Kupershmidt Equation Research Articles

Derivation of Sawada-Kotera and Kaup-Kupershmidt Equations KdV Flow Equations from Modified Nonlinear Schrödinger Equation (MNLS)ns from Modfied Nonlinear Schrödinger Equation (MNLS)

Mathematical models of problems that arise in almost every branch of science are nonlinear evolution equations (NLEE). As a result, nonlinear evolution equations have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear evolution equations and solution methods. The main reason for this situation is that nonlinear evolution equations involve the problem of nonlinear wave propagation. In this study, (1 + 1) dimensional fifth-order nonlinear Korteweg-de Vries (fKdV) type equations were obtained by applying the multi-scale method known as the perturbation method for the modified nonlinear Schrödinger (MNLS) equation. Thus, we showed the relationship between KdV equations and MNLS type equations.

Non-Linear Plasma Wave Dynamics: Investigating Chaos in Dynamical Systems

This work addresses the significant issue of plasma waves interacting with non-linear dynamical systems in both perturbed and unperturbed states, as modeled by the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBK-BK) Equations. We investigate analytical solutions and the subsequent emergence of chaos within these systems. Initially, we apply advanced mathematical techniques, including the transform method and the G′G2 method. These methods allow us to derive new precise solutions and enhance our understanding of the non-linear processes dominating plasma wave dynamics. Through a systematic analysis, we identify the conditions under which the system transitions from orderly patterns to chaotic behavior. This investigation provides valuable insights into the fundamental mechanisms of non-linear wave propagation in plasmas. Our results highlight the dynamic interplay between non-linearity and variation, leading to chaos, which may be useful in predicting and potentially controlling similar phenomena in practical applications.

Classifications of bosonic supersymmetric third and fifth order systems

This manuscript explores the extensions and classifications of the bosonic supersymmetric systems. For the third order bosonic superfield equations, four types of integrable supersymmetric extensions are identified, including the B-type (trivial) supersymmetric modified Korteweg–de Vries equation, the supersymmetric Sharma–Tasso–Olver equation, and an A-type (non-trivial) supersymmetric potential Korteweg–de Vries equation. In the case of the fifth order bosonic supersymmetric systems, nine kinds of extensions are discovered, with six being B-type and three being A-type. Notably, several equations such as the supersymmetric Sawada–Kotera equation, the supersymmetric Kaup–Kupershmidt equation and the supersymmetric Fordy–Gibbons equation are classified as B-type extensions. Despite this classification, these supersymmetric systems are shown to be connected to linear integrable couplings. The findings have implications for various fields including string theory and dark matter and highlight the importance of understanding bosonic supersymmetric systems. The obtained supersymmetric systems are solved via bosonization method. Applying the bosonization procedure to every one of supersymmetric systems, one can find various dark equation systems. These dark equation systems can be solved by means of the solutions of the classical equations and some graded linear couplings including homogeneous and nonhomogeneous symmetry equations.

Localized wave solutions to $$(2+1)$$-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation

Localized wave solutions to $$(2+1)$$-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation

Novel analysis of nonlinear seventh-order fractional Kaup–Kupershmidt equation via the Caputo operator

In this work, we use two unique methodologies, the homotopy perturbation transform method and Yang transform decomposition method, to solve the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation, in particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics, and so on. The provided methods are used to solve the fractional nonlinear seventh-order KK problem along with the Yang transform and fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and fractional Caputo derivative. He’s and Adomian polynomials are useful to manage nonlinear terms. It is shown that the suggested approximate solution converges to the exact one. In these approaches, the results are calculated as convergent series. The key advantage of the recommended approaches is that they provide highly precise results with little computational work. The suggested approach results are compared to the precise solution. By comparing the outcomes with the precise solution using graphs and tables we can verify the efficacy of the offered strategies. Also, the outcomes of the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective for resolving differential equations.

Dynamical behavior analysis and soliton solutions of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations

This article investigates the dynamical behavior analysis and soliton solutions of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations in fluid flow dynamics and plasma waves. Firstly, the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations are simplified into the ordinary differential equations. Secondly, two-dimensional planar dynamical system and its phase portraits are given by utilizing the theory of planar dynamical system analysis. Finally, the soliton solutions of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations are presented by using the fourth-order complete discriminant system. In order to better reveal the soliton propagation phenomenon of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations, some three-dimensional, two-dimensional, and contour graphs of the solutions were drawn.

