Modified Korteweg-de Vries Equation Research Articles

The inverse scattering transform and soliton solutions of a combined modified Korteweg–de Vries equation

Abstract The inverse scattering transform is developed for a combined modified Korteweg–de Vrie equation through the technique of Riemann–Hilbert problems. From special Riemann–Hilbert problems with an identity jump matrix, soliton solutions are generated, which corresponds to the inverse scattering problems with reflectionless coefficients. A specific example of two-soliton solutions is explicitly presented, together with its 3d plots, contour plots and x-curve plots.

Numerical solutions of the modified KdV Equation with collocation method

Numerical solutions of the modified KdV Equation with collocation method

The design of conservative finite element discretisations for the vectorial modified KdV equation

We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg–de Vries equation with periodic boundary conditions. We demonstrate that the scheme conserves energy up to solver tolerance. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting.

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.

A τ-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter

A matrix spectral problem is researched with an arbitrary parameter. Through zero curvature equations, two hierarchies are constructed of isospectral and nonisospectral generalized derivative nonlinear schrödinger equations. The resulting hierarchies include the Kaup-Newell equation, the Chen-Lee-Liu equation, the Gerdjikov-Ivanov equation, the modified Korteweg-de Vries equation, the Sharma-Tasso-Olever equation and a new equation as special reductions. The integro-differential operator related to the isospectral and nonisospectral hierarchies is shown to be not only a hereditary but also a strong symmetry of the whole isospectral hierarchy. For the isospectral hierarchy, the corresponding τ -symmetries are generated from the nonisospectral hierarchy and form an infinite-dimensional symmetry algebra with the K-symmetries.

Normal form approach to near-linear dynamics of modified KdV equation

In this paper, we implement normal form reduction to the periodic modified Korteweg–de Vries (mKdV) equation to investigate the behavior of a solution when a subtle high-frequency initial data is given. We use differentiation by parts to decompose the equation into resonant and non-resonant parts and provide some nonlinear estimates for each term. If a subtle high-frequency initial data is given, a solution of the mKdV equation can be approximated by a solution of the linearized mKdV equation for large times.

Plasma Parameters Effects on Dust Acoustic Solitary Waves in Dusty Plasmas of Four Components

The presence and propagation of dust-acoustic solitary waves in dusty plasma contains four components such as negative and positive dust species beside ions and electrons are studied. Both the ions and electrons distributions are represented applying nonextensive formula. Employing the reductive perturbation method, an evolution equation is derived to describe the small-amplitude dust-acoustic solitons in the considered plasma system. The used reductive perturbation stretches lead to the nonlinear KdV and modified KdV equations with nonlinear and dispersion coefficients that depend on the parameters of the plasma. This study represents that the presence of compressive or/and rarefactive solitary waves depends mainly on the value of the first-order nonlinear coefficient. The structure of envelope wave is undefined for first-order nonlinear coefficient tends to vanish. The coexistence of the two types of solitary waves appears by increasing the strength of nonlinearity to the second order using the modified KdV equation.

An improved lattice hydrodynamic model considering the driver’s desire of driving smoothly

Besides the forward vehicles’ traffic situation, driving behaviors may be influenced by the driver’s desire of driving smoothly, which is embodied by the fact that many drivers like to adjust current traffic situation to the optimal steady state. With the consideration of the effect of driver’s desire of driving smoothly, an improved lattice hydrodynamic model is proposed in this paper. The analytical stability criterion is derived by the linear analysis method. And then the modified KdV equation is deduced to describe the characteristic of traffic jams near the critical point. In addition, the numerical simulations are presented to investigate the effect of the factor contained herein on the traffic stability. The analytical and numerical results all show that traffic jams can be effectively relieved when the factor of driver’s desire of driving smoothly is incorporated into the traffic model.

Ultrashort spatiotemporal optical solitons in waveguide arrays: the effect of combined linear and nonlinear couplings

We investigate the propagation of Gaussian spatiotemporal inputs in arrays of parallel optical waveguides, assuming both linear and nonlinear non-dispersive coupling between adjacent guides. The numerical simulations employ a discrete version of the modified Korteweg–de Vries equation that adequately describe the propagation of few-cycle spatiotemporal solitons in coupled waveguide arrays. It is shown that depending on the numerical values of the key parameters of the spatiotemporal Gaussian input, the combined effect of linear and nonlinear couplings leads to either the diffraction and dispersion spreading effect or to the formation of different types of (2 + 1)-dimensional ultrashort spatiotemporal solitons such as one or many single-channel few-cycle solitons, and half-cycle (single-humped) ones.

