Drinfeld Sokolov Research Articles

Residual symmetry and interactions between solitons and periodic waves of the Drinfeld–Sokolov–Satsuma–Hirota system

In this paper, the Drinfeld–Sokolov–Satsuma–Hirota (DSSH) system is studied by using residual symmetry and the consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the DSSH system is localized to Lie point symmetry in a properly prolonged system, based on which we get a new Bäcklund transformation for this system. New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system. Moreover, the DSSH system proves to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.

N = 2 supersymmetric extension on multi-component D type Drinfeld–Sokolov hierarchy

In this paper, we define a N=2 multi-component supersymmetric B type 2KP(MS2BKP) hierarchy and its additional symmetries. These additional flows form a B type multi SW1+∞ Lie algebra. Under a reduction, we derive a N=2 multi-component supersymmetric D type Drinfeld–Sokolov hierarchy which has a multi super Virasoro algebraic structure.

Negative flows for several integrable models

A construction of negative flows for integrable systems based on the Lax representation and squared eigenfunctions is proposed. Examples considered include the Boussinesq equation and its reduction to the Sawada–Kotera and Kaup–Kupershmidt equations; one of the Drinfeld–Sokolov systems and its reduction to the Krichever–Novikov equation.

Computational approach and dynamical analysis of multiple solitary wave solutions for nonlinear coupled Drinfeld–Sokolov–Wilson equation

In this article, the coupled Drinfeld–Sokolov–Wilson equation under investigation base on computational approach named extension of modified rational expansion approach. By this computational approach secured various kinds of solitons named kink solitons, dark solitons, anti kink solitons, periodic solitons, bright solitons, kink dark solitons, kink bright solitons, anti kink dark solitons, anti kink bright solitons for coupled DSW system. The calculated results are interesting, precise, novel, and different which have not been secured in the previous research. We represent the graphical behaviour of determined solutions by 2-D, 3-D and contour through computational software. The determined results may helpful and play significant role in the study of nonlinear behaviour in various branches of physical sciences such as laser optics, communication system, fluid dynamics, transfer of heat, shallow water waves, acoustics, optics and many others. The constructed results prove that proposed computational approach is concise, powerful, and straightforward for securing the solitons results of other nonlinear partial differential equations.

A Comparative Study of Time Fractional Nonlinear Drinfeld–Sokolov–Wilson System via Modified Auxiliary Equation Method

The time-fractional nonlinear Drinfeld–Sokolov–Wilson system, which has significance in the study of traveling waves, shallow water waves, water dispersion, and fluid mechanics, is examined in the presented work. Analytic exact solutions of the system are produced using the modified auxiliary equation method. The fractional implications on the model are examined under β-fractional derivative and a new fractional local derivative. Extracted solutions include rational, trigonometric, and hyperbolic functions with dark, periodic, and kink solitons. Additionally, by specifying values for fractional parameters, graphs are utilized to comprehend the fractional effects on the obtained solutions.

Analysis of the exact solutions of nonlinear coupled Drinfeld–Sokolov–Wilson equation through [formula omitted]-model expansion method

The coupled Drinfeld–Sokolov–Wilson (DSW) equation is a system of nonlinear partial differential equations (NLPDEs) in (1+1)-dimensions that play a fundamental role in soliton theory and integrable systems. Due to its ability to accurately describe wave phenomena in dispersive water waves, coupled DSW equation has wide-ranging applications in fluid dynamics, plasma physics, and mathematical physics. This study investigates the solitary wave solution of coupled DSW equation by using ϕ6-model expansion. This technique reveals the diverse types of solutions, including bright, dark, singular, and periodic singular solitons. The 3D, contour, and 2D plots are also illustrated to demonstrate the physical behavior of the obtained solutions. The findings of this study can contribute to the development of new analytical and numerical tools for solving other nonlinear equations in the future. The inclusion of constraint conditions in the ϕ6-model expansion technique enhances its applicability and reliability, rendering it a valuable approach for investigating nonlinear systems in various scientific domains.

Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water waves

We adopted the Lie symmetry technique to obtain some new exact waveform solutions for the coupled Drinfeld Sokolov Wilson (DSW) system. This system emerges from phenomena such as water waves, theoretical physics, fluid dynamics, biology, and chemical sciences. We use the Jacobi elliptic function (JEF) method to deal with some of the reduced systems during the symmetry reductions, which reveals rational, exponential, and hyperbolic functions based waveform solutions. The properties of these solutions indicate that these are periodic waves, solitary waves, double periodic waves, and shock wave solutions. Several of the answers that have been found are entirely new and might be important for researchers to recognize. By assigning specific values to parameters, some chosen solutions are graphically validated. This method can be applied more broadly to handle several different types of nonlinear evolution systems.

