Modified Korteweg-de Vries Equation Research Articles

Rational Solutions to the ABS List: Transformation Approach

In the paper we derive rational solutions for the lattice potential modified Korteweg-de Vries equation, and Q2, Q1($\delta$), H3($\delta$), H2 and H1 in the Adler-Bobenko-Suris list. B\"acklund transformations between these lattice equations are used. All these rational solutions are related to a unified $\tau$ function in Casoratian form which obeys a bilinear superposition formula.

Two wave mode higher-order modified KdV equations

PurposeThe purpose of this paper is concerned with developing two-mode higher-order modified Korteweg-de Vries (KdV) equations. The study shows that multiple soliton solutions exist for essential conditions related to the nonlinearity and dispersion parameters.Design/methodology/approachThe proposed technique for constructing a two-wave model, as presented in this work, has been shown to be very efficient. The employed approach formally derives the essential conditions for soliton solutions to exist.FindingsThe examined two-wave model features interesting results in propagation of waves and fluid flow.Research limitations/implicationsThe paper presents a new and efficient algorithm for constructing and studying two-wave-mode higher-order modified KdV equations.Practical implicationsA two-wave model was constructed for higher-order modified KdV equations. The essential conditions for multiple soliton solutions to exist were derived.Social implicationsThe work shows the distinct features of the standard equation and the newly developed equation.Originality/valueThe work is original and this is the first time for two-wave-mode higher-order modified KdV equations to be constructed and studied.

Numerical Analysis of Plasma KdV Equation: Time-Fractional Approach

Numerical solution of the fractional order Modified Korteweg-de Vries equation governing the dynamics, is approximated using a novel space spectral time fractional finite difference tool. A spectral technique for space and a multi-step finite difference scheme for time are designed and implemented. The spatial spectral discretization error and the stability bounds are discussed. The nonlinear phenomena of plasma waves are well demonstrated with the aid of graphical analysis.

On solitary wave solutions of a class of nonlinear partial differential equations based on the function [formula omitted

The method of simplest equation is applied for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. The used simplest equation is fξ2=n2(f2−f(2n+2)/n). The developed methodology is illustrated on examples of classes of nonlinear partial differential equations that contain: (i) only monomials of odd grade with respect to participating derivatives; (ii) only monomials of even grade with respect to participating derivatives; (iii) monomials of odd and monomials of even grades with respect to participating derivatives. The obtained solitary wave solution for the case (i) contains as particular cases the solitary wave solutions of Korteweg–deVries equation and of a version of the modified Korteweg–deVries equation. One of the obtained solitary wave solutions for the case (ii) is a solitary wave solution of the classic version of the Boussinesq-type equation.

Some New Exact Solutions of the Modified KdV Equation Using Lie Point Symmetry Method

In this paper, we obtain some new exact solutions of the modified Korteweg de Vries equation using Lie point symmetry method. The new obtained solutions are doubly periodic and periodic solutions. We confirm that this equation possesses many unusual features of distinct solutions. To our knowledge, the obtained solutions are new.

New doubly-periodic solutions for the new integrable non local modified KdV equation

In this paper, new doubly-periodic solutions will be found for a new integrable nonlocal modified Korteweg-de Vries equation (mKdV). We adopt a technique using a travelling wave transformation which results in different solutions represented in Jacobi elliptic functions.

Initial-boundary value problems of the coupled modified Korteweg–de Vries equation on the half-line via the Fokas method

In this paper, we implement the Fokas method in order to study initial-boundary value problems of the coupled modified Korteweg–de Vries equation formulated on the half-line, with Lax pairs involving matrices. This equation can be considered as a generalization of the modified KdV equation. We show that the solution can be written in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions and , which are respectively determined by the initial values and boundary values at . Finally, the associated Dirichlet to Neumann map of the equation is analyzed in detail. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Abstract The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV–nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions.

The Role Of Painleve II In Predicting New Liquid Crystal Self-Assembly Mechanisms

We prove the existence of a new class of solutions, called shadow kinks, of the Painleve II equation \({{\frac {{\rm d}^{2} w}{{\rm d}z^{2}}}=2w^{3} +zw+\alpha,}\) where \({\alpha < 0}\) is a constant. Shadow kinks are sign changing solutions which satisfy \({ w(z) \sim -{\sqrt {-z/2}}\ {\rm as}\ z \to - \infty}\) and \({w(z) \sim -{\frac {\alpha}{z}} \ {\rm as}\ z \to \infty.}\) These solutions play a critical role in the prediction of a new class of topological defects, one dimensional shadow kinks and two dimensional shadow vortices, in light-matter interaction experiments on nematic liquid crystals. These new defects are physically important since it has recently been shown (Wang et al. in Nat Mater 15:106–112, 2016) that topological defects are a “template for molecular self-assembly” in liquid crystals. Connections with the modified KdV equation are also discussed.

