Fifth-order KdV Equation Research Articles

Novel nonlinear wave transitions and interactions for (2+1)-dimensional generalized fifth-order KdV equation

The (2 + 1)-dimensional generalized fifth-order KdV (2GKdV) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKdV equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped (XS/YS), asymmetric M-Shaped (MS), W-Shaped (WS), Space-Curved (SC) and Oscillation M-Shaped (OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.

Hybrid soliton, breather waves and solution molecules of the (2+1)-dimensional generalized fifth-order KdV equation

In this paper, we take the (2+1)-dimensional generalized fifth-order KdV (fKdV) equation as an example. We obtain many wave solutions, including hybrid soliton, breather waves and solution molecules. In order to solve hybrid soliton, we give two transformation forms. For Transform one, real and complex parameters lead to different wave solutions. Soliton solutions are obtained when the parameter values are real, but some breather waves appear when the parameter values are complex. When the parameter value changes, the number of solitons and the position of the waves will also change. For Transform two, we assign the parameter values to real values and obtain the corresponding two-, three- and four-soliton solutions. The most important thing is that, on the basis of Transform one, we add different constraint conditions according to different coordinate systems, and finally get a very interesting phenomenon: soliton molecules. In addition, we give not only the three-dimensional plots but also the density plots. In general, these results will enrich the existing literature on the (2+1)-dimensional generalized fKdV equation and help to understand this kind of KdV equation.

New derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons

The computation of the amplitude, α, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg–de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, ε2≪1, a new derivation of the “beyond all orders in ε” amplitude, α, is presented. It is shown by asymptotic matching techniques, extended to higher orders in ε, that the value of α can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [], not only sheds new light on the computation of α but also greatly facilitates its numerical determination to a remarkable precision for so small values of ε, which are beyond the capabilities of standard numerical methods. Published by the American Physical Society 2024

New Exact and Numerical Experiments for the Caudrey-Dodd-Gibbon Equation

In this study, an exact and a numerical method namely direct algebraic method and collocation finite element method are proposed for solving soliton solutions of a special form of fifth-order KdV (fKdV) equation that is of particular importance: Caudrey-Dodd-Gibbon (CDG) equation. For these aims, homogeneous balance method and septic B-spline functions are used for exact and numerical solutions, respectively. Next, it is proved by applying von-Neumann stability analysis that the numerical method is unconditionally stable. The error norms $L_{2}$ and $L_{\infty }$ have been computed to control proficiency and conservation properties of the suggested algorithm. The obtained numerical results have been listed in the tables. The graphs are modelled so that easy visualization of properties of the problem. Also, the obtained results indicate that our method is favourable for solving such problems.

Stability and dynamics of regular and embedded solitons of a perturbed Fifth-order KdV equation

Families of symmetric embedded solitary waves of a perturbed Fifth-order Korteweg–de Vries (FKdV) system were treated in Choudhury et al. (2022) using perturbative and reversible systems techniques. Here, the stability of those solutions, which was not considered in the earlier paper, is detailed. In addition, the results of Choudhury et al. (2022) are extended to the case of asymmetric solitary waves, as well as their stability. Finally, other novel multi-humped regular solitary waves of this system are derived using convergent infinite series solutions for the homoclinic orbits of the FKdV-traveling wave equation.

Some Exact Solutions for the Fifth-Order Kdv Equation with Constant Coefficients by the Lie Symmetries

Some Exact Solutions for the Fifth-Order Kdv Equation with Constant Coefficients by the Lie Symmetries

Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval

Abstract We have established the existence and uniqueness of the local solution for (0.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 , \left\{\begin{array}{ll}{\partial }_{t}u+{\partial }_{x}^{5}u-u{\partial }_{x}u=0,& 0\lt x\lt 1,\hspace{1.0em}t\gt 0,\\ u\left(x,0)=\varphi \left(x),& 0\lt x\lt 1,\\ u\left(0,t)={h}_{1}\left(t),u\left(1,t)={h}_{2}\left(t),\hspace{0.33em}{\partial }_{x}u\left(1,t)={h}_{3}\left(t),& \\ {\partial }_{x}u\left(0,t)={h}_{4}\left(t),\hspace{0.33em}{\partial }_{x}^{2}u\left(1,t)={h}_{5}\left(t),& t\gt 0,\end{array}\right. in the study of Zhao and Zhang [Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval, J. Math. Anal. Appl. 470 (2019), 251–278]. A question arises naturally: Can the local solution be extended to a global one? This article will address this question. First, through a series of logical deductions, a global a priori estimate is established, and then the local solution is naturally extended to a global solution.

Important Theories and Applications Arise in Physics and Engineering using a New Technique: The Natural Decomposition Method

In the current research, we used the natural Adomian decomposition method (NADM), which is a unique technique, to find new exact solutions to partial and ordinary differential equations, such as the Sawada-Kotera equation, the extended fifth-order Korteweg-de Vries (efKdV) equation, and the Caudrey-Dodd-Gibbon equation. We also provided proofs for newly discovered theorems related to natural transformation. Our solutions to both linear and nonlinear differential equations are provided in series form, and the suggested method has high convergence. To handle differential equations, including nonlinear and linear differential equations, the proposed scheme in this work can be simply viewed as an alternative to those found in the literature.

