Modified Korteweg-de Vries Research Articles

Analysis of soliton interactions of modified Korteweg-de Vries equation using conserved quantities

In this paper, the conservative quantities are used to develop an approximate method to calculate the merged waveform shape of the solitary waves described by modified Korteweg–de Vries (mKdV) equation. With this method, we can efficiently and effectively capture the physics of the complicated merging phenomena when two solitary waves described by the nonlinear evolution partial differential equation merge at the maximum without the need to solve the equation in detail. This offers a simple and robust tool to analyse the interactions between solitons and to benchmark the results obtained by the asymptotic and numerical methods. It is expected that the approximate analysis demonstrated in this paper can be applied to a series of nonlinear evolution equations to simulate various solitary wave interaction problems. In future, our goal is to extend this simple method to other nonlinear wave evolution phenomena.

The deformed modified Korteweg–de Vries equation: Multi-soliton solutions and their interactions

The deformed modified Korteweg–de Vries equation: Multi-soliton solutions and their interactions

Semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg–de Vries equations

The ubiquitous Korteweg–de Vries equation is widely used in oceanic, atmospheric dynamics, and fluid mechanics and can explain almost all kinds of physical phenomena of waves, while explicit solutions are studied in nonlinear mechanics and optics. We investigate explicit solutions for the semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg–de Vries (cmKdV) equations, which describe sound waves in an inharmonic lattice. A semi-discrete four-component Frobenius-coupled cmKdV equation is presented by utilizing a zero-curvature equation. We use two different reduction methods to get a semi-discrete local Frobenius-coupled cmKdV equation and a semi-discrete nonlocal Frobenius-coupled cmKdV equation. In the paper, we study the above two new types of semi-discrete Frobenius-coupled cmKdV equations. Then the bi-Hamiltonian structures of the above two types of equations are constructed, which show that the two types of equations possess Liouville integrability. The covariant properties of the both new equations are shown by constructing their corresponding Darboux transformation (DT), respectively. Soliton solutions, semi-rational soliton solutions, and rogue wave solutions of the above two types of equations are constructed by N-fold DT and generalized (n,N−n)-fold DT. Three-dimensional plots and density profiles of these explicit solutions are presented. The characteristics of these figures demonstrate the relationship between the above two types of equations.

A six-component integrable hierarchy and its Hamiltonian formulation

The aim of this paper is to construct a six-component integrable hierarchy associated with a matrix spatial spectral problem of arbitrary order. The adopted method is the zero curvature formulation. The corresponding Hamiltonian formulation is furnished by using the trace identity, which guarantees the Liouville integrability for the resulting hierarchy. Two illustrative examples of integrable equations of lower orders are six-component coupled nonlinear Schrödinger equations and modified Korteweg–de Vries equations.

AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials

The aim of this paper is to derive a kind of integrable hierarchies with four potentials, which possess bi-Hamiltonian structures, from reduced AKNS matrix spectral problems. The associated recursion operators are worked out explicitly. The Lax pair formulation and the trace identity are basic tools in the analysis. Two nonlinear examples in the resulting integrable hierarchies are integrable nonlinear Schrödinger type equations and integrable modified Korteweg–de Vries type equations.

Improved Uniqueness of Multi-breathers of the Modified Korteweg–de Vries Equation

Improved Uniqueness of Multi-breathers of the Modified Korteweg–de Vries Equation

Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation

AbstractWe analyse the case of a dense modified Korteweg–de Vries (mKdV) soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann–Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally, we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.

The Whitham Modulation Solution of the Complex Modified KdV Equation

This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler–Poisson–Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory.

Dust acoustic solitary waves in dusty plasma with nonuniform temperature

In this paper, the dust acoustic wave is studied in conditions where the dust temperature is not constant. This model includes negatively charged dust particles and thermal ions. The reductive perturbation technique is used and the new point in this study is the nonuniformity of the plasma temperature. The nonconstancy of the temperature is entered as a first-order perturbation in the calculations, and finally the modified Korteweg–de Vries (mKdV) equation is derived. The solution of the modified KdV equation clarifies the change in soliton shape of the wave when it moves in the perturbation plasma. We show how the soliton wave undergoes deformation and amplitude reduction during propagation when it encounters a region with a different temperature. The output of this research can be effective to better understand the wave behavior in a real plasma with nonuniform temperature.

Integrability of linear structure of components of modified Korteweg–De Vries hierarchy: multiple soliton solutions and breathers solutions

PurposeThe purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation, 5th order and 7th order mKdV equations.Design/methodology/approachThe authors investigate Painlevé integrability of the constructed linear structure.FindingsThe Painlevé analysis demonstrates that established sum of integrable models retains the integrability of each component.Research limitations/implicationsThe research also presents a set of rational schemes of trigonometric and hyperbolic functions to derive breather solutions.Practical implicationsThe authors also furnish a variety of solitonic solutions and complex solutions as well.Social implicationsThe work formally furnishes algorithms for extending integrable equations that consist of components of a hierarchy.Originality/valueThe paper presents an original work for developing Painlevé integrable model via using components of a hierarchy.

