Modified Korteweg-de Vries Equation Research Articles

Solid angle car following model**Project supported by the National Key R&D Program of China (Grant No. 2018YFB1601000).

Existing traffic flow models give little consideration on vehicle sizes. We introduce the solid angle into car-following theory, taking the driver’s perception of the leading vehicle’s size into account. The solid angle and its change rate are applied as inputs to the novel model. A nonlinear stability analysis is performed to analyze the asymmetry of the model and the size effect of the leading vehicle, and the modified Korteweg–de Vries equation is derived. The solid angle model can explain complex traffic characteristics and provide an important basis for modeling nonlinear traffic phenomena.

Magnetic field quantization in pulsars

Magnetic field quantization is an important issue for degenerate environments such as neutron stars, radio pulsars and magnetars etc., due to the fact that these stars have a magnetic field higher than the quantum critical field strength of the order of $4.4\times 10^{13}~\text{G}$ , accordingly, the cyclotron energy may be equal to or even much more than the Fermi energy of degenerate particles. We shall formulate here the exotic physics of strongly magnetized neutron stars, known as pulsars, specifically focusing on the outcomes of the quantized magnetic pressure. In this scenario, while following the modified quantum hydrodynamic model, we shall investigate both linear and nonlinear fast magnetosonic waves in a strongly magnetized, weakly ionized degenerate plasma consisting of neutrons and an electron–ion plasma in the atmosphere of a pulsar. Here, linear analysis depicts that sufficiently long, fast magnetosonic waves may exist in a weakly dispersive pulsar having finite phase speed at cutoff. To investigate one-dimensional nonlinear fast magnetosonic waves, a neutron density expression as a function of both the electron magnetic and neutron degenerate pressures, is derived with the aid of Riemann’s wave solution. Consequently, a modified Korteweg–de Vries equation is derived, having a rarefractive solitary wave solution. It is found that the basic properties such as amplitude, width and phase speed of the fast magnetoacoustic waves are significantly altered by the electron magnetic and the neutron degenerate pressures. The results of this theoretical investigation may be useful for understanding the formation and features of the solitary structures in astrophysical compact objects such as pulsars, magnetars and white dwarfs etc.

Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation

We give the asymptotics of the Fourier transform of self-similar solutions for the modified Korteweg-de Vries equation. In the defocussing case, the self-similar profiles are solutions to the Painlevé II equation; although they were extensively studied in physical space, no result to our knowledge describe their behavior in Fourier space. These Fourier asymptotics are crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flow.Our result is obtained through a fixed point argument in a weighted W1,∞ space around a carefully chosen, two term ansatz, and we are able to relate the constants involved in the description in Fourier space with those of the description in physical space.

Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation

Based on Darboux transformation, we generate breather solutions sitting on zero background via module resonance for complex modified KdV equation. Then, some novel soliton molecules, breather molecules and breather-soliton molecules are obtained theoretically by module resonance and velocity resonance. Further, we study the higher-order breather-positons with zero background on the basis of degenerate Darboux transformation and module resonance for the first time. We discuss the dynamics of higher-order breather-positons in detail and propose related propositions. Finally, we find new elastic interactions between smooth positons and breather-positons by means of degenerate Darboux transformation and partial module resonance.

The periodic solutions of the discrete modified KdV equation

The periodic solutions of the discrete modified KdV equation

The integrable property of a higher-order Zakharov–Shabat hierarchy

In this letter, we derive the isospectral and nonisospectral soliton hierarchies of a higher-order Zakharov–Shabat spectral problem where nonlinear Schrödinger equation and modified KdV equation are included. Furthermore, it is discussed that the new integrable properties are found about the higher-order Zakharov–Shabat hierarchy, including the master symmetry and the time dependent symmetry.

Solitary Wave Solutions of Some Nonlinear Physical Models Using Riccati Equation Approach

Riccati equation approach is used to look for exact travelling wave solutions of some nonlinear physical models. Solitary wave solutions are established for the modified KdV equation, the Boussinesq equation and the Zakharov-Kuznetsov equation. New generalized solitary wave solutions with some free parameters are derived. The obtained solutions, which includes some previously known solitary wave solutions and some new ones, are expressed by a composition of Riccati differential equation solutions followed by a polynomial. The employed approach, which is straightforward and concise, is expected to be further employed in obtaining new solitary wave solutions for nonlinear physical problems.

Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them

This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are taken into account. The mathematical model phenomenon is constructed by means of the method of two-scale asymptotic expansion. Structural damping in the shells and surrounding elastic media did not allow discovery of the exact solution of the problem of the deformation waves propagation. This leads to the need for numerical methods. A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank–Nicholson scheme for the heat equation. In the absence of the structural damping and surrounding media influences, and under the similar initial conditions for both shells, the velocity and amplitude of the wave do not change. The result of the numerical experiment coincides with the exact solution, which is found in the case of the absence of the structural damping and surrounding media influences; therefore, the difference scheme is adequate to the generalized modified Korteweg–de Vries equations system. There is energy is transferred in the presence of the fluid, between the shells. The presence of inertia of the fluid motion leads to a decrease in the velocity of the deformation wave.

The periodic solutions of the discrete modified KdV equation with a self-consistent source

This work is devoted to the application of inverse spectral problem for integration of the periodic solutions of the discrete modified KdV equation with a self-consistent source. The effective method of solution of the inverse spectral problem for the discrete linear Ablowitz-Ladik equation is presented.

The long-wave potential-vorticity dynamics of coastal fronts

This paper studies the propagation of free, long waves on a potential vorticity front in the presence of a vertical coast, using a -layer, quasi-geostrophic model with piecewise-constant potential vorticity. The coastal boundary induces flow through image vorticity and a Kelvin wave, either of which can reinforce or oppose the Rossby wave dynamics at the front. The behaviour of the front depends strongly on the relative strengths of these three mechanisms, which are explicit in our model. The richest behaviour, which includes kink solitons (under-compressive shocks) and compound-wave structures, occurs in the regime where vortical effects are dominant. The evolution of the front is described by a fully nonlinear finite-amplitude equation including first-order dispersive effects, which is related to the modified Korteweg–de Vries equation. The different behaviours are classified using the canonical example of the Riemann problem, which we analyse using El’s technique of ‘dispersive shock-fitting’. Contour-dynamic simulations show that the dispersive long-wave theory captures the behaviour of the full quasi-geostrophic system to a high degree of accuracy.

Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg-de Vries equation

Two nonlocal versions of the semi-discrete modified Korteweg-de Vries equation are derived by different nonlocal reductions from a coupled equation set in the Ablowitz–Ladik hierarchy. Different kinds of exact solutions in terms of double Casoratians to the reduced equations are obtained by imposing constraint conditions on the double Casorati determinant solutions of the coupled equation set. Dynamics of the soliton solutions for the real and complex nonlocal semi-discrete modified Korteweg-de Vries equations are analyzed and illustrated by asymptotic analysis.

Discrete Crum’s Theorems and Lattice KdV-Type Equations

We develop Darboux transformations (DTs) and their associated Crum’s formulas for two Schrodinger-type difference equations that are themselves discretized versions of the spectral problems of the KdV and modified KdV equations. With DTs viewed as a discretization process, classes of semidiscrete and fully discrete KdV-type systems, including the lattice versions of the potential KdV, potential modified KdV, and Schwarzian KdV equations, arise as the consistency condition for the differential/difference spectral problems and their DTs. The integrability of the underlying lattice models, such as Lax pairs, multidimensional consistency, τ-functions, and soliton solutions, can be easily obtained by directly applying the discrete Crum’s formulas.

Three-dimensional modified Korteweg-de Vries equation in a magnetised relativistic plasma with positron beam and vortex-like electron distribution

The nonlinear features of Ion Acoustic (IA) waves are studied in a fully relativistic three-dimensional (3-D) plasma system with consideration of effect of both positron beam and trapped electrons. We consider a set of 3-D magnetised hydrodynamic equations with pressure expansion for our plasma model along with kinetic Vlasov equation for electrons. Applying the perturbative expansion technique, a Modified Korteweg-de Vries (m-KdV)-like equation is derived, exhibiting the evolution of small amplitude IA waves in plasma. The modified coefficient of nonlinear term in K-dV equation has arrived due to impact of vortex-like distribution of electrons. An analytical and numerical investigation of the nonlinear evolution equations is exhibited with external magnetic field effects, the time derivative pressure expansion as well as other parameters like relativistic effect, mass variation, beam velocity and temperature effect have been taken into consideration. The presence of vortex like trapped electron distribution and positron beam governs the influence of soliton structure quite significantly. The present result should help us to understand the experiments that involve particle trapping and also the salient features of astrophysical environment like ionospheric plasma together with situations in plasma describing the electrostatic solitary structures usually seen in antimatter-related environment in interplanetary region.

