Konopelchenko Dubrovsky Equation 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.

WITHDRAWN: The construction of first integral, soliton wave solutions, chaos analysis and sensitive demonstration of Konopelchenko–Dubrovsky equation

WITHDRAWN: The construction of first integral, soliton wave solutions, chaos analysis and sensitive demonstration of Konopelchenko–Dubrovsky equation

Bifurcation phase portraits and nonlinear wave solutions for the modified Konopelchenko-Dubrovsky equation

The bifurcation phase portraits and exact nonlinear wave solutions of the modified Konopelchenko-Dubrovsky (KD) equation are studied by using the bifurcation method. The bifurcation phase portraits of the modified KD equation are given when a>0 and a<0. When different combinations of the parameters a, b, k and n, eighteen new exact nonlinear wave solutions for the modified KD equation are obtained.

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.

Qualitative analysis and wave propagation for Konopelchenko-Dubrovsky equation

We are interested in studying some qualitative analysis and wave propagation and their applications to the Konopelchenko-Dubrovsky equation.Using the qualitative theory of planar systems and a complete discriminant system, we propose a new method. The proposed method is more efficient because it only creates real bounded wave solutions, which are desirable in real-world applications, as well as it is applicable to large classes of nonlinear partial differential equations. We present an algorithm for the proposed method to facilitate and clarify its applicability. Some new solutions are introduced using this method, and they are classified as periodic and solitary wave solutions. These solutions are graphically represented by displaying 3D and 2D graphic representations, as well as the contour plot. Furthermore, we investigate the impact of the included parameters on the obtained solutions. We numerically examine the 2D and 3D phase portraits after permitting a certain periodic additional term to arise, revealing the existence of quasi-periodic behavior.

New perspective to the fractal Konopelchenko–Dubrovsky equations with M-truncated fractional derivative

In this work, for the first ever, the fractal Konopelchenko–Dubrovsky equations is defined by using a new fractional derivative called [Formula: see text]-truncated fractional derivative. The main goal of this work is to seek new type of fractal solitary wave solutions for the fractal Konopelchenko–Dubrovsky equations by a novel mathematical scheme, which is called variational sech-function method. The forms of these new fractal solitary wave solutions are different from those in the existing literature. Ultimately, the fractal dynamic behavior of these derived fractal solitary wave solutions is illustrated via a number of 3D and 2D simulation graphs with different parameters and fractal dimensions. The proposed new method can be employed to other nonlinear wave equations in mathematical physics with the same fractional derivative.

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).

Exact solutions of space–time fractional KdV–MKdV equation and Konopelchenko–Dubrovsky equation

Abstract In the present study, we deal with the space–time fractional KdV–MKdV equation and the space–time fractional Konopelchenko–Dubrovsky equation in the sense of the conformable fractional derivative. By means of the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method, many exact solutions are obtained, which include hyperbolic function solutions, trigonometric function solutions and rational solutions. The results show that the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method is an efficient technique for solving nonlinear fractional partial equations. We also provide some graphical representations to demonstrate the physical features of the obtained solutions.

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.

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.