Anatomy of modified Korteweg–de Vries equation for studying the modulated envelope structures in non-Maxwellian dusty plasmas: Freak waves and dark soliton collisions

In this work, we study, in a systematic way, dust-acoustic modulated envelope structures such as rogue waves (unstable waves) and dark soliton (stable waves) collisions in a complex plasma with nonthermal ions and Boltzmann electrons. In the present plasma system, we can have both negative and positive potential structures associated with the nonlinear dust-acoustic structures. Therefore, we derived the modified Korteweg–de Vries (mKdV) equation, by using the reductive perturbation technique, to describe the nonlinear structures at critical plasma parameters. For studying the properties of the modulated envelope structures, the mKdV equation transformed to a nonlinear Schrödinger equation. Depending on the modulational instability analysis, the stability and instability regions for the propagating nonlinear modulated waves have been determined precisely. After that, the properties of the dust-acoustic rogue waves are examined within the instability regions. Moreover, the effects of physical parameters, such as the ion-to-electron temperature ratio and the ion nonthermal parameter on the profile of dust-acoustic rogue waves are examined. Furthermore, our investigations extended to study the head-on collisions of two-dark solitons in the stability regions. Using the extended Poincaré–Lighthill–Kuo perturbation method, the dark solitons in the present plasma system develop according to two quasi-Korteweg–de Vries equations. After that, the phase shifts induced by the face-to-face collisions between two-dark solitons are obtained analytically. Also, the effects of the above physical parameters on the phase shifts are reported. The results may have relevance in space and laboratory dusty plasmas.

Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

We consider the fifth-order modified Korteweg–de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in [Formula: see text], [Formula: see text], via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173(2) (2008) 265–304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation.

Effect of the driver’s desire for smooth driving on the car-following model

To reduce the fuel consumption caused by the acceleration or deceleration, many drivers wish to drive cars as smoothly as possible, which means that driving behavior is influenced by the driver’s desire for smooth driving. In this paper, with the consideration of the driver’s desire for smooth driving, an improved car-following model is presented by incorporating the difference of the steady and history velocities into optimal velocity model. To show the effect of the factor discussed herein, the improved car-following model is investigated by the analytical and numerical methods. The linear stability criterion is obtained by the linear analysis method. And then the modified KdV equation is derived to describe the propagating behavior of traffic jams near the critical point. The amplitude of the traffic jam can be relieved by strengthening driver’s desire for smooth driving. The numerical simulations are explored to verify the validity of the analytical results and the effect of the driver’s desire for smooth driving. The analytical and numerical results demonstrate that the driver’s desire for smooth driving plays a positive role in improving traffic stability and flux.

Do N-soliton breathers exist for the Hirota equation models?

We reveal the method for systematic construction of unexpected bound soliton states (localized breathers on the zero background) and formulate specific conditions for their appearance both in the framework of the canonical and the generalized nonautonomous Hirota models. These conditions show that both all parameters of solitons forming the breather and the dispersion and nonlinearity variations cannot be chosen independently, they are related by the exact integrability of the model. Novel N-soliton breathers derived for the nonautonomous Hirota equation with vanishing boundary conditions differ in that they generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain, or losses. We apply the specific conditions for the breather formation to derive novel soliton breather solutions of the nonlinear Schrödinger and the complex modified Korteweg–de Vries equations. Consequently, we demonstrate that the way is opened to generalize and apply the presented method for other evolution equations of the Ablowitz–Kaup–Newell–Segur hierarchy.

Nonplanar dust-acoustic waves and chaotic motions in Thomas Fermi dusty plasmas

The properties of linear and nonlinear nonplanar dust acoustic (DA) solitary waves and chaotic behavior are investigated in an unmagnetized Thomas Fermi dusty plasma, whose components are degenerate electrons, ions, and negatively charged inertial cold dust grains. A linear dispersion relation is obtained and solved numerically. It has been observed that linear excitation characteristics are influenced by radial distance r, geometric term ν, and ion-to-electron Fermi temperature ratio σi. We have also noted that the addition of a geometrical term in dispersion relation gives damping along the radial axis. A modified Korteweg-de Vries (KdV) equation is derived by employing the reductive perturbation technique, and its numerical solutions are obtained. The modified KdV equation is discussed for cold dust grains in planar and nonplanar frameworks. Upon the introduction of external periodic perturbation, the perturbed modified KdV equation is studied in planar geometry via some qualitative and quantitative approaches. The perturbed KdV equation can give rise to the periodic, quasiperiodic, and chaotic motions for DA waves. The strength of the external perturbation and dust concentration h play the major role of the switching parameter in the transition of dynamic motion. The developed chaos can be weakened with the variation of dust concentration h. It has been observed that the dust concentration affects the dynamics of DA waves in planar geometry which is an important observation in this study.