A gradient reproducing kernel based stabilized collocation method for the 5th order Korteweg–de Vries equations

This paper presents a gradient reproducing kernel based stabilized collocation method (GRK-SCM) to solve the generalized nonlinear 5th order Korteweg–de Vries (KdV) equations. By introducing gradient reproducing kernel (GRK) approximations, one can circumvent the intricacy of high-order derivatives in RK approximations, while still satisfying high-order consistency requirements. This leads to the high accuracy and efficiency. GRKs are very good approximation candidates for addressing high order partial differential equations that require higher order derivatives formulated as strong form. The stabilized collocation method (SCM) effectively meets high-order integration constraints which can achieve accurate integration in the subdomains. This ensures both high precision and optimal convergence. Von Neumann analysis is utilized to establish the stability criteria for GRK-SCM when combined with forward difference temporal discretization. Numerical solutions for Sawada–Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are studied where the solitary wave migration and collision, and periodic waves are represented. A fifth order forced KdV equation is also considered. The presented method's high accuracy and convergence are demonstrated through comparative studies with analytical solutions, and investigations of invariants and corresponding errors determine the good conservation properties of this algorithm.

Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation

In this paper we study the algebraic properties of a new integrable differential-difference equation. This equation can be seen as a deformation of the modified Narita-Itoh-Bogoyavlensky equation and has the Kaup-Kupershmidt equation in its continuous limit. Using its Lax representation we explicitly construct a recursion operator for this equation and prove that it is a Nijenhuis operator. Moreover, we present the bi-Hamiltonian structures for this new equation.

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.

Symmetry Analysis and Wave Solutions of Time Fractional Kupershmidt Equation

This study employs the Lie symmetry technique to explore the symmetry features of the time fractional Kupershmidt equation. Specifically, we use the Lie symmetry technique to derive the symmetry generators for this equation, which incorporates a conformal fractional derivative. We use the symmetry generators to transform the fractional partial differential equation into a fractional ordinary differential equation, thereby simplifying the analysis. The obtained reduced equation is of fourth order nonlinear ordinary differential equation. To find the wave solutions, F/G‐expansion process has been used to obatin different types of solutions of the time‐fractional Kuperschmidt equation. The obtained wave solutions are hyperbolic and trigonometric in nature. We then use Maple software to visually depict these wave solutions for specific parameter values, providing insights into the behaviour of the system under investigation. Peak and kink wave solutions are achieved for the given problem.

Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method

Abstract Finding analytical solutions for nonlinear partial differential equations is physically meaningful. The Kaup-Kupershmidt (KK) equation is studied in this article. The KK equation is of fifth order, such that several solitary solutions are obtained. In this article, however, the modified auxiliary function approach is applied to this model to find solitary solutions. These solutions are written in terms of Jacobi functions. Therefore, the obtained solutions can be implemented graphically to show different patterns for appropriate parameters.

Application of the Variational Iteration Method for the time-fractional Kaup-Kupershmidt Equation and the Boussinesq-Burger equation

The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems.

Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics

In this study, we utilize the potent generalized Kudryashov method to address the intricate obstacles presented by fractional differential equations in the field of mathematical physics. Specifically, our focus centers on obtaining novel exact solutions for three pivotal equations: the time-fractional seventh-order Sawada-Kotera-Ito equation, the time-fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation, and the time-fractional seventh-order Kaup–Kupershmidt equation. The generalized Kudryashov method, celebrated for its versatility and efficacy in addressing intricate nonlinear problems, plays a central role in our research. This method not only simplifies the equations but also unveils their inner dynamics, rendering them amenable to meticulous analysis. It is worth noting that our fractional derivatives are defined in the context of the conformable fractional derivative, providing a solid foundation for our mathematical investigations. One notable aspect of our study is the visual representation of our findings. Graphical representations of the yielded solutions enliven intricate mathematical structures, providing a concrete insight into the dynamics and behaviors of said equations. This paper highlights the proficiency of the generalized Kudryashov method in resolving complex issues presented by fractional differential equations. Our study not only broadens the range of mathematical methods but also enhances our comprehension of the intriguing realm of nonlinear physical phenomena.