Analysis and simulation of arbitrary order shallow water and Drinfeld–Sokolov–Wilson equations: Natural transform decomposition method

Within the context of fractional calculus, we investigate novel mathematical possibilities. In this context, using the fractional dispersion relations for the fractional wave equation, we explore a class of the generalized fractional wave equation numerically. Some important classes of differential equations in the theory of wave studies are Drinfeld–Sokolov–Wilson and Shallow Water equations. In this effort, the natural transform decomposition technique has been implemented to investigate the explicit result of fractional-order coupled schemes of Drinfeld–Sokolov–Wilson and Shallow Water coupled systems. The proposed method is obtained by coupling the Natural transform with the Adomian decomposition process. The current technique significantly works to find the approximate solution without any discretization or constraining parameter assumptions. The obtained numerical and graphical outcomes by the devised technique are compared with the available exact result to verify the convergence of the method. For mathematical calculations, the Mathematica software package is used.

Investigating the Dynamics of Time-Fractional Drinfeld–Sokolov–Wilson System through Analytical Solutions

This study addresses a nonlinear fractional Drinfeld–Sokolov–Wilson problem in dispersive water waves, which requires appropriate numerical techniques to obtain an approximative solution. The Adomian decomposition approach, the homotopy perturbation method, and Sumudu transform are combined to tackle the problem. The Caputo manner is used to describe fractional derivative, and He’s polynomials and Adomian polynomials are employed to address nonlinearity. By following these approaches, we obtain solutions in the form of convergent series. We verify and demonstrate the effectiveness of our suggested strategies by examining the assumed model in terms of fractional order. We use plots for various fractional orders to represent the physical behavior of the suggested technique solutions, and show a numerical simulation. The results demonstrate that the suggested algorithms are systematic, simple to use, effective, and accurate in analyzing the behavior of coupled nonlinear differential equations of fractional order in related scientific and engineering fields.

CONSTRUCTION OF FRACTAL SOLITON SOLUTIONS FOR THE FRACTIONAL EVOLUTION EQUATIONS WITH CONFORMABLE DERIVATIVE

In this paper, the fractional evolutions are described by using the conformable derivative for the first time. We implement fractional functional variable method (FFVM) to obtain some new kinds of fractal soliton wave solutions for these fractional evolution equations. The simplicity and effectiveness of this proposed method are tested on the fractional Drinfeld–Sokolov system and fractional cubic Klein–Gordon equation. The FFVM provides a new perspective to construct exact fractal soliton wave solutions of complex fractional nonlinear evolution equations in mathematical physics.

A New Study for the Investigation of Nonlinear Fractional Drinfeld–Sokolov–Wilson Equation

In this paper, we present an idea of a new iterative procedure (NIP) to examine the approximate solution of the nonlinear fractional Drinfeld–Sokolov–Wilson (DSW) equation. We first use Mohand transform (MT) to the problem and obtain a recurrence relation without any assumption or restrictive variable. This relation is now very easy to handle and suitable for the study of the homotopy perturbation method (HPM). We observe that HPM produces the iterations in the form of convergence series that becomes very close to the precise solution. The fractional derivative is considered in the Caputo sense. We also demonstrate the graphical solution to show that NIP is a very simple, straightforward, and efficient tool for nonlinear problems of fractional derivatives.

A NEW FRACTAL TRANSFORM FOR THE APPROXIMATE SOLUTION OF DRINFELD–SOKOLOV–WILSON MODEL WITH FRACTAL DERIVATIVES

This study examines the development of a novel approach known as the fractal Elzaki transform method (F[Formula: see text]TM) to investigate the approximation solution of the nonlinear fractal Drinfeld–Sokolov–Wilson (NFDSW) model. We adopt He’s fractal derivative to change the fractal model into its differential parts and then apply the Elzaki transform to obtain the recurrence relation. We utilize the framework of homotopy perturbation method to handle the nonlinear components of this recurrence relation and thus we can obtain the successive iterations very easily. The derived findings are performed in the form of series and the rate of convergence shows the remarkable solutions due to its fast convergence. The numerical example illustrates that F[Formula: see text]TM is very easy to implement and a fascinating tool for fractal models.

Applications of Two Methods in Exact Wave Solutions in the Space-Time Fractional Drinfeld–Sokolov–Wilson System

The fractional differential equations (FDEs) are ubiquitous in mathematically oriented scientific fields, such as physics and engineering. Therefore, FDEs have been the focus of many studies due to their frequent appearance in several applications such as physics, engineering, signal processing, systems identification, sound, heat, diffusion, electrostatics and fluid mechanics, and other sciences. The perusal of these nonlinear physical models through wave solutions analysis, corresponding to their FDEs, has a dynamic role in applied sciences. In this paper, the exp-function method and the rational G ′ / G -expansion method are presented to establish the exact wave solutions of the space-time fractional Drinfeld–Sokolov–Wilson system in the sense of the conformable fractional derivative. The fractional Drinfeld–Sokolov–Wilson system contains fractional derivatives of the unknown function in terms of all independent variables. This system describes the shallow water wave models in fluid mechanics. These presented methods are a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences, especially in physics.

Fredholm Pfaffian $$\tau $$-Functions for Orthogonal Isospectral and Isomonodromic Systems

We extend the approach to \(\tau \)-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld–Sokolov reductions and isomonodromic deformations systems. The combinatorial expansion of the \(\tau \)-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and dependence on the time parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the determinantal \(\tau \)-functions, both in the Drinfeld–Sokolov and isomonodromic case, become squares of \(\tau \)-functions of Pfaffian type. The results are illustrated by several examples, consisting of polynomial \(\tau \)-functions of orthogonal Drinfeld–Sokolov type and isomonodromic ones with four regular singular points.

