Liver and Hearing Loss models by employing three fractional operators called Atangana–Baleanu–Caputo, Caputo and Caputo–Fabrizio derivatives. Th...
The current work is based on performing the non‐classical and classical Lie group analysis for a system of nonlinear time‐fractional Heisenberg equation. Here, we apply analysis of the Lie group symmetry as one of the powerful tools dealing with the large class of fractional order differential equations in the Riemann–Liouville (RL) sense. Indeed,...
This manuscript deals with the existence, uniqueness and stability of solutions to the boundary value problem (BVP) of Riemann-Liouville (RL) fractional differential equations (FDEs) in the variable exponent Lebesgue spaces (Lp(.)). The generalized intervals and piece-wise constant functions are utilized to extract the aims of current paper. The va...
In this work, we consider the generalized nonlinear dispersive mK(m,n) equation with a recently defined local derivative in the temporal direction. Different types of exact solutions are extracted by Nucci’s reduction technique. Combinations of the exponential, trigonometric, hyperbolic, and logarithmic functions constitute the exact solutions espe...