Darboux transformation and explicit solutions for the Kaup-Kupershmidt equation

In this paper, the N-fold Darboux transformation of the Kaup-Kupershmidt equation associated with 3×3 order spectral problem is constructed by means of the gauge transformation. By applying the Darboux transformation, some soliton solutions of this equation are derived.

Soliton molecules and few-cycle-pulse solitons structures for the generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation

Soliton molecules and few-cycle-pulse solitons structures for the generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation

Multi-peak and rational soliton propagations for (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup Kupershmidt model in fluid mechanics, ocean dynamics and plasma physics

This paper retrieves the investigation of rational solitons via symbolic computation with logarithmic transformation and ansatz functions approach for the [Formula: see text]-dimensional generalized Konopelchenko–Dubrovsky–Kaup-Kupershmidt (GKDKK) equation in fluid mechanics, ocean dynamics and plasma physics. We find two categories of M-shaped rational solitons and their dynamics will be revealed through graphs by choosing the suitable values of involved parameters. In addition, two categories of M-shaped rational solitons and their interactions with kink waves will be analyzed. Furthermore, homoclinic breathers, multi-wave and kink cross rational solitons will be investigated. The periodic, rational, dark, bright, Weierstrass elliptic function and positive soliton solutions will also be retrieved with the aid of Sub-ODE approach. Moreover, stability characteristics of solutions will be evaluated.

Fractional evaluation of Kaup-Kupershmidt equation with the exponential-decay kernel

<abstract><p>In this paper, we investigate the semi-analytical solution of Kaup-Kupershmidt equations with the help of a modified method known as the new iteration transformation technique. This method combines the Yang transform and the new iteration technique. The nonlinear terms can be calculated straightforwardly by a new iteration method. The numerical simulation results have been presented to demonstrate the reliability and validity of the proposed approach. The result confirms that the suggested technique is the best tool for dealing with any nonlinear problems arising in technology and science. In addition, in terms of figures for varying fractional order, the physical behavior of new iteration transformation technique solutions has been shown and the numerical simulation is also exhibited. The solutions of the new iteration transformation technique reveal that the projected technique is reliable, competitive and powerful for studying complex nonlinear fractional type models.</p></abstract>

Direct and inverse scattering problems of the modified Sawada–Kotera equation: Riemann–Hilbert approach

It is known that both the Sawada–Kotera equation and the Kaup–Kupershmidt equation are related with the same modified equation by different Miura transformations. There is singularity at the origin in the spectral problems of the Sawada–Kotera equation and the Kaup–Kupershmidt equation. Instead, this work investigates the forward and inverse scattering problems of the modified Sawada–Kotera equation by Riemann–Hilbert approach to avoid the singularity at the origin. The Riemann–Hilbert problem along with the reconstructing formula of the modified Sawada–Kotera equation are proposed. Moreover, the properties of the reflection coefficients are analysed rigorously. The results in this paper make an important step toward the long-time asymptotics of the modified Sawada–Kotera equation.

Solution of generalised fuzzy fractional Kaup–Kupershmidt equation using a robust multi parametric approach and a novel transform

This work considers a fuzzy fractional model of the Kaup–Kupershmidt equation with uncertain initial conditions. The uncertain behaviour of capillary gravity and dispersive waves is studied using a robust double parametric fuzzy approach termed the q-Homotopy analysis method with a generalised novel transform. The proposed method’s accuracy and efficiency have been explained by comparing the observation of fuzziness in the solution with several prior accessible results for the crisp case using the double parametric approach. This methodology provides a viable alternative to existing methods for various fuzzy fractional models.

A new perspective for analytical and numerical soliton solutions of the Kaup–Kupershmidt and Ito equations

In present study, it is aimed to implement the analytical method and a numerical approach based on finite element method to get approximate soliton solutions of two special forms of fifth-order KdV equation (fKdV) that are of particular importance: Kaup–Kupershmidt (K–K) and Ito equations. To reach these aims, the simplest equation method and septic B-spline collocation method are introduced. L2 and L∞ error norms are computed for single soliton solutions to demonstrate the proficiency and accuracy of the present method. The method is shown to be unconditionally stable by performing the von-Neumann stability analysis. The obtained numerical results have been displayed with tables. Additionally, to show the analytical and numerical behaviors of the single soliton, all figures are drawn in 2D and 3D. Analytical and numerical results ensure that the methods are more suitable and systematically handle the solution procedures of nonlinear equations arising in mathematical physics.