Korteveg-de Vries Equation Research Articles

The complex MKDV equation with step-like initial data: Large time asymptotic analysis

In this paper, we study large-time asymptotics for the complex modified Korteveg–de Vries equation with step-like initial data. It is shown that the step-like initial problem can be described by a matrix Riemann–Hilbert problem. Further we apply the steepest descent method to obtain different large-time asymptotics in the Zakharov–Manakov region, a plane wave region, and a slow decay region.

Consistent Riccati expansion solvability, symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg–de Vries equation in fluid dynamics of internal solitary waves* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11775047, 11775146, and 11865013).

We study a forced variable-coefficient extended Korteweg–de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.

On the Solution of Korteweg de Vries Equation by Laplace Homotopy Pertubation Method

In this paper, we applied Laplace Homotopy Pertubation method (LHPM) for solving Korteveg de Vries equation. It was found that this method yields a very rapid convergence of the solution series which leads the solution in a closed form.

Symmetry-invariant conservation laws of partial differential equations

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, theb-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

Painleve analysis for a forced Korteveg-de Vries equation arisen in fluid dynamics of internal solitary waves

In this paper, Painleve analysis is used to test the Painleve integrability of a forced variable-coefficient extended Korteveg-de Vries equation which can describe the weakly-non-linear long internal solitary waves in the fluid with continuous stratification on density. The obtained results show that the equation is integrable under certain conditions. By virtue of the truncated Painleve expansion, a pair of new exact solutions to the equation is obtained.

Korteveg-de Vries solitons in a cold quark-gluon plasma

The relativistic heavy ion program developed at RHIC and now at LHC motivated a deeper study of the properties of the quark gluon plasma (QGP) and, in particular, the study of perturbations in this kind of plasma. We are interested on the time evolution of perturbations in the baryon and energy densities. If a localized pulse in baryon density could propagate throughout the QGP for long distances preserving its shape and without loosing localization, this could have interesting consequences for relativistic heavy ion physics and for astrophysics. A mathematical way to proove that this can happen is to derive (under certain conditions) from the hydrodynamical equations of the QGP a Korteveg-de Vries (KdV) equation. The solution of this equation describes the propagation of a KdV soliton. The derivation of the KdV equation depends crucially on the equation of state (EOS) of the QGP. The use of the simple MIT bag model EOS does not lead to KdV solitons. Recently we have developed an EOS for the QGP which includes both perturbative and non-perturbative corrections to the MIT one and is still simple enough to allow for analitycal manipulations. With this EOS we were able to derive a KdV equation for the cold QGP.

Riemann–Hilbert problem to the modified Korteveg–de Vries equation: Long-time dynamics of the steplike initial data

We consider the modified Korteveg–de Vries equation on the line. The initial data are the pure step function, i.e., q(x,0)=0 for x≥0 and q(x,0)=c for x<0, where c is an arbitrary real number. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t→∞. Using the steepest descent method and the so-called g-function mechanism, we deform the original oscillatory matrix Riemann–Hilbert problem to explicitly solving model forms and show that the solution of the initial-value problem has a different asymptotic behavior in different regions of the xt-plane. In the regions x<−6c2t and x>4c2t, the main terms of asymptotics of the solution are equal to c and 0, respectively. In the region −6c2t<x<4c2t, asymptotics of the solution takes the form of a modulated elliptic wave of finite amplitude.

Ion-acoustic solitons in Bi-Ion dusty plasma

The propagation of ion-acoustic solitons in a warm dusty plasma containing two ion species is investigated theoretically. Using an approach based on the Korteveg de Vries equation, it is shown that the critical value of the negative ion density that separates the domains of existence of compression and rarefaction solitons depends continuously on the dust density. A modified Korteveg de Vries equation for the critical density is derived in the higher order of the expansion in the small parameter. It is found that the nonlinear coefficient of this equation is positive for any values of the dust density and the masses of positive and negative ions. For the case where the negative ion density is close to its critical value, a soliton solution is found that takes into account both the quadratic and cubic nonlinearities. The propagation of a solitary wave of arbitrary amplitude is investigated by the quasi-potential method. It is shown that the range of dust densities around the critical value within which solitary waves with positive and negative potentials can exist simultaneously is relatively wide.

