Order Dispersive Equation Research Articles

On the study of a nonlinear higher order dispersive wave equation: its mathematical physical structure and anomaly soliton phenomena

This paper introduces a systematic approach to investigate a higher order nonlinear dispersive wave equation for modeling different wave modes. We present both the conventional KdV-type soliton and anomaly type solitons for the equation. We also show the conservation laws and Hamiltonian structures for the equation. Our results suggest that the underlying equation has more interacting soliton phenomena than one would have known for the classical KdV and Boussinesq equation.

An Effective Model for Wire Medium Based on the 4th Order Dispersion Equation

In this paper, the wire medium made of parallel thin wires is studied by using the FDTD method incorporating the sub-cell technique of thin wire. The conventional effective model for the wire medium is based on the 2th order dispersion equations; and it becomes inaccurate in describing the EM behavior of wire medium when the transversal spatial harmonics becomes high. A new effective model is proposed based on the 4th order dispersion equations and surface fitting, which guarantees the modeling accuracy for higher spatial harmonics.

Local Well-Posedness in Sobolev Spaces With Negative Indices for A Seventh Order Dispersive Equation

This paper is concerned with the Cauchy problem of a seventh order dispersive equation. We prove local well-posedness with initial data in Sobolev spacesHs(R) for negative indices of s>−114.

Motion of a vortex filament with axial flow in the half space

We consider a nonlinear third order dispersive equation which models the motion of a vortex filament immersed in an incompressible and inviscid fluid occupying the three dimensional half space. We prove the unique solvability of initial–boundary value problems as an attempt to analyze the motion of a tornado.

An LDG numerical approach for Boussinesq type modelling

This work describes the development of a Boussinesq type model based on a set of fully nonlinear and first order dispersive equations in one horizontal dimension. The numerical discretisation is carried out using a discontinuous Galerkin framework. In particular, the basis of the scheme is established through a local discontinuous Galerkin (LDG) method for the spatial discretisation while for the time discretisation a high order total variation diminishing Runge–Kutta procedure is utilised. The numerical tool is validated by means of the simulation of two classic benchmark test cases. Firstly, the propagation over a flat bed of three solitary waves with different nonlinearity is considered. The presence of a dispersive tail of short waves is detected behind nonlinear solitons and confirmed by a RANS model. Secondly, two wave dispersion scenarios over a submerged bar are reproduced. The impact of different orders of accuracy is assessed on the computed solution. The model shows good behaviour for weakly dispersive waves up to kh values close to π. Finally, the numerical efficiency of the code is illustrated by a CPU time analysis for given combinations of the order and the mesh resolution. Test cases for a multi-layer system will be considered in future work.

Effects of finite beam and plasma temperature on the growth rate of a two-stream free electron laser with background plasma

A fluid theory is used to derive the dispersion relation of two-stream free electron laser (TSFEL) with a magnetic planar wiggler pump in the presence of background plasma (BP). The effect of finite beams and plasma temperature on the growth rate of a TSFEL has been verified. The twelve order dispersion equation has been solved numerically. Three instabilities, FEL along with the TS and TS-FEL instabilities occur simultaneously. The analysis in the case of cold BP shows that when the effect of the beam temperature is taken into account, both instable bands of wave-number and peak growth rate in the TS instability increase, but peak growth of the FEL and TS-FEL instabilities decreases. Thermal motion of the BP causes to diminish the TS instability and it causes to decrease the FEL and TS-FEL instabilities. By increasing the beam densities and lowering initial velocities (in the collective Raman regime), growth rate of instabilities increases; however, it has opposite behavior in the Campton regime.

Phragmén–Lindelöf alternative of exponential type for the solutions of a fourth order dispersive equation

In a recent paper (R. Quintanilla, G. Saccomandi. Quarterly Applied Mathematics, 44, (2006) 547–560) we investigated the spatial behavior of a linear equation of fourth order which models several mechanical situations where dispersive e¤ects are taken into account. We proved a polynomial decay estimate for the solutions of this equation. In the present note we improve this result and we show a Phragmén–Lindelöf alternative of exponential type.

Effects of beam temperature and density variation on the growth rate of a two-stream free electron laser

The effect of beam temperature and density variation on the growth rate of a two-stream free electron laser (TSFEL) with planer wiggler pump has been verified. With the aid of momentum transfer, Poisson, continuity, and Maxwell’s equations, the dispersion relation was derived. The eighth order dispersion equation has been solved numerically. The two FEL along with the TS instabilities occur simultaneously. The analysis shows that when the effect of the temperature is taken into account, both instable bands of wave-number and peak growth rate in TS instability increase, but peak growth of FEL instabilities decreases. The growth rate increases with an increase in the strength of the wiggler field. In addition, the growth sensitivity to density variation has been noticed.

A Curve Flow on an Almost Hermitian Manifold Evolved by a Third Order Dispersive Equation

We consider a curve flow for maps from a real line into a compact almost Hermitian manifold, which is governed by a third order nonlinear dispersive equation. This article shows short-time existence of a solution to the initial value problem for the equation. The difficulty comes from the lack of the Kahler condition on the target manifold, since the covariant derivative of the almost complex structure causes a loss of one derivative in our equation and thus the classical energy method breaks down in general. In the present article, we can overcome the difficulty by constructing a gauge transformation on the pull-back bundle for the map to eliminate the derivative loss essentially, which is based on the local smoothing effect of third order dispersive equations on the real line.