Riemann-Hilbert problem for the fifth-order modified Korteweg–de Vries equation with the prescribed initial and boundary values

In this paper, we investigate the fifth-order modified Korteweg–de Vries (mKdV) equation on the half-line via the Fokas unified transformation approach. We show that the solution u(x, t) of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ. The jump matrix L(x, t, θ) has an explicit representation dependent on x, t and it can be represented exactly by the two pairs of spectral functions y(θ), z(θ) (obtained from the initial value u 0(x)) and Y(θ), Z(θ) (obtained from the boundary conditions v 0(t), ). Furthermore, the two pairs of spectral functions y(θ), z(θ) and Y(θ), Z(θ) are not independent of each other, but are related to the compatibility condition, the so-called global relation.

An improved microscopic traffic model for heterogeneous vehicles using the vehicle's mass effect

This study aims to develop a traffic model for heterogeneous vehicle movement, which introduces the vehicle's heterogeneity by considering the internal mass effect. We explore the behavioral characteristics of the flow field generated by the proposed model and provide a comparative analysis of the conventional model. A linear stability condition is deduced to showcase the model's capacity to neutralize flow. Nonlinear analysis is employed to derive the modified Korteweg-de Vries (mKdV) equation and its corresponding analytical solution, enabling the observation of traffic flow behavior in proximity to the neutral stability condition. A numerical simulation is then conducted, considering cyclic boundary conditions. The results indicate that the mass effect tends to absorb traffic jams provided no time delay is imposed.

The jamming transition of multi-lane lattice hydrodynamic model with passing effect

Multi-lane environments are not uncommon on the highway. In multi-lane scenarios, vehicles can not only switch freely between lanes but also overtake each other. However, as basic driving behavior, the passing effect under multi-lane environments has been neglected in traffic flow research. To fill this gap, we present a modified multi-lane lattice hydrodynamic model considering the passing effect. Based on the reduction perturbation method, the corresponding stability norm of the proposed model is obtained, which reveals that the total number of lanes is positively correlated with the stability of traffic flow. When the stability condition does not hold, to further examine the formation and transmission process of traffic jams near the neutral stability curve, we carry out a nonlinear stability analysis of the model. The modified Korteweg-de Vries (mKdV) equation and the existence of the above mKdV equation are deduced, respectively. When the passing ratio is sufficiently small, i.e., the above existence holds, we observe that the jamming transition emerges between uniform flow and kink jam; once the passing ratio exceeds the threshold, the jamming transition occurs among uniform flow and kink-Bando traffic wave through the chaotic phase. We also conduct numerical simulations to verify the theoretical derivation.

Hybrid structures of localized waves for the coupled extended mKdV equation

For the coupled extended modified Korteweg–de Vries (mKdV) equation, we investigate hybrid structure waves composed of localized waves. This equation can be considered as the generalization of the mKdV equation. Firstly, hybrid structures of localized waves can be derived in view of the generalized Darboux transformation. Next, we get different interactions between those hybrid waves with the parameters changing. After that, it is found that under a certain condition, the hybrid solutions will degenerate into rational solutions. In particular, we find an interesting rational rogue wave with the novel dynamical behavior. In addition, we show the striking dynamics of hybrid solutions by analyzing each figure. These new solutions can help us investigate integrable systems better.

On the Bäcklund transformation of a generalized Harry Dym type equation

We connect the generalized Harry Dym type equation with the modified Korteweg–de Vries equation via reciprocal and gauge transformations. This allows us to deduce the N-Bäcklund transformation for the generalized Harry Dym type equation with the aid of the Bäcklund transformation for the modified Korteweg–de Vries equation. As an application, some soliton solutions and their interactions for the generalized Harry Dym type equation are discussed.

Integration of the mKdV Equation with nonstationary coefficients and additional terms in the case of moving eigenvalues

In this paper, we consider the Cauchy problem for the non-stationary modified Korteweg–de Vries equation with an additional term and a self-consistent source in the case of moving eigenvalues. Also, the evolution of the scattering data of the Dirac operator is obtained, the potential of which is the solution of the loaded modified Korteweg–de Vries equation with a self-consistent source in the class of rapidly decreasing functions. Specific examples are given to illustrate the application of the obtained results.

Breather and nondegenerate solitons in the two-component modified Korteweg–de Vries equation

In this paper, we investigate the nonlinear dynamics of an interesting class of vector solitons in the two-component modified Korteweg–de Vries (mKdV) equation. We construct the nondegenerate solitons and the breather solutions of the two-component mKdV equation by applying a non-standard form of the Hirota direct method. Our study shows that the nondegenerate solitons and the breather solutions of the system consist of three profiles: single-hump, double-hump and flat-top, and the collisions among nondegenerate solitons are always standard inelastic collisions. An explicit form of the general breather solution of the two-component mKdV equation is presented.

Darboux Transformation and Soliton Solutions for a Nonlocal Two-Component Complex Modified Korteweg–de Vries Equation

AbstractIn this paper, a nonlocal two-component complex modified Korteweg–de Vries (mKdV) equation is investigated and its Darboux transformation is derived. By using the Darboux transformation, soliton solutions are obtained which include dark soliton solution, quasi-W-type soliton, quasi-M-type soliton and periodic solution.

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

Higher-Order Matrix Spectral Problems and Their Integrable Hamiltonian Hierarchies

Starting from a kind of higher-order matrix spectral problems, we generate integrable Hamiltonian hierarchies through the zero-curvature formulation. To guarantee the Liouville integrability of the obtained hierarchies, the trace identity is used to establish their Hamiltonian structures. Illuminating examples of coupled nonlinear Schrödinger equations and coupled modified Korteweg–de Vries equations are worked out.