The dynamics of fractional KdV type equations occurring in magneto-acoustic waves through non-singular kernel derivatives

To study magneto-acoustic waves in plasma, we will use a numerical method based on the Natural Transform Decomposition Method (NTDM) to find the approximative solutions of nonlinear fifth-order KdV equations. The method combines the familiar Natural transform (NT) with the standard Adomian decomposition method. The fractional derivatives considered are the Caputo–Fabrizio and the Atangana–Baleanu derivatives in the sense of Caputo derivatives. Adomian polynomials may be employed to tackle nonlinear terms. In this method, the solution is calculated as a convergent series, and it is demonstrated that the NTDM solutions converge to the exact solutions. A range of two- and three-dimensional figures have been used to illustrate the dynamic behavior of the derived solutions. The tables provide a visual representation of numerical data. The physical behavior of the derived solutions about fractional order is further demonstrated by several simulations. When addressing nonlinear wave equations in science and engineering, the NTDM offers a broad range of applications. Several examples are given to highlight the importance of this work and to demonstrate the simplicity and trustworthiness of the method.

Investigating soliton dynamics: Contemporary computational and numerical approaches for analytical and approximate solutions of the CDG model

This investigation employs contemporary computational and numerical techniques to derive analytical and approximate soliton solutions for the Caudrey–Dodd–Gibbon model, which represents a significant variation of the fifth-order Korteweg–de Vries equation. Diverse analytical solutions are constructed, employing distinct formats such as exponential, trigonometric, and hyperbolic functions. Simulations, including two-dimensional, three-dimensional, contour, polar, and discrete plots, are presented to illustrate the real-world behavior of a single soliton. Furthermore, these solutions are utilized to evaluate the essential conditions for implementing the proposed numerical scheme. The agreement between the computed and approximate solutions is demonstrated through various techniques. These results unequivocally establish the superiority of these methods for solving nonlinear mathematical physics problems.

Analytical and numerical investigation of soliton wave solutions in the fifth-order KdV equation within the KdV-KP framework

This study investigates the fifth-order Korteweg–de Vries (fKdV) equation, which is a generalized form of the classical KdV equation governing weakly nonlinear waves in dispersive media. By employing analytical and numerical techniques, the Khater II and variational iteration methods are utilized to obtain accurate soliton wave solutions. The combination of these approaches ensures the reliability and precision of the analytical outcomes. Graphical representations, including two-dimensional, three-dimensional, and contour plots, visually illustrate the results. Additionally, the stability of the constructed solutions is assessed through the characterization of the Hamiltonian system, providing valuable insights into the soliton wave solutions’ stability properties. This research integrates analytical and numerical methods to explore the fKdV equation within the KdV-KP framework, offering novel soliton wave solutions and enhancing their interpretation through graphical representations while assessing their stability using the Hamiltonian system’s characterization.

(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

We study the problem of gravity surface waves for an ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure to derive the Boussinesq equations for a given relation between the orders of four expansion parameters, the amplitude parameter α, the long-wavelength parameter β, the transverse wavelength parameter γ, and the bottom variation parameter δ. Such an approach, known in (1+1)-dimensional theory, is extended for the first time for (2+1)-dimensional shallow water waves. Using this procedure, we derived the only possible (2+1)-dimensional extensions of the Korteweg–de Vries equation, the fifth-order KdV equation and the Gardner equation in three cases of the relationship between these parameters. All these equations are non-local. When the bottom is flat, the (2+1)-dimensional KdV equation can be transformed to the Kadomtsev–Petviashvili equation in a fixed reference frame and next to the classical KP equation in a moving frame. Thus, the Kadomtsev–Petviashvili equation gained a derivation from the fundamental laws of hydrodynamics. We have found soliton, cnoidal, and superposition solutions (essentially one-dimensional) to the (2+1)-dimensional Korteweg–de Vries equation and the Kadomtsev–Petviashvili equation.

On soliton solutions, periodic wave solutions and asymptotic analysis to the nonlinear evolution equations in (2+1) and (3+1) dimensions

In this paper, the (2+1)-dimensional generalized fifth-order KdV equation and the extended (3+1)-dimensional Jimbo-Miwa equation were transformed into the Hirota bilinear forms with Hirota direct method. In this process, the Hirota bilinear operator played a significant role. Based on the Hirota bilinear forms, the single soliton solutions and the single periodic wave solutions of these two types of equations were obtained respectively. Meanwhile, the figures of the single soliton solutions and the single periodic wave solutions were plotted. Furthermore, the results shed light on that when the amplitude of water wave approaches 0, the single periodic wave solutions tend to the single soliton solutions. The conclusion has been generalized from (2+1)-dimensional equations to (3+1)-dimensional equations.

