Spectral stability for subsonic traveling pulses of the Boussinesq `abc' system

We consider the spectral stability of certain traveling wave solutions of the Boussinesq `abc' system. More precisely, we consider the explicit $sech^2(x)$ like solutions of the form (\vp(x-w t), \psi(x- w t)=(\vp, const. \vp), exhibited by M. Chen (1998) and we provide a complete rigorous characterization of the spectral stability in all cases for which $a=c0$

TITULACION DE ANTICUERPOS CONTRA TOXOPLASMA CON EL METODO DE HEMOAGLUTINACION INDIRECTA EN MEDICOS VETERINARIOS DE CENTROS DE ATENCION VETERINARIA EN LA CIUDAD DEL CUSCO – 2014

HISTORIA DE LA TOXOPLASMOSIS DEFINICIÓN ETIOLOGÍA ESTADIOS DE DESARROLLO HUÉSPEDES Y LOCALIZACIÓN CICLO BIOLÓGICO BIOLOGÍA MOLECULAR DISTRIBUCIÓN GEOGRÁFICA EPIDEMIOLOGÍA MECANISMOS DE TRANSMISIÓN PATOGENIA SINTOMATOLOGÍA PROFILAXIS Y CONTRO

Elliptic solutions and solitary waves of a higher order KdV--BBM long wave equation

We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass $\wp$ functions of both third and fifth-order KdV--BBM (Korteweg-de Vries--Benjamin, Bona \& Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV--BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstrass elliptic $\wp$ functions. For the fifth-order KdV--BBM equation we show that a parameter $\gamma=\frac {1}{12}$, for which the equation has a Hamiltonian, represents a restriction for which there are constraint curves that never intersect a region of unbounded solitary waves, which in turn shows that only dark or bright solitons and no unbounded solutions exist. Motivated by the lack of a Hamiltonian structure for $\gamma\neq\frac{1}{12}$ we develop $H^k$ bounds, and we show for the non Hamiltonian system that dark and bright solitons coexist together with unbounded periodic solutions. For nonzero boundary conditions, due to the complexity of the nonlinear algebraic system of coefficients of the elliptic equation we construct Weierstrass solutions for a particular set of parameters only.Comment: 13 pages, 6 figure

The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters $\epsilon$ and $\delta$ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.Comment: 24 pages, to appear in Discrete and Continuous Dynamical System

Artificial boundary conditions for the linearized Benjamin-Bona-Mahony equation

International audienceWe consider various approximations of artificial boundary conditions for linearized Benjamin-Bona-Mahoney equation. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomorphic function). In this paper, we derive explicit transparent boundary conditions both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation. We prove consistency, stability and convergence of the numerical scheme and provide many numerical experiments to show the efficiency of our tranparent boundary conditions

On the ill-posedness result for the BBM equation

We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in Hs(R), s < 0 in the sense that the ow-map u0 7! u(t) that associates to initial data u0 the solution u cannot be continuous at the origin from Hs(R) to even D0(R) at any _xed t > 0 small enough. This result is sharp.Fundação para a Ciência e a Tecnologia (FCT

The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters is an element of and delta measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation

Investigation of Novel Piecewise Fractional Mathematical Model for COVID-19

The outbreak of coronavirus (COVID-19) began in Wuhan, China, and spread all around the globe. For analysis of the said outbreak, mathematical formulations are important techniques that are used for the stability and predictions of infectious diseases. In the given article, a novel mathematical system of differential equations is considered under the piecewise fractional operator of Caputo and Atangana&ndash;Baleanu. The system is composed of six ordinary differential equations (ODEs) for different agents. The given model investigated the transferring chain by taking non-constant rates of transmission to satisfy the feasibility assumption of the biological environment. There are many mathematical models proposed by many scientists. The existence of a solution along with the uniqueness of a solution in the format of a piecewise Caputo operator is also developed. The numerical technique of the Newton interpolation method is developed for the piecewise subinterval approximate solution for each quantity in the sense of Caputo and Atangana-Baleanu-Caputo (ABC) fractional derivatives. The numerical simulation is drawn against the available data of Pakistan on three different time intervals, and fractional orders converge to the classical integer orders, which again converge to their equilibrium points. The piecewise fractional format in the form of a mathematical model is investigated for the novel COVID-19 model, showing the crossover dynamics. Stability and convergence are achieved on small fractional orders in less time as compared to classical orders

A comparison of solutions of a Boussinesq system and the Benjamin-Bona-Mahony equation.

We compare solutions of a regularized Boussinesq model system for water waves to solutions of the Benjamin-Bona-Mahony equation. Both the system and the equation are obtained as formal approximations to the Euler equations of motion in which terms of order epsilon2 are neglected, where epsilon is a small parameter related to the wave amplitude and the inverse wavelength. For each choice of initial free-surface profile in an appropriate Sobolev space, we show that the Boussinesq system has a unidirectional solution which exists on a time interval 0 &le; t &le; T of length T = O( 1/e ), and that on this time interval the solution agrees with the corresponding solution of the Benjamin-Bona-Mahony equation to within Cepsilon2t, where C is independent of epsilon and t. Therefore the solution of the system agrees with the solution of the equation to within the accuracy of either model