Polygons of differential equations for finding exact solutions

Abstract A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of polygons corresponding to nonlinear differential equations. It allows one to express exact solutions of the equation studied through solutions of another equation using properties of the basic equation itself. The ideas of power geometry are used and developed. Our approach has a pictorial interpretation, which is illustrative and effective. The method can be also applied for finding transformations between solutions of differential equations. To demonstrate the method application exact solutions of several equations are found. These equations are: the Korteveg–de Vries–Burgers equation, the generalized Kuramoto–Sivashinsky equation, the fourth-order nonlinear evolution equation, the fifth-order Korteveg–de Vries equation, the fifth-order modified Korteveg–de Vries equation and the sixth-order nonlinear evolution equation describing turbulent processes. Some new exact solutions of nonlinear evolution equations are given.

Study of the SAR signature of internal waves by nonlinear parametric autoregressive models

In this paper, we propose a method for estimating the nonlinear differential equation governing the propagation of internal waves (IWs) in a stratified water column of the ocean. This estimation of the kind of nonlinear differential equation and the coefficients of this equation is performed from profiles of the synthetic aperture radar signature at the ocean surface. Theoretically, the equation is based upon the Korteveg-de Vries (KdV) equation or one of its improved versions. The estimation of the differential equation version is based on nonlinear polynomial autoregressive models up to the third order. The estimation of the parameters of these models is performed simply by solving a set of linear equations using higher order statistics. This general approach allows us to identify the best model of propagation, i.e., the kind of nonlinearity as well as the structure of each kernel (linear, quadratic, cubic), without determining an analytic solution to the equation of propagation. Moreover, the physical oceanic parameters (such as the thermocline depth) are deduced from the estimated coefficients of the KdV equation (assuming an underlying model of the water column). Results on simulated profiles generally lead to an exact model identification (i.e., the one used for simulation with the exact order of nonlinearity and the exact structure of each kernel) and lead to satisfying geophysical parameter estimates. Finally, the geophysical parameters estimated from three sets of ERS-1 profiles are generally coherent with the parameters observed in the IW propagation mechanism but have not been validated by "in situ" measurements.

Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlevé equation

Abstract Fourth-order analogue to the second Painleve equation is studied. This equation has its origin in the modified Korteveg–de Vries equation of the fifth-order when we look for its self-similar solution. All power and non-power expansions of the solutions for the fourth-order analogue to the second Painleve equation near points z = 0 and z = ∞ are found by means of the power geometry method. The exponential additions to solutions of the equation studied are determined. Comparison of the expansions found with those of the six Painleve equations confirm the conjecture that the fourth-order analogue to the second Painleve equation defines new transcendental functions.

Dynamics of Solitary Waves Through Taper-Thin Elastic Tube with Localized Deformation

In order to elucidate the dynamical behaviour of nonlinear waves in tapered-thin and elastic tubes with localized deformations, we have developed a mathematical model based on equations of inviscid fluid flow in a slowly tapered cone tube. Numerical simulation of the resulting perturbed Korteveg–de Vries equation shows that the amplitude of the velocity wave decreases along the tube as the results of the decrease of the radius and the increase of the rigidity; we also find that the velocity of the pulse velocity wave increases. In the localized deformation area, after some fluctuations, the wave suffers important variations consisting firstly (in the case of constriction) by a decrease and then by an increase. The severity and the length of the variation depend respectively on the amplitude and the gradient scale of the localized deformation. Some biological implications are given.

