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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 ...

#### Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity

(Elsevier, 2015-12)

In this paper we investigate traveling wave solutions of a non-linear differential equation describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of ...

#### 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 ...

#### 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 ...

#### 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 ...

#### Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity

(Elsevier, 2020-11-15)

In this work we prove local existence of strong solutions to the initial-value problem arising in one-dimensional strain-limiting viscoelasticity, which is based on a nonlinear constitutive relation between the linearized ...

#### Numerical computation of solitary wave solutions of the Rosenau equation

(Elsevier, 2020-11)

We construct numerically solitary wave solutions of the Rosenau equation using the Petviashvili iteration method. We first summarize the theoretical results available in the literature for the existence of solitary wave ...

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