Nonlinear Korteweg-de Vries Research Articles

An Adaptive Semi-Analytical Approach in Solving Nonlinear Korteweg-De Vries Equations

This paper introduces a novel method named the Adaptive Hybrid Reduced Differential Transform Method (AHRDTM) for solving Nonlinear Korteweg-De Vries Equations (NKdVEs). AHRDTM provides convergent semi-analytical solutions over long-time frames by generating subintervals of varying lengths, significantly reducing the number of time-steps and processing time needed for solutions, distinguishing it from the traditional multistep approach of RDTM. Notably, AHRDTM avoids the need for perturbation, linearization or discretization, enhancing its adaptability and reliability. The findings demonstrate that AHRDTM provides highly accurate and efficient solutions for NKdVEs. Additionally, the method is straightforward, significantly reduces the computational effort required to solve NKdVE problems and shows promise for application to a wide range of partial differential equations (PDEs). The efficacy of AHRDTM is illustrated through tables and graphical representations

Elzaki residual power series approach for semi-analytic approximation of time-fractional nonlinear equations occurring in magneto-acousticwaves

Abstract The study of magneto-sound propagation in plasma, capillary gravity water waves, and the travel of long waves under the effect of gravity in shallow water are all important applications of the 5th-order Korteweg–de Vries equation. This research article uses the Elzaki residual power series method(ERPSM), an appropriate innovative technique, to explore time-fractional nonlinear 5th-order Korteweg–de Vries equations. Three instances of the time-fractional 5th-order Korteweg–de Vries equations in order to demonstrate the effectiveness and relevance of the proposed method has been examined. This work's originality is expressed by an analysis of the fractional behavior of the considered equations.There is a discernible difference in the solutions that are produced for various fractional orders, which may have distinct ramifications for further work. Readers can employ the hybrid approaches combining with our proposed scheme as a future study direction to get better results. Additionally, appropriate numerical simulations were setup to confirm the accuracy and dependability of the suggested strategy. The results collected indicate that the suggested approach is highly satisfying and looks at the intricate nonlinear problems that occur in innovation and science.

The Nonlinear Schrödinger Equation Derived from the Third Order Korteweg-de Vries Equation Using Multiple Scales Method

Nonlinear equations of evolution (NLEE) are mathematical models used in various branches of science. As a result, nonlinear equations of evolution 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 equations of evolution and solution methods. Although the origin of nonlinear equations of evolution dates back to ancient times, there have been significant developments regarding these equations from the past to the present. The main reason for this situation is that nonlinear equations of evolution involve the problem of nonlinear wave propagation. In recent years, equations of formation have become increasingly important in applied mathematics. This work focuses on the perturbation approach, often known as many scales, for nonlinear evolution equations. The article focuses on the analysis of the (1+1) dimensional third-order nonlinear Korteweg-de Vries (KdV) equation using the multiple scales method, which resulted in obtaining nonlinear Schrödinger (NLS) type equations.

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.

Solitonic Analysis of the Newly Introduced Three-Dimensional Nonlinear Dynamical Equations in Fluid Mediums

The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features.

A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations

A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations

Higher order contribution to ion-acoustic Korteweg–de Vries (KdV) soliton in unmagnetized quantum plasmas with arbitrary degeneracy of electrons

The propagation of nonlinear ion-acoustic Korteweg–de Vries (KdV) solitons and dressed solitons (with higher order contribution term) in unmagnetized quantum plasmas with arbitrary degeneracy of electrons is investigated using quantum hydrodynamic model. The reductive perturbation method is used to derive the first order nonlinear ion-acoustic KdV equation and also higher order contribution of inhomogeneous differential equation is obtained for the dressed ion-acoustic wave soliton in arbitrary degenerate electron plasmas. Moreover, the higher order inhomogeneous differential equation is solved (nonsecular solution is obtained) using the renormalization method. The conditions for the validity of the higher order correction are also discussed in detail for ion-acoustic solitons in unmagnetized arbitrary degenerate electrons plasma case. The effects of chosen values of electron temperature, equilibrium fugacity and corresponding electron density of a physical quantum plasma system and quantum diffraction parameter H (for two conditions i.e., H < 2 or H > 2) on the amplitude and width of the KdV and dressed ion-acoustic wave solitons for both electrostatic potential hump (compressive) or dip (rarefactive) structures are investigated. The numerical plots are also presented for illustration with numerical values of a physical partially degenerate electrons plasma system exist in the astrophysical or laboratory conditions.

