Dimensional Konopelchenko Dubrovsky Research Articles

Wronskian solution, Bäcklund transformation and Painlevé analysis to a (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

Abstract The main work of this paper is to construct the Wronskian solution and investigate the integrability characteristics of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation. Firstly, the Wronskian technique is used to acquire a sufficient condition of the Wronskian solution. According to the Wronskian form, the soliton solution is obtained by selecting the elements in the determinant that satisfy the linear partial differential systems. Secondly, the bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived directly via the Hirota bilinear method and the Bell polynomial theory, respectively. Finally, Painlevé analysis proves that this equation possesses the Painlevé property, and a Painlevé-typed Bäcklund transformation is constructed to solve a family of exact solutions by selecting appropriate seed solution. It shows that the Wronskian technique, Bäcklund transformation, Bell polynomial and Painlevé analysis are applicable to obtain the exact solutions of the nonlinear evolution equations, e.g., soliton solution, single-wave solution and two-wave solution.

Novel soliton molecules, asymmetric solitons, W-shape and the breather wave solutions to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

Novel soliton molecules, asymmetric solitons, W-shape and the breather wave solutions to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

Solitons, breathers and rational solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

Solitons, breathers and rational solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

Dynamical behavior of solitons of the (2+1)-dimensional Konopelchenko Dubrovsky system.

Utilizing nonlinear evolution equations (NEEs) is common practice to establish the fundamental assumptions underlying natural phenomena. This paper examines the weakly dispersed non-linear waves in mathematical physics represented by the Konopelchenko-Dubrovsky (KD) equations. The [Formula: see text]-expansion method is used to analyze the model under consideration. Using symbolic computations, the [Formula: see text]-expansion method is used to produce solitary waves and soliton solutions to the [Formula: see text]-dimensional KD model in terms of trigonometric, hyperbolic, and rational functions. Mathematica simulations are displayed using two, three, and density plots to demonstrate the obtained solitary wave solutions' behavior. These proposed solutions have not been documented in the existing literature.

Dynamical and physical characteristics of soliton solutions to the (2+1)-dimensional Konopelchenko–Dubrovsky system

Abstract Soliton solutions of the Konopelchenko–Dubrovsky (KD) equation using four analytical methods are established. The KD system is used to study the portrays in physics with weak dispersion. The investigated results are obtained in different forms such as trigonometric, hyperbolic, and exponential functions. For the physical behavior of the concerned nonlinear system, some solutions are plotted graphically via assigning the certain values to the parameters. Mathematica software 11.11 is used to handle all results as well as figures. Hence, searched results have rewarding recompenses in nonlinear science.

Wave profile analysis of the (2 + 1)-dimensional Konopelchenko–Dubrovsky model in mathematical physics

The (2 + 1)-dimensional Konopelchenko-Dubrovsky (KD) model and the modified version of the new Kudryashov (MVNK) technique are chosen in the current research to obtain the traveling wave solutions (TWSs). The obtained solutions represent the rich range of explicit solutions to the studied model. As a result, TWSs to the stated model are expressed as the different types of wave profiles such as the kink shape, bell shape, anti-bell shape, and W-shape wave profiles. The Hamiltonian function is found from the stated model and shown it as three dimensional, contour and phase plane in this manuscript. The effects of wave velocity and other parameters on the wave profile are also discussed. The obtained wave profiles are typically useful in applications how waves interact with high-dimensional systems in new, specialized structures. Additionally, the direction and position of solitons for changing other parameters can offer a clear-cut explanation of all the different features of wind and water waves. It is seen that the mentioned scheme is effective, potential and easy in mathematical physics. Finally, this study may be opened up brand-new avenues for further study and application in the fields of mathematical physics.

Transverse spectral instabilities in Konopelchenko–Dubrovsky equation

AbstractWe study the transverse spectral stability of the one‐dimensional small‐amplitude periodic traveling wave solutions of the (2+1)‐dimensional Konopelchenko–Dubrovsky (KD) equation. We show that these waves are transversely unstable with respect to two‐dimensional perturbations that are periodic in both directions with long wavelength in the transverse direction. We also show that these waves are transversely stable with respect to perturbations which are either mean‐zero periodic or square‐integrable in the direction of the propagation of the wave and periodic in the transverse direction with finite or short wavelength. We discuss the implications of these results for special cases of the KD equation—namely, KP‐II and mKP‐II equations.