Analytical approximate solutions of fractional nonlinear Drinfeld–Sokolov–Wilson model using modified Mittag–Leffler function

In this article, we introduce a modification of the generalized Mittag–Leffler function method (GMLFM) in order to provide appropriate analytical-approximate solutions for time-fractional nonlinear partial differential equations (PDEs). Moreover, we implement the proposed method on the Drinfeld–Sokolov–Wilson (DSW) model which is considered one of the most important mathematical models describing mathematical physics and dispersive water waves. We use Caputo fractional derivative (CFD) as a time-fractional operator for the mathematical formulation of this model. We introduce the results obtained by modified generalized Mittag Leffler function method (MGMLFM) when the fractional operator takes different values to show the impact of fractional order on the solution of the fractional DSW model, as well, we compare our results when α=1 with the exact solution to prove the efficiency of the used method. Furthermore, a simulation of our outcomes in some graphical and tabular forms is presented to be clearer for the reader. This simulation detects that by decreasing the value of time t the obtained solutions of the DSW system are remarkably close to the given exact solutions. Furthermore, decreasing the fractional operator from α=1 rapidly increases the amplitude of the solitary wave of the model. Also, results reveal that the absolute error decreases as spatial variable x increases at small-time t. The outcomes of this article confirm that this modification made to the GMLFM leads us to an effective, easy and accurate method to solve linear and nonlinear fractional PDEs which need little effort for computations.

A new construction of the Drinfeld–Sokolov hierarchies

The Drinfeld–Sokolov hierarchies are integrable hierarchies associated with every affine Lie algebra. We present a new construction of such hierarchies, which only requires the computations of a formal Laurent series.

THE COMPARATIVE REPORT ON DYNAMICAL ANALYSIS ABOUT FRACTIONAL NONLINEAR DRINFELD–SOKOLOV–WILSON SYSTEM

The new extended direct algebraic technique is used to attain the new exact solitary wave solutions of the nonlinear fractional couple Drinfeld–Sokolov–Wilson system (DSWS). This DSWS has a lot of applications in water waves, fluid mechanics, quantum mechanics, chemistry, and biology. This system is developed in the perspective of fractional theory of calculus, for this purpose Riemann–Liouville fractional derivative (RLFD), [Formula: see text]-fractional differential operator and Atangana–Baleanu fractional derivative (ABFD) are utilized. The attained solutions contain some new families such as trigonometric functions, hyperbolic functions, rational functions, logarithmic functions, and exponential functions. The graphically analysis and comparison of solutions are plotted by taking appropriate values of the constant parameters to enhance the understanding of physical real-life phenomena. The approach is said to be compatible and effective as a scientific approach for the examination of the various spacetime fractional models in engineering and physics concerning real-life issues.

Virasoro constraints for Drinfeld–Sokolov hierarchies and equations of Painlevé type

We construct a tau cover of the generalized Drinfeld–Sokolov hierarchy associated with an arbitrary affine Kac–Moody algebra with gradations s ⩽ 1 $\mathrm{s}\leqslant \mathbb {1}$ and derive its Virasoro symmetries. By imposing the Virasoro constraints we obtain solutions of the Drinfeld–Sokolov hierarchy of Witten–Kontsevich and of Brezin–Gross–Witten types, and of those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on the space of solutions of such ordinary differential equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé-type equations.

Investigation of Fractal Fractional nonlinear Drinfeld–Sokolov–Wilson system with Non-singular Operators

The aim of this article is to study the Drinfeld–Sokolov–Wilson equation considered in fractal-fractional sense with exponential decay and Mittag-Leffler type kernels. The Laplace transform combined with Adomian decomposition method is used to calculate the general solution of the system in series form. The convergence of the obtained series solution is also presented. For validity of our results, we consider a numerical example with suitable initial conditions. The results reveal that by decreasing fractal parameter enhances the amplitude of the soliton solution of the system. Moreover, decreasing fractional parameter, rapidly increases the amplitude of solitary wave of the system. We also observed that for small value of time t, the wave solutions of the system are very close to each other, while increasing in time t enhances the system when one of the parameters fractional or fractal is equal to one. We also studied the error analysis of the system reveals that the absolute error decreases as spatial variable x increases at small time t, while, increase in iterations, reduces the absolute error in the system.

New Iterative Method for Solving a Coupled System of Fractional‐Order Drinfeld–Sokolov–Wilson (FDSW) and Fractional Shallow Water (FSW) Equations

In this study, the new iterative method has been applied to a coupled system of fractional‐order Drinfeld–Sokolov–Wilson (FDSW) and fractional shallow water (FSW) equations. The fractional‐order derivatives are taken in the Caputo sense whose order is between 0 and 1. The suggested method is capable to handle the FDEs without any transformation and discretization. The obtained results have been compared with the exact solution and with the q‐homotopy analysis transform method. The outcomes show the efficiency and effectiveness of NIM by comparing through tables and graphs.