Modified Kortweg-de Vries Research Articles

Large‐time asymptotics to the focusing nonlocal modified Kortweg‐de Vries equation with step‐like boundary conditions

AbstractWe investigate the large‐time asymptotics of solution for the Cauchy problem of the nonlinear focusing nonlocal modified Kortweg‐de Vries (MKdV) equation with step‐like initial data, that is, as , as , where A is an arbitrary positive real number. We first develop the direct scattering theory to establish the basic Riemann‐Hilbert (RH) problem associated with step‐like initial data. Thanks to the symmetries , of nonlocal MKdV equation, we investigate the asymptotics as and , respectively. Our main technique is to use the steepest descent analysis to deform the original matrix‐valued RH problem to corresponded regular RH problem, which could be explicitly solved. Finally, we obtain the different large‐time asymptotic behaviors of the solution of the Cauchy problem for focusing nonlocal MKdV equation in different space‐time sectors , , , and on the whole ‐plane.

Characteristics of supernonlinear and coexistence features for electron-acoustic waves in an adiabatic quantum plasma

Bifurcation of quantum electron-acoustic waves (QEAWs) is studied in an adiabatic quantum plasma with nonextensively distributed hot electrons. Using reductive perturbation technique, modified Kortweg de Vries (KdV) equation is derived with dual power nonlinearity for highly nonlinear QEAWs. Using Galilean transformation the modified KdV equation is reduced to a planar dynamical system with three equilibrium points. Applying phase plane analysis periodic wave solutions and supernonlinear periodic wave solutions for QEAWs are perceived. It is found that the amplitude of the nonlinear structures is indirectly proportional to the ratio of number densities ( $$\mu $$ ), the cold to hot electron temperature ratio ( $$\sigma $$ ) and the nonextensivity parameter (q), we have explained physically that confronts our results. In addition, coexistence of superperiodic and quasiperiodic phenomena, quasiperiodic and chaotic phenomena and chaotic, superperiodic and quasiperiodic phenomena for QEAWs are observed for appropriate initial values.

Analysis of an extended two-lane lattice hydrodynamic model considering mixed traffic flow and self-stabilization effect

PurposeThe purpose of this paper is to explore the impact of the mixed traffic flow, self-stabilization effect and the lane changing behavior on traffic flow stability.Design/methodology/approachAn extended two-lane lattice hydrodynamic model considering mixed traffic flow and self-stabilization effect is proposed in this paper. Through linear analysis, the stability conditions of the extended model are derived. Then, the nonlinear analysis of the model is carried out by using the perturbation theory, the modified Kortweg–de Vries equation of the density of the blocking area is derived and the kink–antikink solution about the density is obtained. Furthermore, the results of theoretical analysis are verified by numerical simulation.FindingsThe results of numerical simulation show that the increase of the proportion of vehicles with larger maximum speed or larger safe headway in the mix flow are not conducive to the stability of traffic flow, while the self-stabilization effect and lane changing behavior is positive to the alleviation of traffic congestion.Research limitations/implicationsThis paper does not take into account the factors such as curve and slope in the actual road environment, which will have more or less influence on the stability of traffic flow, so there is still a certain gap with the real traffic environment.Originality/valueThe existing two-lane lattice hydrodynamic models are rarely discussed in the case of mixed traffic flow. The improved model proposed in this paper can better reflect the actual traffic, which can also provide a theoretical reference for the actual traffic governance.

Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes

In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.

Wave Propagation Analysis in a One – Dimensional Lattice with Long Range Interactions and Its Continuum Model

Fermi-Pasta-Ulam (FPU) lattice known as a nonlinear lattice includes nonlinear terms explicitly. Therefore, we can evaluate nonlinear effects of FPU lattice straightforwardly. It includes, however, only the nearest neighbor interactions. In some cases, long range interaction plays an important role in dynamical properties in real crystals. We take the second nearest neighbor interactions into account in FPU lattice. We derive the nonlinear wave equation (modified Kortweg-de Vries – type equation) from the nonlocal FPU lattice as a continuum limit. Linear dispersion relationship of the wave equation is discussed. We also find as exact solution of soliton in the nonlinear wave equation analytically. We reveal that the velocity of the soliton is related to the second nearest neighbor interactions. We analytically evaluate the temporal evolution of the soliton and estimate the spectrum of the solitary wave using two dimensional FFT in case of different interactions. The result shows that the second nearest neighbor interactions affects the appearance of the lean of peak.

Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients

Abstract In this paper, Hirota’s bilinear method is extended to a new modified Kortweg–de Vries (mKdV) hierarchy with time-dependent coefficients. To begin with, we give a bilinear form of the mKdV hierarchy. Based on the bilinear form, we then obtain one-soliton, two-soliton and three-soliton solutions of the mKdV hierarchy. Finally, a uniform formula for the explicit N-soliton solution of the mKdV hierarchy is summarized. It is graphically shown that the obtained soliton solutions with time-dependent functions possess time-varying velocities in the process of propagation.

Modified Kortweg-de Vries equation approach to zero-energy states of graphene

We utilize the relation between soliton solutions of the modified Kortweg-de Vries (mKdV) and the combined mKdV-KdV equation and the Dirac equation to construct electrostatic fields which yield exact zero-energy states in graphene. The examples considered here are fairly general and it has also been shown that some of these examples reproduce the previously known results when parameters are chosen suitably.

Nonlinear dust acoustic waves in inhomogeneous four-component dusty plasma with opposite charge polarity dust grains

The reductive perturbation technique is employed to investigate the propagation properties of nonlinear dust acoustic (DA) waves in a four-component inhomogeneous dusty plasma (4CIDP). The 4CIDP consists of both positive- and negative-charge dust grains, characterized by different mass, temperature, and density, in addition to a background of Maxwellian electrons and ions. The inhomogeneity caused by nonuniform equilibrium values of particle densities, fluid velocities, and electrostatic potential leads to a significant modification to the nature of nonlinear DA solitary waves. It is found that this model reveals two DA wave velocities, one slow, λs, and the other is fast, λf. The nonlinear wave evolution is governed by a modified Kortweg-de Vries equation, whose coefficients are space dependent. Both the two soliton types; compressive and rarefactive are allowed corresponding to λs. However, only compressive soliton is created corresponding to λf. The numerical investigations illustrate the dependence of the soliton amplitude, width, and velocity on the plasma inhomogeneities in each case. The relevance of these theoretical results with 4CIDPs observed in a multi-component plasma configurations in the polar mesosphere is discussed.

On the exact solutions of a modified Kortweg de Vries type equation and higher-order modified Boussinesq equation with damping term

In this paper, we obtain exact solutions of two nonlinear evolution equations, namely the modified Kortweg de Vries equation and the higher-order modified Boussinesq equation with damping term. The method employed to obtain the exact solutions is the (G � /G)-expansion method. Traveling wave solutions of three types are obtained

The general analytical and numerical solution for the modified KdV equation with convergence analysis

We investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge–Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge–Kutta discontinuous Galerkin method by some numerical illustrations. Copyright © 2012 John Wiley & Sons, Ltd.

N-fold Darboux transformation and double-Wronskian-typed solitonic structures for a variable-coefficient modified Kortweg-de Vries equation

Under investigation in this paper is a variable-coefficient modified Kortweg-de Vries (vc-mKdV) model describing certain situations from the fluid mechanics, ocean dynamics and plasma physics. N-fold Darboux transformation (DT) of a variable-coefficient Ablowitz–Kaup–Newell–Segur spectral problem is constructed via a gauge transformation. Multi-solitonic solutions in terms of the double Wronskian for the vc-mKdV model are derived by the reduction of the N-fold DT. Three types of the solitonic interactions are discussed through figures: (1) Overtaking collision; (2) Head-on collision; (3) Parallel solitons. Nonlinear, dispersive and dissipative terms have the effects on the velocities of the solitonic waves while the amplitudes of the waves depend on the perturbation term.

Painlevé property, Lax pair and Darboux transformation of the variable-coefficient modified Kortweg-de Vries model in fluid-filled elastic tubes

With the consideration on the artery as a thin walled prestressed elastic tube with variable radius, a variable-coefficient modified Kortweg-de Vries (vc-mKdV) equation is obtained by the long wave approximation for the blood which is assumed as the incompressible non-viscous fluid. In the present paper, we firstly investigate the Painlevé property of the vc-mKdV equation. Furthermore, with the Ablowitz-Kaup-Newell-Segur procedure and symbolic computation, the Lax pair of the vc-mKdV equation is constructed, by virtue of which we construct the Darboux transformation and a new soliton solution. Finally, the features of the new solution are discussed to illustrate the influences of the constant and variable coefficients in the solitonic propagation.