Solutions of the modified Ostrovskii equation with cubic non-linearity

Approximate solutions of the stationary Ostrovskii equation [1] with cubic non-linearity, describing, in particular, wave processes in rotating media, are constructed. A new integral invariant of this equation is found, making it possible to close the system of approximate equations and to eliminate excessive arbitrariness in the parameters. It is shown that there is a family of periodic stationary solutions in the form of a combination of a sawtooth wave with smoothed peaks and troughs, in which there are solitons of the modified Korteveg-de Vries equation of different polarity. This family of solutions is well described by approximate analytical theory, in which the role of a perturbation is essentially played by the ratio of the characteristic size of the soliton to the period of the sawtooth wave. The analytical solutions constructed are in good agreement with numerical solutions. The use of such solutions as the starting data for numerical calculations within the framework of the non-stationary equation has made it possible to establish the degree to which they are s tationary and their stability to small perturbations.

An efficient numerical method for solving the Korteveg–de Vries equation in a class of discontinuous functions

In this study, an efficient numerical method is proposed and implemented for solving the well-known one-dimensional Korteveg–de Vries (KdV) equation in a class of discontinuous functions. To this end, an auxiliary problem is formulated and it is shown that the solution of the auxiliary problem is the weak solution of KdV equation. Furthermore, some properties of the solution to the auxiliary problem are established. Since no smoothness assumptions are imposed on the unknown function, the numerical method proposed in this paper gives better results as compared to the conventional finite difference methods. Indeed, several simulations are presented in which the analytical solution and the computed numerical solution are in perfect agreement. Further numerical experiments with initial conditions of varying regularities are provided.

Nonlinear Effects in Nuclear Cluster Problem

Some nonlinear aspects of a cluster phenomenon in nuclei are considered using cubic Nonlinear Schrödinger equation and Korteveg de Vries equation. We discuss the following possible nonlinear effects: i) describing clusters as solitons; ii) an anomalous large angle scattering of α-particles by light and intermediate nuclei; iii) stable vortical objects; iv) and dynamical clusterization in the presence of instability.

The properties of electrosound waves at electron trapping

The influence of electron trapping on the propagation of an electrosound wave is investigated. It is shown, that on the stage of modulational instability the trapped particles prevent the spatial localization of the HF field. Under certain conditions the trapped particles destroy the stationary soliton. For nonstationary envelope waves the nonlinearity, connected with trapped particles, leads to a Korteveg-de Vries equation.

Perturbation of the two-soliton solution to the KdV equation

The Cauchy problem for the perturbed Korteveg-de Vries equation with the two-soliton initial data is considered. Differential equations for the slow deformation of the parameters—amplitudes and phase shifts—are derived. It is shown that the phase shift of the slow soliton depends on the deformation of the fast solition.

Long non-linear strain waves in layered elastic half-space

Elastic strain wave propagation in a thin non-linearly elastic layer superimposed on non-linearly elastic half-space is studied. Two layer-half-space contact models are considered. It is found that the Benjamin-Ono equation can be derived for description of longitudinal non-linear strain waves, when the contact between the layer and the half-space is provided only by means of the normal stresses and displacements. When the full contact problem is considered the more complicated integro-differential equation is derived. It is found that long non-linear periodical and solitary strain waves as well as envelope waves may exist in the first case, while only envelope wave solutions are found to the full contact problem. Linear wave analysis shows that the Korteveg-de Vries equation, often usable, is unlikely to be an adequate model for longitudinal surface strain waves. Application of the results obtained to experiments devoted to superconductivity threshold control in thin metal films as well as to generation of acoustic solitons in layered half-space is discussed.

(Super)Korteveg-de Vries equation as a (super)conformal field theory.

The solution to the Korteveg--de Vries equation is interpreted as a conformal field of conformal dimension two in a system having a nonvanishing central charge. Its N = 1, 2 supersymmetrized extensions are discussed.

On the nature of the Gardner transformation

It is shown that every higher Korteveg-de Vries equation can be included in a one-parameter family of integrable equations.