Exact traveling wave solutions for (2+1)-dimensional Konopelchenko-Dubrovsky equation by using the hyperbolic trigonometric functions methods

Abstract In this research, the extended rational sinh-cosh method and the modified extended tanh-function method for mathematically constructing traveling wave solutions to the (2+1)-dimensional integro-differential Konopelchenko-Dubrovsky evolution equation are successfully employed to obtain specific appropriate solutions for the first time. A traveling wave transformation was utilized to turn the provided model into a third-order nonlinear ordinary differential equation. Solitary and periodic wave solutions for the model under investigation are obtained in terms of various complex hyperbolic trigonometric and rational functions. Several of the aforementioned solutions have been represented by two- and three-dimensional graphics with appropriate arbitrary parameters to highlight their physical implications. Two-dimensional graphs have presented the influence of time evolution on the solution’s structures.

Dynamical behavior of analytical soliton solutions, bifurcation analysis, and quasi-periodic solution to the (2+1)-dimensional Konopelchenko–Dubrovsky (KD) system

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko–Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closed-form invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, 3D, 2D combined line graph and their contour graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

Lump, lump-trigonometric, breather waves, periodic wave and multi-waves solutions for a Konopelchenko–Dubrovsky equation arising in fluid dynamics

In this paper, we get certain the lump-trigonometric solutions and rogue waves with predictability of a (2+1)-dimensional Konopelchenko–Dubrovsky equation in fluid dynamics with the assistance of Maple based on the Hirota bilinear form. We first construct a general quadratic form to get the general lump solution for the referred model. At the same time the lump-trigonometric solutions are concluded with plenty of solutions, in which the lump solution localized in all directions in space. The analytical solutions obtained are employed in the investigation of the impacts of the parameters on type of solutions. Moreover, when the lump solution is cut by twin-solitons, the special rogue waves are also introduced. Furthermore, we obtain a new sufficient solutions containing breather wave, cross-kink, periodic-kink, multi-waves and solitary wave solutions. The developed models in this work will serve as the basis for comparisons for the analytical solutions of the subsequent works on the performance of mentioned equation. It is worth noting that the emerging time and place of the rogue waves depend on the moving path of lump solution.

The [formula omitted]-dressing method for the (2+1)-dimensional Konopelchenko–Dubrovsky equation

A novel systematical solution procedure of the (2+1)-dimensional Konopelchenko–Dubrovsky equation is employed on the basis of the ∂̄-dressing method. The eigenfunctions and Green’s function of the Lax pair play a fairly important role in constructing the scattering equation. By analyzing the analyticity of the eigenfunctions and Green’s function, a new ∂̄ problem is introduced to help explore the solution with the help of Cauchy–Green formula and choosing the proper spectral transformation. Furthermore, we can work out the solution formally of the Konopelchenko–Dubrovsky equation from inverse spectral problem once the time evolution of the spectral data is determined.

Numerical Soliton Solutions of Fractional Modified (2 + 1)-Dimensional Konopelchenko–Dubrovsky Equations in Plasma Physics

Abstract In this paper, the time-fractional modified (2 + 1)-dimensional Konopelchenko–Dubrovsky equations have been solved numerically using the Kansa method, in which the multiquadrics is used as radial basis function. To achieve this, a numerical scheme based on finite difference and Kansa method has been proposed. The stability and convergence of the proposed time-discretized scheme are theoretically proven. Also, the solitary wave solutions have been obtained by using Kudryashov technique. The computed results are compared with the exact solutions as well as with the soliton solutions obtained by Kudryashov technique to show the accuracy of the proposed method.

Solutions to the Konopelchenko-Dubrovsky equation and the Landau-Ginzburg-Higgs equation via the generalized Kudryashov technique

The (2 + 1)-dimensional Konopelchenko-Dubrovsky (KD) equation and the Landau-Ginzburg-Higgs (LGH) equation describe the nonlinear waves with weak scattering and long-range interactions between the tropical, mid-latitude troposphere, the interaction of equatorial and mid-latitude Rossby waves etc. This article studies the KD and LGH models stated earlier using the generalized Kudryashov technique. We obtained a variety of analytical solutions including unknown parameters. The figures of some of the obtained solutions are sketched with certain parameters. The derived results demonstrate the efficiency and reliability of the generalized Kudryashov technique for establishing systematic solutions to nonlinear evolution equations (NLEEs).

On some novel exact solutions to the time fractional (2 + 1) dimensional Konopelchenko–Dubrovsky system arising in physical science

AbstractThe purpose of this article is to construct some novel exact travelling and solitary wave solutions of the time fractional (2 + 1) dimensional Konopelchenko–Dubrovsky equation, and two different forms of integration schemes have been utilized in this context. As a result, a variety of bright and dark solitons, kink- and antikink-type solitons, hyperbolic functions, trigonometric functions, elliptic functions, periodic solitary wave solutions and travelling wave solutions are obtained, and the sufficient conditions for the existence of solution are also discussed. Moreover, some of the obtained solutions are illustrated as two- and three-dimensional graphical images by using computational software Mathematica. These types of solutions have a wide range of applications in applied sciences and mathematical physics. The proposed methods are very useful for solving nonlinear partial differential equations arising in physical science and engineering.