Darboux Transformation and Soliton Solutions for a Variable-Coefficient Modified Kortweg-de Vries Model from Fluid Mechanics, Ocean Dynamics, and Plasma Mechanics

This paper is to investigate a variable-coefficient modified Kortweg-de Vries (vc-mKdV) model, which describes some situations from fluid mechanics, ocean dynamics, and plasma mechanics. By the Ablowitz–Kaup–Newell–Segur procedure and symbolic computation, the Lax pair of the vc-MKdV model is derived. Then, based on the aforementioned Lax pair, the Darboux transformation is constructed and a new one-soliton-like solution is obtained as well. Features of the one-soliton-like solution are analyzed and graphically discussed to illustrate the influence of the variable coefficients in the solitonlike propagation.

Planar and nonplanar dust acoustic solitary waves in electron–positron–ion–dust plasmas

Planar and nonplanar dust acoustic (DA) solitary waves are studied in an unmagnetized plasma consisting of electrons, positrons, ions and dust species. This is done by deriving the modified Kortweg–de Vries (mKdV) equation under the small amplitude perturbation expansion method. It is found that in the presence of positrons, DA hump-like solitary waves may also exist under suitable conditions. It is also observed that the positron concentration and the ratio of ion-to-electron temperature significantly modify the propagation characteristics of DA solitary waves. The range of plasma parameters for which we obtain hump-like solitons are found numerically. The present investigation may have relevance in both laboratory and astrophysical plasmas.

Parametrically controlling solitary wave dynamics in the modified Korteweg-de Vries equation

We demonstrate the control of solitary wave dynamics of modified Kortweg-de Vries (MKdV) equation through the temporal variations of the distributed coefficients. This is explicated through exact cnoidal wave and localized soliton solutions of the MKdV equation with variable coefficients. The solitons can be accelerated and their propagation can be manipulated by suitable variations of the above parameters. In sharp contrast with nonlinear Schr\"{o}dinger equation, the soliton amplitude and widths are time independent.

On solitary wave solutions for the two-dimensional nonlinear modified Kortweg–de Vries–Burger equation

Abstract In this paper, we construct an analytic solution of the nonlinear modified Kortweg–de Vries–Burger equation. By applying a special solution of Cole–Hopf transformation to an ordinary differential equation, we propose a bounded travelling wave solution u(x, y, t) = U(ξ) where ξ = kx + ly − wt, to the two-dimensional nonlinear modified Kortweg–de Vries–Burger equation. We use Jacobian and rational methods to calculate the solitary wave solutions for the two-dimensional modified Kortweg–de Vries–Burger equation.

On some classes of mKdV periodic solutions

We obtain exact periodic solutions of the positive and negative modified Kortweg-de Vries (mKdV) equations. We examine the dynamical stability of these solitary wave lattices through direct numerical simulations. While the positive mKdV breather lattice solutions are found to be unstable, the two-soliton lattice solution of the same equation is found to be stable. Similarly, a negative mKdV lattice solution is found to be stable. We also touch upon the implications of these results for the KdV equation.

Low-frequency electromagnetic solitary and shock waves in an inhomogeneous dusty magnetoplasma

It is shown that the nonlinear dynamics of one-dimensional Shukla mode [Phys. Lett. A 316, 238 (2003)] is governed by a modified Kortweg–de Vries–Burgers equation. The latter admits stationary solutions in the form of either a solitary wave or a monotonic/oscillatory shock. The present nonlinear waves may help to understand the salient features of localized density and magnetic field structures in molecular dusty clouds as well as in low-temperature laboratory dusty plasma discharges.

Dynamics of nonlinear dust lattice waves in the presence of ion plasma waves

The parametric interaction between nonlinear dust lattice waves and ion plasma waves in a plasma sheath is considered. It is found that the interaction is governed by a pair of equations comprising a modified Kortweg–de Vries equation for the driven (by the ion plasma wave ponderomotive force) nonlinear dust lattice waves and a nonlinear Schrödinger equation for the modulated ion plasma wave envelops. The pair admits stationary solutions in the form of a dust density cavity in which localized ion plasma wave electric field envelope is trapped.

Modified Kortweg–de Vries hierarchies in multiple–time variables and the solutions of modified Boussinesq equations

We study solitary-wave and kink-wave solutions of a modified Boussinesq equation through a multiple-time reductive perturbation method. We use appropriated modified Korteweg-de Vries hierarchies to eliminate secular producing terms in each order of the perturbative scheme. We show that the multiple-time variables needed to obtain a regular perturbative series are completely determined by the associated linear theory in the case of a solitary-wave solution, but requires the knowledge of each order of the perturbative series in the case of a kink-wave solution. These appropriate multiple-time variables allow us to show that the solitary-wave as well as the kink-wave solutions of the modified Botussinesq equation are actually respectively a solitary-wave and a kink-wave satisfying all the equations of suitable modified Korteweg-de Vries hierarchies.