Soliton and breather solutions for a fifth-order modified KdV equation with a nonzero background

We investigate an integrable fifth-order modified KdV equation, which contains the fifth-order dispersion and relevant higher-order nonlinear terms, on the line with a nonzero constant background via the Riemann–Hilbert method. As an application, the explicit soliton and breather solutions are obtained. Some figures are also presented to depict the soliton and breather behaviors of the fifth-order modified KdV equation.

Dynamics of various waves in nonlinear Schrödinger equation with stimulated Raman scattering and quintic nonlinearity

In this work, we consider a nonlinear Schrodinger equation with stimulated Raman scattering and quintic nonlinearity, which works as a model for the propagation of ultrashort pulse in optical fiber and also corresponds to one-dimensional anisotropic Heisenberg ferromagnetic spin chain. Introducing a Galilean transformation, we transform the model into a fifth-order complex modified KdV equation, the integrability of which has been considered in the AKNS technique framework. The N-fold Darboux transformation for the model is derived in terms of the gauge transformation of the associated $$3 \times 3$$ matrix spectral problem. As applications, abundant intriguing types of nonlinear waves are obtained in zero and nonzero boundary conditions. The dynamical evolution of these nonlinear waves can be well controlled under stimulated Raman scattering and quintic nonlinearity management. Beside that, we also find some novel and interesting conversion phenomena, where a high-order term plays a pivotal role.

Asymptotics of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane

The large-time behavior of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane is established. Our approach uses the unified transform method of Fokas and the nonlinear steepest descent method of Deift and Zhou.

A new two-lane lattice hydrodynamic model with the introduction of driver’s predictive effect

This paper devises a new two-lane lattice hydrodynamic model (TLHM) to explore driver’s predictive effect (DPE) on traffic oscillation. First, a linear approach is conducted to analytically predict the DPE on traffic performance. Theoretical analysis shows that with the help of DPE, the traffic flow stability will be gradually enhanced. Then, nonlinear analysis is implemented to explore the characteristics of traffic oscillation when sensitivity coefficient is near the critical point. The modified KdV equation derived from the new model and its analytical solution related kink–antikink density waves are obtained. Finally, numerical experiments show that the DPE can effectively dampen the growth of oscillation, which is well consistent with the theoretical analysis of the new model.

Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy

We construct the hierarchy of a multi-component generalisation of modified KdV equation and find exact solutions to its associated members. The construction of the hierarchy and its conservation laws is based on the Drinfel'd-Sokolov scheme, however, in our case the Lax operator contains a constant non-regular element of the underlying Lie algebra. We also derive the associated recursion operator of the hierarchy using the symmetry structure of the Lax operators. Finally, using the rational dressing method we obtain the one-soliton solution and we find the one-breather solution of general rank in terms of determinants.

Self-Similar Dynamics for the Modified Korteweg–de Vries Equation

Abstract We prove a local well-posedness result for the modified Korteweg–de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an asymptotic description of small solutions as $t \to +\infty $.

Employing the dynamics of poles in the complex plane to describe properties of rogue waves: case studies using the Boussinesq and complex modified Korteweg–de Vries equations

The dynamics and properties of rogue waves of two classical evolution equations are studied in terms of trajectories of the poles of the exact solutions, by analytically continuing the spatial variable to be complex. The Boussinesq equation describes the motion of hydrodynamic waves in two opposite directions in the shallow water regime. The complex modified Korteweg–de Vries equation is relevant for wave packets in nonlinear media governed by a higher-order nonlinear Schrodinger equation, for the special case where second-order dispersion and cubic self-interaction are absent. On examining the movement of poles of the exact solutions for rogue waves, the real parts of the poles correlate well with the locations of maximum displacements in the physical space. This phenomenon holds for high precision numerically for the first-, second- and third-order rogue waves of the Boussinesq equation. A similar principle is also valid for the first- and second-order rogue waves of the complex modified Korteweg–de Vries equation. The imaginary parts of the poles can generate useful information too. For these two evolution models, a smaller imaginary part in the complex plane is associated with a larger amplitude of the rogue wave in physical space. An empirical formula is proposed which works well for the three lowest orders of rogue waves of the Boussinesq equation.