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

Instability analysis and face‐to‐face collision of dust acoustic waves in dusty plasmas: Perspective Jupiter environment

AbstractThe collisionless unmagnetized five components dusty plasmas have been considered to study the significance of the face‐to‐face collision of single‐ and multi‐solitons, phase shifts due to face‐to‐face collision, and instability of the dust acoustic waves in Jupiter's environment. To study the consequence of face‐to‐face collision extended Poincaré–Lighthill–Kuo method is used to derive the two‐sided nonlinear Korteweg–De Vries (KdV) equation. Besides, to investigate the dust acoustic wave's instability, the nonlinear Schrödinger equation has been derived following the procedure of Slathia et al. The multi‐soliton solutions are determined by employing the Hirota bilinear method. The concerned parameters in the issue play a crucial role in the phase shift (generated by the face‐to‐face collision); the collisional process of single‐, double‐, triple‐, and quadruple‐soliton; and the generation of dust acoustic solitons. Furthermore, the plasma parameters are crucial in the formation of dark solitons and envelop shocks.

Mathematical frameworks for investigating fractional nonlinear coupled Korteweg-de Vries and Burger’s equations

This article utilizes the Aboodh residual power series and Aboodh transform iteration methods to address fractional nonlinear systems. Based on these techniques, a system is introduced to achieve approximate solutions of fractional nonlinear Korteweg-de Vries (KdV) equations and coupled Burger’s equations with initial conditions, which are developed by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. As a result, the Aboodh residual power series and Aboodh transform iteration methods for integer-order partial differential equations may be easily used to generate explicit and numerical solutions to fractional partial differential equations. The results are determined as convergent series with easily computable components. The results of applying this process to the analyzed examples demonstrate that the new technique is very accurate and efficient.

The characteristics of ion acoustic solitons in nonthermal regularized kappa distributed plasmas

Nonthermal equilibrium is an intrinsic characteristic of space and astrophysical plasmas, and in many space environments, the velocity distributions of charged particles with suprathermal tails may be well be fitted by the Kappa function, which becomes the Maxwellian distribution for κ→∞. Various studies of ion or dusty acoustic solitons, thus, have considered the Kappa distributed electrons in the model calculations. However, the Kappa velocity distribution (KVD) is theoretically not applicable for κ≤3/2. Alternatively, the recently proposed regularized Kappa distribution with two free parameters, κ and α, have been shown to be mathematically and physically smooth for all κ values, which may recover the standard KVD for α=0 and the Maxwellian distribution for κ→∞ and α=0. In this study, we examine the characteristics of ion acoustic solitons based on the linear, weakly nonlinear Korteweg–de Vries (KdV) and fully nonlinear theories with the regularized Kappa distributed electrons and warm ion fluids. These approaches may give rise to the dispersion relation with modified characteristic speed of acoustic waves, the analytical KdV solutions, and the Sagdeev's potential as well as the fully nonlinear solutions. It is shown that the model results are mathematically and physically valid for κ≤3/2 and the formulations with the charges being free parameters are applicable for general acoustic solitons.

Various wave solutions to the nonlinear fractional Korteweg-de Vries Zakharov-Kuznetsov equation by a new approach to the two-variable expansion scheme

Solving nonlinear differential equations is crucial for the design, optimization, and characterization of engineering systems. However, investigating such equations poses significant challenges and requires the development of new mathematical and computational techniques. In this article, we extend the two-variable (G′/G,1/G) -expansion approach and establish scores of general analytical solutions, consisting of different functions and arbitrary parameters, to the time-fractional Korteweg–de Vries Zakharov-Kuznetsov (KdV-ZK) equation. The KdV-ZK equation provides a powerful model for modulating nonlinear waves in an assortment of physical systems, including plasma physics, atmospheric science, fluid dynamics, soliton theory, and optical fibers. The proposed technique allows extracting several novel solutions to the KdV-ZK equation, which is essential for gaining new insights into physical problems. In addition, the suggested extension shortens the calculation process and enhances computational efficiency. The obtained solitons have potential applications in several scientific and technological fields. The beta fractional derivative and the consistent wave variable are considered to restructure the stated fractional nonlinear equation. This study investigates the merits and demerits of the fractional-order derivative β on the dynamics of the system through the representation of two-dimensional graphs, which vary according to different values of β. The enhanced computational efficiency might assist the researchers in exploring a wider range of phenomena, facilitating further extensive investigations, and contributing to overall progress in the fields of science and engineering.

