Dimensional Modified Korteweg-de Vries Research Articles

Analytical soliton solutions and wave profiles of the (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation

In this research paper, the improved generalized Riccati equation mapping (IGREM) method is employed to get analytical soliton solutions of (3+1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation. By giving appropriate parametric values, we obtain soliton solutions of various types, such as kink soliton, singular soliton, and periodic-singular soliton. The description provided in physical variables allows for the study of genuine multispecies plasmas, plasma models, and frequency regimes. These solutions are essential in illuminating several physical phenomena in engineering and other applied sciences. In addition, 3D and 2D graphs of some selected solutions are sketched for the best physical characterization of the obtained results. This method can also be implemented in many non-linear equations in contemporary research areas. Comparison with previous studies shows that the discovery of singular soliton is novel. Therefore, our research represents a significant advancement in the field and contributes to expanding the knowledge regarding soliton behavior. The applications of this study include electron and ion-acoustic modes in regular plasmas with hot Boltzmann electron species, as well as ion and dust-acoustic modes, depending on the specific modeling of the heavier components.

Dynamical behavior of the fractional coupled Konopelchenko–Dubrovsky and (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznestsov equations

Dynamical behavior of the fractional coupled Konopelchenko–Dubrovsky and (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznestsov equations

Bifurcation of Travelling Wave Solutions for (3+1)-dimensional mKdV-ZK Equation

Since the (3+1)-dimensional modified Korteweg de Vries- Zakharov–Kuznetsov (mKdV-ZK equation) was proposed, many methods have been used to find its exact solution, but some solutions are always missed in the calculation process. In this paper, the mKdV-ZK equation is studied by the planar dynamical systems method, the bifurcation diagram and phase diagram under different parameters are obtained, and the solution of the equation is numerically simulated. Finally, the traveling wave solutions of the mKdV-ZK equation is constructed under different parameters.

Soliton solutions of some nonlinear evolution equations in shallow water theory

This paper proposes some soliton solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation, a (3+1) dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov and a Variant Boussinesq equations. We presented a exponential function method (EFM) for solving the considered evolution equations. The solitary, cuspon and periodic wave solutions are acquired and presented graphically. The soliton solutions of considered equations play a typical role for expressing kinds of wave transmission in any natural instance, particularly in shallow wave kinetics. The results suggest that EFM is effective and useful technique to handle non-linear engineering problems in oceans.

The d-bar formalism for the modified Veselov-Novikov equation on the half-plane

We study the modified Veselov-Novikov equation (mVN) posed on the half-plane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural \((2+1)\)-dimensional generalization of the \((1+1)\)-dimensional modified Korteweg-de Vries equation in the sense as to how the Novikov-Veselov equation is related to the Korteweg-de Vries equation. In this paper, by means of the Fokas method, we present the so-called global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the \(d\)-bar formalism, also known as Pompieu's formula. In addition, we characterize the \(d\)-bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions.

Study of the dynamical nonlinear modified Korteweg–de Vries equation arising in plasma physics and its analytical wave solutions

In this article we have discussed the analytical analysis of two dimensional modified Korteweg–de Vries (mK–dV) equation arising in plasma physics that governs the ion-acoustic solitary waves for their asymptotic behavior because of the trapping of electrons using auxiliary equation mapping method. By using this technique we have obtained some quite general and new variety of exact traveling wave solutions which are collecting some kind of semi half bright, bright, dark, semi half dark, doubly periodic, combined, periodic, half hark and half bright via three parametric values which is the primary key point of difference of our technique. These results are highly applicable to develop new theories of quantum mechanics, biomedical problems, soliton dynamics, plasma physics, nuclear physics, optical physics, fluid dynamics, electromagnetism, industrial studies, mathematical physics, biomedical problems, and in many other natural and physical sciences. For detailed physical dynamical representation of our results we have shown them with graphs in different dimensions via Mathematica 10.4 to get more understanding of different new dynamical shapes of solutions.

A coupled [formula omitted]-dimensional mKdV system and its nonlocal reductions

In this paper, we propose a coupled (2+1)-dimensional modified Korteweg-de Vries (mKdV) system, which admits two kinds of nonlocal reductions. By constructing the Darboux transformation for the considered equation, a variety of exact solutions, such as soliton, soliton-type, kink, kink-type, rational, rogue wave and lump solutions are given explicitly.

