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Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations
(American Institute of Mathematical Sciences, 2019-05)
We consider the Cauchy problem defined for a general class of nonlocal wave equations modeling bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution ...
Comparison of nonlocal nonlinear wave equations in the long-wave limit
(Taylor & Francis, 2020-11-17)
We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear ...
Some remarks on the stability and instability properties of solitary waves for the double dispersion equation
(Estonian Academy of Sciences, 2015)
In this article we give a review of our recent results on the instability and stability properties of travelling wave solutions of the double dispersion equation utt − uxx + auxxxx − buxxtt = −(|u|p−1u)xx for p > 1, a ≥ b ...
Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: The KdV, BBM and CH equations
(Estonian Academy of Sciences, 2015)
We consider unidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral with a suitable kernel function. We first give a brief review of ...
Existence and stability of traveling waves for a class of nonlocal nonlinear equations
(Elsevier, 2015-05-01)
In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: utt−Luxx=B(±|u|p−1u)xx, p>1. The main characteristic of this class of ...
Derivation of the Camassa-Holm equations for elastic waves
(Elsevier, 2015-06-05)
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 ...
Derivation of generalized Camassa-Holm equations from Boussinesq-type equations
(Informa Group, 2016)
In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive ...
Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations
(Elsevier, 2014-01)
In this paper we study the global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, View the MathML source, where the nonlocality enters through two ...
Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations
(EDP Sciences, 2018-09-13)
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 Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
(AIMS, 2016-11)
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 ...
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