Novel conditions for soliton breathers of the complex modified Korteweg–de Vries equation with variable coefficients

We reveal novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg–de Vries equation with variable coefficients and the loss (or gain) term (vc cmKdV). Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. This finding allows us to systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.

Nonlocal modified KdV equations and their soliton solutions by Hirota Method

We study the nonlocal modified Korteweg–de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz–Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz–Musslimani reduction formulas, we find one-, two-, and three-soliton solutions of nonlocal mKdV and nonlocal complex mKdV equations. The soliton solutions of these equations are of two types. We give one-soliton solutions of both types and present only first type of two- and three-soliton solutions. We illustrate our solutions by plotting their graphs for particular values of the parameters.

Dust‐ion‐acoustic solitary waves in a magnetized dusty pair‐ion plasma with Cairns‐Gurevich electrons and opposite polarity dust particles

The nonlinear dust‐ion‐acoustic (DIA) solitary structures have been studied in a dusty plasma, including the Cairns‐Gurevich distribution for electrons, both negative and positive ions, and immobile opposite polarity dust grains. The external magnetic field directed along the z‐axis is considered. By using the standard reductive perturbation technique and the hydrodynamics model for the ion fluid, the modified Zakharov–Kuznetsov equation was derived for small but finite amplitude waves and was provided the solitary wave solution for the parameters relevant. Using the appropriate independent variable, we could find the modified Korteweg–de Vries equation. By plotting some figures, we have discussed and emphasized how the different plasma values, such as the trapping parameter, the positive (or negative) dust number density, the non‐thermal electron parameter, and the ion cyclotron frequency, can influence the solitary wave structures. In addition, using the bifurcation theory of planar dynamical systems, we have extracted the centre and saddle points and illustrated the phase portrait of such a system for some particular plasma parameters. Finally, we have graphically investigated the behaviour of the solitary energy wave by changing the plasma values as well as by calculating the instability criterion; we have also discussed the growth rate of the solitary waves. The results could be useful for studying the physical mechanism of nonlinear propagation of DIA solitary waves in laboratory and space plasmas where non‐thermal electrons, pair‐ions, and dust particles can exist.

Reduction to modified KdV and its KP-like generalization via phase modulation

The main observation of this paper is that the modified Korteweg–de Vries equation has its natural origin in phase modulation of a basic state such as a periodic travelling wave, or more generally, a family of relative equilibria. Extension to 2 + 1 suggests that a modified Kadomtsev–Petviashvili (or a Konopelchenko–Dubrovsky) equation should emerge, but our result shows that there is an additional term which has gone heretofore unnoticed. Thus, through the novel application of phase modulation a new equation appears as the 2 + 1 extension to a previously known one. To demonstrate the theory it is applied to the cubic-quintic nonlinear Schrödinger (CQNLS) equation, showing that there are relevant parameter values where a modified KP equation bifurcates from periodic travelling wave solutions of the 2 + 1 CQNLS equation.

Nonlocal Symmetry and Explicit Solution of the Alice-Bob Modified Korteweg-de Vries Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11705077 and 11775104, Natural Science Foundation of Zhejiang Province under Grant No. LY14A010005, and Scientific Research Foundation of the First-Class Discipline of Zhejiang Province (B) (No. 201601)

Nonlocal symmetry and explicit solution of the integrable Alice-Bob modified Korteweg-de Vries (AB-mKdV) equation is discussed, which has been established by the aid of the shifted parity and delayed time reversal to describe two-place events. Based on the Lax pair which contains the two-order partial derivative, the Lie symmetry group method is successfully applied to find the exact invariant solution for the AB-mKdV equation with nonlocal symmetry by introducing one suitable auxiliary variable. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to some specific functions are derived. Figures show the physical phenomenon, that is, “the shifted parity and delayed time reversal to describe two-place events”.

Dynamics of new higher-order rational soliton solutions of the modified Korteweg–de Vries equation

In this paper, we propose a generalised perturbation \((n, N-n)\)-fold Darboux transformation (DT) of the modified Korteweg–de Vries (mKdV) equation using the Taylor expansion and a parameter limit procedure. We apply the generalised perturbation \((1, N-1)\)-fold DT to find the new explicit higher-order rational soliton (RS) solutions in terms of determinants of the mKdV equation. These higher-order RS solutions are different from those known soliton results in terms of hyperbolic functions which are obtained from the classical iterated DT. The dynamics behaviours of the first-, second-, third-, and fourth-order RS solutions are shown graphically. The wave propagation characteristics and stability are also discussed using numerical simulations. We find that the initial constant seed solution plays an important role on the wave propagation stability of RS. Through Miura transformation, we give some complex higher-order rational solutions of the Korteweg–de Vries (KdV) equation which are different from the known results. The relevant structures also are discussed using some figures. The method used can also be extended to seek explicit rational solutions of other nonlinear integrable equations.