Generalized (2 + 1)-dimensional mKdV-Burgers equation and its solution by modified hyperbolic function expansion method

Abstract We investigate a new higher-dimensional nonlinear dynamics model to describe the generation and evolution of Rossby waves. We derive a generalized (2 + 1)-dimensional modified Korteweg-de Vries (mKdV)-Burgers equation by considering the quasi-geostrophic potential vorticity equation under the generalized β approximation with dissipation and external source in barotropic fluids, and by utilizing multiple scales and the perturbation expansion method. Qualitative analysis yields that the generalized β and shear basic flow can induce the nonlinear Rossby solitary waves, and dissipation has effects on the propagation of Rossby waves as well. Then, we analyze the conservation laws for the mass and energy of Rossby solitary waves. Moreover, the asymptotic kink-shaped solitary wave solution for the generalized (2 + 1)-dimensional mKdV-Burgers equation is explored by modified hyperbolic function expansion method. The solitary wave solutions and figures are analyzed, the results show that the dissipation effect causes the speed and amplitude of Rossby solitary waves to decrease exponentially and the width of Rossby solitary waves to increase exponentially with time.

New exact solution for the (3+1) conformable space–time fractional modified Korteweg–de-Vries equations via Sine-Cosine Method

ABSTRACTIn this research work, we established exact solution for the conformable space–time fractional (3 + 1) dimensional modified Korteweg de Vries equations (mKdV). A Sine- Cosine method is used for obtaining travelling wave solutions for these models with minimal algebra. We can conclude that the proposed scheme is reliable and efficient as its required minimal algebra without using sophisticated Mathematical tools (maple, Mathematica and others). The goal has been achieved with minimal computational cost and the present solutions obtained will serve as new solutions to the modified Korteweg–de-Vries (mKdV) equations.

Constructing (2+1)-dimensional supersymmetric integrable systems from the Hirota formalism in the superspace**Project supported by the National Natural Science Foundation of China (Grant Nos. 11605102, 11475052, 11675055, and 11626140).

The supersymmetric extensions of two integrable systems, a special negative Kadomtsev–Petviashvili (NKP) system and a (2+1)-dimensional modified Korteweg–de Vries (MKdV) system, are constructed from the Hirota formalism in the superspace. The integrability of both systems in the sense of possessing infinitely many generalized symmetries are confirmed by extending the formal series symmetry approach to the supersymmetric framework. It is found that both systems admit a generalization of type algebra and a Kac-Moody–Virasoro type subalgebra. Interestingly, the first one of the positive flow of the supersymmetric NKP system is another supersymmetric extension of the (2+1)-dimensional MKdV system. Based on our work, a hypothesis is put forward on a series of (2+1)-dimensional supersymmetric integrable systems. It is hoped that our work may develop a straightforward way to obtain supersymmetric integrable systems in high dimensions.

Bright soliton solutions, power series solutions and travelling wave solutions of a (3+1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili equation

In this paper, we consider the (3[Formula: see text]+[Formula: see text]1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili (mKdV-KP) equation, which can be used to describe the nonlinear waves in plasma physics and fluid dynamics. By using solitary wave ansatz in the form of sech[Formula: see text] function and a direct integrating way, we construct the exact bright soliton solutions and the travelling wave solutions of the equation, respectively. Moreover, we obtain its power series solutions with the convergence analysis. It is hoped that our results can provide the richer dynamical behavior of the KdV-type and KP-type equations.

Multiple soliton solutions for the (3+1) conformable space–time fractional modified Korteweg–de-Vries equations

In the present paper, multiple exact soliton solutions for the old and the newly introduced (3+1)-dimensional modified Korteweg–de-Vries equation (mKdV) will be sought. The mKdV equations considered feature fractional derivative orders in both the space and time variables. A variety of soliton solutions ranging from hyperbolic to periodic function solutions will be constructed using simple ansatze for the equations. Finally, the algebraic equations to be obtained along the way and graphical representations will be carried out by utilizing the Mathematica software.

Finite Genus Solutions to the Generalised Kaup-Newell Hierarchy and Two 2+1 Dimensional Modified Korteweg-de Vries Equations

Abstract A generalised Kaup-Newell (gKN) hierarchy is introduced, which starts with a system of first-order ordinary differential equations and includes the Gerdjikov-Ivanov equation. By introducing an appropriate generating function, its related Hamiltonian systems and algebraic curve are given. The Hamiltonian systems are proved to be integrable, then the gKN hierarchy is solved by Hamiltonian flows. The algebraic curve is provided with suitable genus, then based on the trace formula and Riemann-Jacobi inversion theorem, finite genus solutions of the gKN hierarchy are obtained. Besides, two 2+1 dimensional modified Korteweg-de Vries (mKdV) equations are also solved.