CONTROLLABILITY OF NONLINEAR THIRD ORDER DISPERSION EQUATION WITH DISTRIBUTED DELAY

This paper is concerned with the exact controllability of nonlinear third order dispersion equation with infinite distributed delay. Sufficient conditions are formulated and proved for the exact controllability of this system. Without imposing a compactness condition on the semigroup, we establish controllability results by using a fixed point analysis approach.

Exact Controllability of Semilinear Third Order Dispersion Equation

In this paper, a family of nonlinear functions is given for the exact controllability of semilinear third order dispersion equation. The obtained result has been illustrated by applying it on nonlinear Korteweg-de Vries (KdV) equation.

Applications of elliptic functions to ion-acoustic plasma waves

New several classes of exact solutions are obtained in terms of the Weierstrass elliptic function for some nonlinear partial differential equations modeling ion-acoustic waves as well as dusty plasmas in laboratory and space sciences. The Weierstrass elliptic function solutions of the Schamel equation, a fifth order dispersive wave equation and the Kawahara equation are constructed. Moreover, Jacobi elliptic function solutions and solitary wave solutions of the Schamel equation are also given. The stability of some periodic wave solutions is computationally studied.

Uniqueness properties of higher order dispersive equations

In this paper we study unique continuation properties of solutions to higher (fifth) order nonlinear dispersive models. The aim is to show that if the difference of two solutions of the equations, u 1 − u 2 , decays sufficiently fast at infinity at two different times, then u 1 ≡ u 2 .

Local Smoothing and Local Solvability for Third Order Dispersive Equations

Two‐dimensional deep water waves can be described by various third order dispersive equations, modifying and generalizing the KdV equations as well as nonlinear Schrödinger equations. We establish local well‐posedness for initial data $u_0\in H^{3/2}(\R^2)$ for many of the proposed models using local smoothing estimates with respect to the forcing term and the initial data.

High-frequency waves in the solar coronal plasma

We derived numerical solutions of a dispersion equation in order to analyze the effect of finite plasma temperature on the highfrequency wave dispersion characteristics in conditions of hot magnetized plasma in the solar corona. Spectra of the high-frequency eigen modes of these plasma were determined in conditions when the electron gyrofrequency is lower than the plasma one and when the eigen modes frequencies are higher than the electron gyrofrequency. The longitudinal wave mode is shown to turn to the Z-mode at refractive index n < 1. At refractive index n � 1, the longitudinal wave frequency increases when n grows, and these waves go to strongly damped ones with an anomalous dispersion. We interpret some spectral features of type II and IV radio bursts in the solar corona.

The Initial Value Problem for Third and Fourth Order Dispersive Equations in One Space Dimension

We present the sufficient conditions for the L2-well-posedness of the initial value problem for third and fourth order dispersive equations in one space dimension. In third order case and particular fourth order case, these conditions are also necessary. Tarama's pseudodifferential operators play crucial role in our proof.

The initial value problem for a third order dispersive equation on the two-dimensional torus

We present the necessary and sufficient conditions for the L 2 L^2 -well-posedness of the initial problem for a third order linear dispersive equation on the two-dimensional torus. Birkhoff’s method of asymptotic solutions is used to prove necessity. Some properties of a system for quadratic algebraic equations associated to the principal symbol play a crucial role in proving sufficiency.

Dispersion Estimates for Third Order Equations in Two Dimensions

Two-dimensional deep water waves and some problems in nonlinear optics can be described by various third order dispersive equations, modifying and generalizing the KdV as well as nonlinear Schrödinger equations. We classify all third order polynomials up to certain transformations and study the pointwise decay for the fundamental solutions, for all third order polynomials pin two variables. We deduce the corresponding Strichartz estimates. These estimates imply global existence and uniqueness for the Shrira system.

Characteristics of TM Surface Waves in a Nonlinear Antiferromagnet–Semiconductor–Superconductor Waveguide Structure

The frequency characteristics of the transverse magnetic (TM) surface waves at mid-infrared frequencies in a layered system, consisting of YBa2Cu3O7−x superconductor film bounded by a nonlinear antiferromagnet (FeF2) cover and a resonant semiconductor plasma (In0.53Ga0.47As) substrate, are fully described. The permeability tensor of the antiferromagnet crystal, for the absence of an applied Zeeman field, has a nonlinear response to the intense rf field. The complex wave number of TM wave is computed by solving the dispersion equation in order to determine the effects of the superconductor and resonant semiconductor on the dispersion characteristics for both the reduced phase and reduced attenuation constants as a function of the temperature and frequency. The power flow through the waveguide structure has also been investigated. We have found that superconductors have the ability to reduce the propagation losses. We have also found that the reduced attenuation constant is highly dependent on the temperature of the superconductor.

Dispersive kinetics of solid thermolysis. Decomposition of basic aluminium ammonium sulphate

The formalism of dispersive kinetics is applied for thermolysis of solids. As an example, the isothermal decomposition of basic aluminium ammonium sulphate in vacuum is analysed. The first order dispersive kinetic equation provides a highly accurate description of thermolysis processes for the three stages of decomposition: partial dehydration at temperatures below 623 K, final dehydration and removal of ammonium in the temperature range 623–873 K, and desulphurisation at temperatures above 873 K.