2,058 research outputs found
Derivation of the Camassa-Holm equations for elastic waves
In this paper we provide a formal derivation of both the Camassa-Holm
equation and the fractional Camassa-Holm equation for the propagation of
small-but-finite amplitude long waves in a nonlocally and nonlinearly elastic
medium. We first show that the equation of motion for the nonlocally and
nonlinearly elastic medium reduces to the improved Boussinesq equation for a
particular choice of the kernel function appearing in the integral-type
constitutive relation. We then derive the Camassa-Holm equation from the
improved Boussinesq equation using an asymptotic expansion valid as
nonlinearity and dispersion parameters tend to zero independently. Our approach
follows mainly the standard techniques used widely in the literature to derive
the Camassa-Holm equation for shallow water waves. The case where the Fourier
transform of the kernel function has fractional powers is also considered and
the fractional Camassa-Holm equation is derived using the asymptotic expansion
technique.Comment: 15 page
Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations
In this article we study global existence and blow-up of solutions for a
general class of nonlocal nonlinear wave equations with power-type
nonlinearities, , where the
nonlocality enters through two pseudo-differential operators and . We
establish thresholds for global existence versus blow-up using the potential
well method which relies essentially on the ideas suggested by Payne and
Sattinger. Our results improve the global existence and blow-up results given
in the literature for the present class of nonlocal nonlinear wave equations
and cover those given for many well-known nonlinear dispersive wave equations
such as the so-called double-dispersion equation and the traditional
Boussinesq-type equations, as special cases.Comment: 17 pages. Accepted for publication in Nonlinear Analysis:Theory,
Methods & Application
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
Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity
We study the initial-value problem for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided
A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity
In one space dimension, a non-local elastic model is based on a single integral law, giving the stress when the strain is known at all spatial points. In this study, we first derive a higher-order Boussinesq equation using locally non-linear theory of 1D non-local elasticity and then we are able to show that under certain conditions the Cauchy problem is globally well-posed
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
Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations
This study deals with the analysis of the Cauchy problem of a general class
of nonlocal nonlinear equations modeling the bi-directional propagation of
dispersive waves in various contexts. The nonlocal nature of the problem is
reflected by two different elliptic pseudodifferential operators acting on
linear and nonlinear functions of the dependent variable, respectively. The
well-known doubly dispersive nonlinear wave equation that incorporates two
types of dispersive effects originated from two different dispersion operators
falls into the category studied here. The class of nonlocal nonlinear wave
equations also covers a variety of well-known wave equations such as various
forms of the Boussinesq equation. Local existence of solutions of the Cauchy
problem with initial data in suitable Sobolev spaces is proven and the
conditions for global existence and finite-time blow-up of solutions are
established.Comment: 17 page
Traveling waves in one-dimensional nonlinear models of strain-limiting viscoelasticity
In this article we investigate traveling wave solutions of a nonlinear
differential equation describing the behaviour of one-dimensional viscoelastic
medium with implicit constitutive relations. We focus on a subclass of such
models known as the strain-limiting models introduced by Rajagopal. To describe
the response of viscoelastic solids we assume a nonlinear relationship among
the linearized strain, the strain rate and the Cauchy stress. We then
concentrate on traveling wave solutions that correspond to the heteroclinic
connections between the two constant states. We establish conditions for the
existence of such solutions, and find those solutions, explicitly, implicitly
or numerically, for various forms of the nonlinear constitutive relation
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