A mathematical fractional model of waves on Shallow water surfaces: The Korteweg-de Vries equation

&lt;abstract&gt;&lt;p&gt;The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear time-fractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When $ \varsigma = 1 $, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems.&lt;/p&gt;&lt;/abstract&gt;

A new $ \alpha $-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation

&lt;abstract&gt;&lt;p&gt;In this work, we consider an $ \alpha $-robust high-order numerical method for the time fractional nonlinear Korteweg-de Vries (KdV) equation. The time fractional derivatives are discretized by the L1 formula based on the graded meshes. For the spatial derivative, the nonlinear operator is defined to approximate the $ uu_x $, and two coupling equations are obtained by processing the $ u_{xxx} $ with the order reduction method. Finally, the nonlinear difference schemes with order ($ 2-\alpha $) in time and order $ 2 $ precision in space are obtained. This means that we can get a higher precision solution and improve the computational efficiency. The existence and uniqueness of numerical solutions for the proposed nonlinear difference scheme are proved theoretically. It is worth noting the unconditional stability and $ \alpha $-robust stability are also derived. Moreover, the optimal convergence result in the $ L_2 $ norms is attained. Finally, two numerical examples are given, which is consistent with the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Zadacha Koshi dlya nagruzhennogo uravneniya Kortevega–de Friza v klasse periodicheskikh funktsiy

The inverse spectral problem method is applied to finding a solution of the Cauchyproblem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions.A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded termsand a derivation of an analog of Dubrovin’s system of differential equations are proposed. Itis shown that the sum of a uniformly convergent function series constructed by solving theDubrovin system of equations and the first trace formula actually satisfies the loaded nonlinearKorteweg–de Vries equation. In addition, we prove that if the initial function is a real π-periodicanalytic function, then the solution of the Cauchy problem is a real analytic function in thevariable x as well, and also that if the number π/n, n ∈ N, n ≥ 2, is the period of the initialfunction, then the number π/n is the period for solving the Cauchy problem with respect to thevariable x.

Reductive perturbation method in magnetized plasma and role of negative ions

An analysis of reductive perturbation method (RPM) is presented to show why the solitary structures of non-linear ion acoustic waves (IAWs) cannot be obtained in magnetized electron ion plasma by employing this technique. In RPM, the non-linear Korteweg–de Vries equation is derived using stretched co-ordinates in the reference frame of the wave phase speed, considering the dispersion to be a higher-order effect that balances the non-linearity to produce a solitary structure. The maximum amplitude |Φm| of the non-linear solitary wave turns out to be larger than one that contradicts the small amplitude approximation. In the presence of negative ions, the maximum amplitude satisfies the condition |Φm|&amp;lt;1. To elaborate these points, the results have been applied to an experimental plasma consisting of positive ions of xenon (Xe+) and negative ions of fluorene (F−) along with electrons. The amplitude and width of the solitary structures depend upon the ratio of the electron to positive ion density (ne0ni0). Since the non-linear coefficient turns out to be negative, rarefied (dip) solitons are formed in the magnetized Xe+−F−−e plasma.

Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform

The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system’s approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues.

Ion acoustic soliton with thermal ions and non-thermal electrons in a high-relativistic electron-positron-ion plasma

Ion acoustic soliton with thermal ions and non-thermal electrons in a high-relativistic electron-positron-ion plasma

Traveling wave solutions to the nonlinear space–time fractional extended KdV equation via efficient analytical approaches

Traveling wave solutions to the nonlinear space–time fractional extended KdV equation via efficient analytical approaches

Bifurcation analysis of dynamical systems with fractional order differential equations via the modified Riemann-Liouville derivative

In this manuscript, the solutions of linear dynamical systems with fractional differential equations via themodified Riemann-Liouville derivative is derived. By using Jumarie type of derivative (JRL), we stated and provedthe Existence and uniqueness theorems of the dynamical systems with fractional order equations. Also a novel stability analysis of fractional dynamical systems by Jumarie type derivative is established and some important stability conditions are determined. The achieved results have various applications in mathematics, plasma physics and almost all branches of physics that have non-conservative forces. Finally, we investigated interesting application of nonlinear space-time fractional Korteweg-de Vries (STFKdV) equation in Saturn F-ring’s region. Moreover, our investigation could be basic interest to explain and interpret the effects of fractional and modification parameters on STFKdV equation. This is novel study on this model by dynamical system (DS) to describe the behavior of nonlinear waves without solve this system.