New solitary solutions of the Gardner equation and Whitham–Broer–Kaup equations by the modified simplest equation method

In this paper, the modified simplest equation method (SEM) is employed to handle the Gardner equation and Whitham-Broer-Kaup (WBK) equations, respectively. Five new solitary solutions are accordingly obtained. The paper proceeds step-by-step with increasing detail about derivation process, first illustrating the algorithm of the modified SEM, which the simplest equation and its exact solution are not pre-defined. Then, new solutions are used to illustrate the connection between the Gardner equation and WBK equations. In addition, new solutions are reconstructed as solutions for the VBEs, ALWEs, the dispersive long wave equations, and the modified KdV equation.

Superregular breathers in a complex modified Korteweg-de Vries system.

We study superregular (SR) breathers (i.e., the quasi-Akhmediev breather collision with a certain phase shift) in a complex modified Korteweg-de Vries equation. We demonstrate that such SR waves can exhibit intriguing nonlinear structures, including the half-transition and full-suppression modes, which have no analogues in the standard nonlinear Schrödinger equation. In contrast to the standard SR breather formed by pairs of quasi-Akhmediev breathers, the half-transition mode describes a mix of quasi-Akhmediev and quasi-periodic waves, whereas the full-suppression mode shows a non-amplifying nonlinear dynamics of localized small perturbations associated with the vanishing growth rate of modulation instability. Interestingly, we show analytically and numerically that these different SR modes can be evolved from an identical localized small perturbation. In particular, our results demonstrate an excellent compatibility relation between SR modes and the linear stability analysis.

Bäcklund transformation and CRE solvability for the negative-order modified KdV equation

The residual symmetry is obtained for the negative-order modified KdV (nmKdV) equation from the truncated Painleve expansion by using symbolic computation. The multiple residual symmetries are presented and localized by introducing auxiliary variables, and then, nth Backlund transformation in terms of determinant is derived. By using the consistent Riccati expansion (CRE) method, the nmKdV equation is proved to be CRE solvable and exact soliton–Weierstrass elliptic function interaction solutions are constructed from the final consistent equation.

Multiple closed form wave solutions to the KdV and modified KdV equations through the rational (G′/G)-expansion method

The purpose of this article is to generate the closed form traveling wave solutions of some nonlinear partial differential equations namely; the KdV equation and the modified KdV equation using the recently established rational (G′/G)-expansion method. By this method, new and further general closed form solutions expressed by three types of functions as for instance hyperbolic, trigonometric and rational function solutions are found. The basic ideas presented in this article and the obtained solutions might be useful to analyze the shallow water waves, ion acoustic waves in plasma, acoustic waves on a crystal lattice and the role of nonlinear dispersion in the formation of structures like liquid drops.

Qualitative analysis of the positron-acoustic waves in electron-positron-ion plasmas with κ deformed Kaniadakis distributed electrons and hot positrons

Qualitative analysis of the positron acoustic (PA) waves in a four-component plasma system consisting of static positive ions, mobile cold positron, and Kaniadakis distributed hot positrons and electrons is investigated. Using the reductive perturbation technique, the Korteweg-de Vries (K-dV) equation and the modified KdV equation are derived for the PA waves. Variations of the total energy of the conservative systems corresponding to the KdV and mKdV equations are presented. Applying numerical computations, effect of parameter (κ), number density ratio (μ1) of electrons to ions and number density (μ2) of hot positrons to ions, and speed (U) of the traveling wave are discussed on the positron acoustic solitary wave solutions of the KdV and mKdV equations. Furthermore, it is found that the parameter κ has no effect on the solitary wave solution of the KdV equation, whereas it has significant effect on the solitary wave solution of the modified KdV equation. Considering an external periodic perturbation, the perturbed dynamical systems corresponding to the KdV and mKdV equations are analyzed by employing three dimensional phase portrait analysis, time series analysis, and Poincare section. Chaotic motions of the perturbed PA waves occur through the quasiperiodic route to chaos.

Classification of Dark Modified KdV Equation**Supported by the Global Change Research Program of China under Grant No. 2015Cb953904, National Natural Science Foundation of China under Grant Nos. 11675054, 11435005, 11175092, and 11205092 and Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213) and K. C. Wong Magna Fund in Ningbo University

The dark Korteweg-de Vries (KdV) systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV (MKdV) systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Furthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The recursion operators of two classes of dark MKdV systems are also given.

Dynamics of the smooth positons of the complex modified KdV equation

We provide the explicit formulae of the smooth positon solutions of the complex modified KdV (mKdV) equation using degenerate Darboux transformation with respect to the eigenvalues. The dynamics of the smooth positons of the complex mKdV are discussed in details using the method, i.e. decomposition of the modulus square. For this kind solution, we show explicitly the decomposition, bent trajectory and variable ‘phase shift’ after collision, which are remarkably different from the singular positons of the real-valued mKdV equation.

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.

CRE Solvability, Nonlocal Symmetry and Exact Interaction Solutions of the Fifth-Order Modified Korteweg-de Vries Equation**Supported by National Natural Science Foundation of China under Grant No. 11505090, and Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009

This paper is concerned with the fifth-order modified Korteweg-de Vries (fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion (CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion (CTE) method, the nonlocal symmetry related to the consistent tanh expansion (CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlevé method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.

Darboux and Binary Darboux Transformations for Discrete Integrable Systems. II. Discrete Potential mKdV Equation

The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.