An Efficient Analytical Approach to Investigate Fractional Caudrey–Dodd–Gibbon Equations with Non-Singular Kernel Derivatives

Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by researchers in important monographs and review articles, there is still a great deal of non-local phenomena that have not been studied and are only waiting to be explored. As a result, we can continually learn about new applications and aspects of fractional modelling. In this study, a precise and analytical method with non-singular kernel derivatives is used to solve the Caudrey–Dodd–Gibbon (CDG) model, a modification of the fifth-order KdV equation (fKdV). The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative and the Atangana–Baleanu derivative in the Caputo sense (ABC). This model illustrates the propagation of magneto-acoustic, shallow-water, and gravity–capillary waves in a plasma medium. The dynamic behaviour of the acquired solutions has been represented in a number of two- and three-dimensional figures. A number of simulations are also performed to demonstrate how the resulting solutions physically behave with respect to fractional order. The significance of the current research is that new solutions are obtained by using a strong analytical approach. Utilizing a fractional derivative operator to solve equivalent models is another benefit of this approach. The results of the present work have similar aspects to the symmetry of partial differential equations.

Time-fractional generalized fifth-order KdV equation: Lie symmetry analysis and conservation laws

The purpose of this study is to apply the Lie group analysis method to the time-fractional order generalized fifth-order KdV (TFF-KdV) equation. We examine applying symmetry analysis to the TFF-KdV equation with the Riemann–Liouville (R–L) derivative, employing the G′/G-expansion approach to yield trigonometric, hyperbolic, and rational function solutions with arbitrary constants. The discovered solutions are unique and have never been studied previously. For solving non-linear fractional partial differential equations, we find that the G′/G-expansion approach is highly effective. Finally, conservation laws for the equation are well-built with a full derivation based on the Noether theorem.

Analytic Computational Method for Solving Fractional Nonlinear Equations in Magneto-Acoustic Waves

In this article, we employ a useful and intriguing method known as the ARA-homotopy transform approach to explore the fifth-order Korteweg-de Vries equations that are nonlinear and time-fractional. The study of capillary gravity water waves, magneto-sound propagation in plasma, and the motion of long waves under the effect of gravity in shallow water have all been influenced by Korteweg-de Vries equations. We discuss three instances of the fifth-order time-fractional Korteweg-de Vries equations to demonstrate the efficacy and applicability of the proposed method. Utilizing, also known as the auxiliary parameter or convergence control parameter, the ARA-homotopy transform technique which is a combination between ARA transform and the homotopy analysis method, allows us to modify the convergence range of the series solution. The obtained results show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.

Complex flow structures beneath rotational depression solitary waves in gravity-capillary flows

Particle trajectories beneath depression solitary gravity-capillary waves in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. A fifth-order Korteweg–de Vries equation that incorporates the surface tension and the vorticity effects is derived asymptotically from the full Euler equations when the Bond number is nearly a critical value that depends on the vorticity. The velocity field in the bulk fluid is approximated which allow us to study the submarine structures beneath depression solitary waves. The dynamical system of particle trajectories is reformulated in the moving wave frame and its features are seen through the streamlines. We show that for large values of the vorticity parameter there are always stagnation points in the bulk fluid, which produce rich cat’s-eyes structures. In this case, particle trajectories can undergo vertical excursions, describing closed orbits or undergo pure horizontal transport. Furthermore, bifurcation diagrams according to the number of stagnation points and complex flow structures beneath depression solitary waves with multiple local extrema are shown.

Improved Soliton Solutions of Generalized Fifth Order Time-Fractional KdV Models: Laplace Transform with Homotopy Perturbation Algorithm

The main purpose of this research is to propose a new methodology to observe a class of time-fractional generalized fifth-order Korteweg–de Vries equations. Laplace transform along with a homotopy perturbation algorithm is utilized for the solution and analysis purpose in the current study. This extended technique provides improved and convergent series solutions through symbolic computation. The proposed methodology is applied to time-fractional Sawada–Kotera, Ito, Lax’s, and Kaup–Kupershmidt models, which are induced from a generalized fifth-order KdV equation. For validity purposes, obtained and existing results at integral orders are compared. Convergence analysis was also performed by computing solutions and errors at different values in a fractional domain. Dynamic behavior of the fractional parameter is also studied graphically. Simulations affirm the dominance of the proposed algorithm in terms of accuracy and fewer computations as compared to other available schemes for fractional KdVs. Hence, the projected algorithm can be utilized for more advanced fractional models in physics and engineering.

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.

Solutions of Time Fractional fKdV Equation Using the Residual Power Series Method

The fifth-order Korteweg-de Vries (fKdV) equation is a nonlinear model in various long wave physical phenomena. The residual power series method (RPSM) is used to gain the approximate solutions of the time fractional fKdV equation in this study. Basic definitions of fractional derivatives are described in the Caputo sense. The solutions of the time fractional fKdV equation with easily computable components are calculated as a quick convergent series. When compared to exact solutions, the RPSM provides good accuracy for approximate solutions. The reliability of the proposed method is also illustrated with the aid of table and graphs. Cleary observed from the results that the suggested method is suitable and simple for similar type of the time fractional nonlinear differential equations.