18 research outputs found
Instability and stability properties of traveling waves for the double dispersion equation
In this article we are concerned with the instability and stability
properties of traveling wave solutions of the double dispersion equation
for ,
. The main characteristic of this equation is the existence of two
sources of dispersion, characterized by the terms and . We
obtain an explicit condition in terms of , and on wave velocities
ensuring that traveling wave solutions of the double dispersion equation are
strongly unstable by blow up. In the special case of the Boussinesq equation
(), our condition reduces to the one given in the literature. For the
double dispersion equation, we also investigate orbital stability of traveling
waves by considering the convexity of a scalar function. We provide both
analytical and numerical results on the variation of the stability region of
wave velocities with , and and then state explicitly the conditions
under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure
The Cauchy Problem for a One Dimensional Nonlinear Peridynamic Model
This paper studies the Cauchy problem for a one-dimensional nonlinear
peridynamic model describing the dynamic response of an infinitely long elastic
bar. The issues of local well-posedness and smoothness of the solutions are
discussed. The existence of a global solution is proved first in the sublinear
case and then for nonlinearities of degree at most three. The conditions for
finite-time blow-up of solutions are established.Comment: To appear in Journal of Differential Equations (added references,
corrected typos, minor revision in Section 2, 18 pages
The Cauchy Problem for a One Dimensional Nonlinear Peridynamic Model
This paper studies the Cauchy problem for a one-dimensional nonlinear
peridynamic model describing the dynamic response of an infinitely long elastic
bar. The issues of local well-posedness and smoothness of the solutions are
discussed. The existence of a global solution is proved first in the sublinear
case and then for nonlinearities of degree at most three. The conditions for
finite-time blow-up of solutions are established.Comment: To appear in Journal of Differential Equations (added references,
corrected typos, minor revision in Section 2, 18 pages
Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations
We study the initial-value problem for a general class of nonlinear nonlocal
coupled wave equations. The problem involves convolution operators with kernel
functions whose Fourier transforms are nonnegative. Some well-known examples of
nonlinear wave equations, such as coupled Boussinesq-type equations arising in
elasticity and in quasi-continuum approximation of dense lattices, follow from
the present model for suitable choices of the kernel functions. We establish
local existence and sufficient conditions for finite time blow-up and as well
as global existence of solutions of the problem.Comment: 11 pages. Minor changes and added reference
Existence and stability of traveling waves for a class of nonlocal nonlinear equations
In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: u_tt−Lu_xx=B(±|u|^(p−1)u)_xx, p>1. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operators L and B. Members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations. For instance, all Boussinesq-type equations and the so-called double-dispersion equation are members of the class. We first establish the existence of traveling wave solutions to the nonlocal wave equations considered. We then obtain results on the orbital stability or instability of traveling waves. For the case L=I, corresponding to a class of Klein-Gordon-type equations, we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow-up
Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations
In this article, we prove the convergence of a semi-discrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution operator in space. The most important characteristic of the numerical method is that it is directly applied to the nonlocal equation by introducing the discrete convolution operator. Starting from the continuous Cauchy problem defined on the real line, we first construct the discrete Cauchy problem on a uniform grid of the real line. Thus the semi-discretization in space of the continuous problem gives rise to an infinite system of ordinary differential equations in time. We show that the initial-value problem for this system is well-posed. We prove that solutions of the discrete problem converge uniformly to those of the continuous one as the mesh size goes to zero and that they are second-order convergent in space. We then consider a truncation of the infinite domain to a finite one. We prove that the solution of the truncated problem approximates the solution of the continuous problem when the truncated domain is sufficiently large. Finally, we present some numerical experiments that confirm numerically both the expected convergence rate of the semi-discrete scheme and the ability of the method to capture finite-time blow-up of solutions for various convolution kernels