New solitary wave structures to the (2 + 1)-dimensional KD and KP equations with spatio-temporal dispersion

The present paper studies the novel generalized (G′/G)-expansion technique to two nonlinear evolution equations: The (2+1)-dimensional Konopelchenko-Dubrovsky (KD) equation and the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation and acquires some new exact answers. The secured answers include a particular variety of solitary wave solutions, such as periodic, compaction, cuspon, kink, soliton, a bright periodic wave, Bell shape soliton, dark periodic wave and various kinds of soliton of the studied equation are achieved. These new particular kinds of solitary wave solutions will improve the earlier solutions and help us understand the physical meaning further and interpret the nonlinear generation of nonlinear wave equations of fluid in an elastic tube and liquid, including small bubbles and turbulence and the acoustic dust waves in dusty plasmas. Additionally, the studied approach could also be employed to obtain exact wave solutions for the other nonlinear evolution equations in applied sciences.

Multiple lump solutions of the (2+1)‐dimensional Konopelchenko–Dubrovsky equation

In this paper, multiple lump solutions of the (2+1)‐dimensional Konopelchenko–Dubrovsky equation are obtained by means of the Hirota bilinear method. With the aid of positive quartic‐quadratic‐functions, we can get the 1‐lump solutions, 3‐lump solutions, and 6‐lump solutions. Via the density plots and three‐dimensional plots, the dynamic properties of multiple lump solutions are discussed by choosing the appropriate parameters. It is expected that our results are valuable for revealing the high‐dimensional dynamic phenomenon of the nonlinear evolution equations.

Propagation of nonlinear waves with a weak dispersion via coupled $$(2+1)$$-dimensional Konopelchenko–Dubrovsky dynamical equation

This work applies the modified extended direct algebraic method to construct some novel exact travelling wave solutions for the coupled $$(2+1)$$ -dimensional Konopelchenko–Dubrovsky (KD) equation. Soliton, periodic, solitary wave, Jacobi elliptic function, new elliptic, Weierstrass elliptic function solutions and so on are obtained, which have several implementations in the field of applied sciences and engineering. In addition, we discuss the dynamics of some solutions like periodic, soliton and dark-singular combo soliton by their evolutionary shapes.

Rational and semi-rational solutions of the modified Kadomtsev–Petviashvili equation and the $$\varvec{(2 + 1)}$$-dimensional Konopelchenko–Dubrovsky equation

General rational and semi-rational solutions of the modified Kadomtsev–Petviashvili (mKP) equation and the Konopelchenko–Dubrovsky equation are obtained based on the bilinear method and the KP hierarchy reduction technique. These solutions are expressed in terms of $$N \times N$$ determinants. The dynamics of the solutions, which exhibit various patterns, are thoroughly analyzed. It is shown that the rational solutions may describe the elastic interaction of a single-peak wave with either a double-peak (M-shape) wave or another single-peak wave for $$N=1$$ . Depending on the choice of parameters, the semi-rational solutions are found to depict the inelastic interaction between two (Y-shape) or three waves for $$N=1$$ . The second-order ( $$N=2$$ ) rational solutions exhibit the elastic interaction of three single-peak waves with either one double-peak wave or another single-peak wave. Inelastic interaction is displayed by proper choices of the parameters for semi-rational solutions. When $$N > 2$$ , similar local dynamical behaviors of the rational and semi-rational solutions have been observed.

Lump soliton wave solutions for the (2+1)-dimensional Konopelchenko–Dubrovsky equation and KdV equation

This paper studies (2+1)-dimensional Konopelchenko–Dubrovsky equation and (2+1)-dimensional KdV equation via a modified auxiliary equation technique. These two systems describe the connection between the nonlinear weaves with a weak scattering and long-range interactions between the tropical, mid-latitude troposphere, the interaction of equatorial and mid-latitude Rossby waves, respectively. We implement a novel technique to these systems to find analytical traveling wave solutions. The performance of this novel method shows its ability for applying on various nonlinear partial differential equations. All solutions obtained are checked by the Maple software system and verified for its high fidelity.

On the Exact Solution of Time-Fractional (2 + 1) Dimensional Konopelchenko–Dubrovsky Equation

The analysis of nonlinear equations performs a significant part in examining the physical phenomena. In this article, the exact solution of time-fractional (2 + 1) dimensional Konopelchenko–Dubrovsky equation (KDE) has been derived by utilising the Kudryashov method. The Kudryashov technique is effectively implemented in order to acquire the analytical solutions of the time-fractional KDE. As exact solution of fractional KDE is not documented previously, the Kudryashov technique is utilized to construct exact solutions via fractional complex transform. The solution thus attained by the above method is illustrated graphically and are discussed in details. This research analysis manifests that the Kudryashov method deemed capable of yielding solutions of such fractional differential equations.