Soliton solutions for a (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a plasma

Under investigation in this paper is a (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation, which describes the nonlinear behaviors of ion-acoustic waves in a magnetized plasma where the cooler ions are treated as a fluid with adiabatic pressure and the hot isothermal electrons are described by a Boltzmann distribution. With the Hirota method and symbolic computation, we obtain the one-, two- and three-soliton solutions for such an equation. We graphically study the solitons related with the coefficient of the cubic nonlinearity [Formula: see text]. Amplitude of the one soliton increases with increasing [Formula: see text], but the width of one soliton keeps unchanged as [Formula: see text] increases. The two solitons and three solitons are parallel, and the amplitudes of the solitons increase with increasing [Formula: see text], but the widths of the solitons are unchanged. It is shown that the interactions between the two solitons and among the three solitons are elastic.

Some Remarks on the Riccati Equation Expansion Method for Variable Separation of Nonlinear Models

Abstract Based on the Riccati equation expansion method, 11 kinds of variable separation solutions with different forms of (2+1)-dimensional modified Korteweg–de Vries equation are obtained. The following two remarks on the Riccati equation expansion method for variable separation are made: (i) a remark on the equivalence of different solutions constructed by the Riccati equation expansion method. From analysis, we find that these seemly independent solutions with different forms actually depend on each other, and they can transform from one to another via some relations. We should avoid arbitrarily asserting so-called “new” solutions; (ii) a remark on the construction of localised excitation based on variable separation solutions. For two or multi-component systems, we must be careful with excitation structures constructed by all components for the same model lest the appearance of some un-physical structures. We hope that these results are helpful to deeply study exact solutions of nonlinear models in physical, engineering and biophysical contexts.

Darboux Transformations and N-soliton Solutions of Two (2+1)-Dimensional Nonlinear Equations

Two Darboux transformations of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka (CDGKS) equation and (2+1)-dimensional modified Korteweg-de Vries (mKdV) equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux transformations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the (2+1)-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the (2+1)-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.

A variety of (3 + 1)-dimensional mKdV equations derived by using the mKdV recursion operator

We establish a new variety of (3+1)-dimensional modified Korteweg–de Vries (mKdV) equations. The recursion operator of the mKdV equation is used to derive these higher-order dimensional integrable mKdV equations. The new integrable equations generate distinct solitons structures and distinct dispersion relations as well.

A new bidirectional generalization of (2+1)-dimensional matrix k-constrained Kadomtsev-Petviashvili hierarchy

We introduce a new bidirectional generalization of (2+1)-dimensional k-constrained Kadomtsev-Petviashvili (KP) hierarchy ((2+1)-BDk-cKPH). This new hierarchy generalizes (2+1)-dimensional k-cKP hierarchy, (tA, τB) and (γA, σB) matrix hierarchies. (2+1)-BDk-cKPH contains a new matrix (1+1)-k-constrained KP hierarchy. Some members of (2+1)-BDk-cKPH are also listed. In particular, it contains matrix generalizations of Davey-Stewartson (DS) systems, (2+1)-dimensional modified Korteweg-de Vries equation and the Nizhnik equation. (2+1)-BDk-cKPH also includes new matrix (2+1)-dimensional generalizations of the Yajima-Oikawa and Melnikov systems. Binary Darboux Transformation Dressing Method is also proposed for construction of exact solutions for equations from (2+1)-BDk-cKPH. As an example the exact form of multi-soliton solutions for vector generalization of the DS system is given.

New Variable Separation Solutions for Two Nonlinear Evolution Equations in Higher Dimensions

Based on the multi-linear variable separation approach, a new direct variable separation algorithm is proposed. The effectiveness of the algorithm is demonstrated by the applications of the (2+1)-dimensional modified Korteweg-de Vries equation and the (3+1)-dimensional BKP equation. The new variable separation solutions which include at least one arbitrary function are derived for these two equations with the aid of Maple.

New variable separation solutions and nonlinear phenomena for the (2+1)-dimensional modified Korteweg–de Vries equation

Variable separation approach, which is a powerful approach in the linear science, has been successfully generalized to the nonlinear science as nonlinear variable separation methods. The (2 + 1)-dimensional modified Korteweg–de Vries (mKdV) equation is hereby investigated, and new variable separation solutions are obtained by the truncated Painlevé expansion method and the extended tanh-function method. By choosing appropriate functions for the solution involving three low-dimensional arbitrary functions, which is derived by the truncated Painlevé expansion method, two kinds of nonlinear phenomena, namely, dromion reconstruction and soliton fission phenomena, are discussed.