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    Derivation of generalized Camassa-Holm equations from Boussinesq-type equations
    (Informa Group, 2016) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.
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    The Camassa-Holm approximation to the double dispersion equation for arbitrarily long times
    (Springer, 2022-09) Erbay, Saadet; Erkip, A.; Kuruk, G.; Natural and Mathematical Sciences; ERBAY, Saadet
    In the present paper we prove the validity of the Camassa-Holm equation as a long wave limit to the double dispersion equation which describes the propagation of bidirectional weakly nonlinear and dispersive waves in an infinite elastic medium. First we show formally that the right-going wave solutions of the double dispersion equation can be approximated by the solutions of the Camassa-Holm equation in the long wave limit. Then we rigorously prove that the solutions of the double dispersion and the Camassa-Holm equations remain close over a long time interval, determined by two small parameters measuring the effects of nonlinearity and dispersion.
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    Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations
    (EDP Sciences, 2018-09-13) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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.
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    A comparison of solutions of two convolution-type unidirectional wave equations
    (Taylor and Francis, 2023) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    In this work, we prove a comparison result for a general class of nonlinear dispersive unidirectional wave equations. The dispersive nature of one-dimensional waves occurs because of a convolution integral in space. For two specific choices of the kernel function, the Benjamin–Bona–Mahony equation and the Rosenau equation that are particularly suitable to model water waves and elastic waves, respectively, are two members of the class. We first prove an energy estimate for the Cauchy problem of the non-local unidirectional wave equation. Then, for the same initial data, we consider two distinct solutions corresponding to two different kernel functions. Our main result is that the difference between the solutions remains small in a suitable Sobolev norm if the two kernel functions have similar dispersive characteristics in the long-wave limit. As a sample case of this comparison result, we provide the approximations of the hyperbolic conservation law.
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    Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations
    (American Institute of Mathematical Sciences, 2019-05) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 integral. We prove a long-time existence result for the nonlocal wave equations with a power-type nonlinearity and a small parameter. As the energy estimates involve a loss of derivatives, we follow the Nash-Moser approach proposed by Alvarez-Samaniego and Lannes. As an application to the long-time existence theorem, we consider the limiting case in which the kernel function is the Dirac measure and the nonlocal equation reduces to the governing equation of one-dimensional classical elasticity theory. The present study also extends our earlier result concerning local well-posedness for smooth kernels to nonsmooth kernels.
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    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) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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.
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    On the full dispersion Kadomtsev–Petviashvili equations for dispersive elastic waves
    (Elsevier, 2022-09) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    Full dispersive models of water waves, such as the Whitham equation and the full dispersion Kadomtsev–Petviashvili (KP) equation, are interesting from both the physical and mathematical points of view. This paper studies analogous full dispersive KP models of nonlinear elastic waves propagating in a nonlocal elastic medium. In particular we consider anti-plane shear elastic waves which are assumed to be small-amplitude long waves. We propose two different full dispersive extensions of the KP equation in the case of cubic nonlinearity and ”negative dispersion”. One of them is called the Whitham-type full dispersion KP equation and the other one is called the BBM-type full dispersion KP equation. Most of the existing KP-type equations in the literature are particular cases of our full dispersion KP equations. We also introduce the simplified models of the new proposed full dispersion KP equations by approximating the operators in the equations. We show that the line solitary wave solution of a simplified form of the Whitham-type full dispersion KP equation is linearly unstable to long-wavelength transverse disturbances if the propagation speed of the line solitary wave is greater than a certain value. A similar analysis for a simplified form of the BBM-type full dispersion KP equation does not provide a linear instability assessment.
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    A semi-discrete numerical method for convolution-type unidirectional wave equations
    (Elsevier, 2021-05-15) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolution-type nonlocality in space is considered. A semi-discrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the Cauchy problem. The method is proved to be uniformly convergent as the mesh size goes to zero. The order of convergence for the discretization error is linear or quadratic depending on the smoothness of the convolution kernel. The discrete problem defined on the whole spatial domain is then truncated to a finite domain. Restricting the problem to a finite domain introduces a localization error and it is proved that this localization error stays below a given threshold if the finite domain is large enough. For two particular kernel functions, the numerical examples concerning solitary wave solutions illustrate the expected accuracy of the method. Our class of nonlocal wave equations includes the Benjamin–Bona–Mahony equation as a special case and the present work is inspired by the previous work of Bona, Pritchard and Scott on numerical solution of the Benjamin–Bona–Mahony equation.
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    Derivation of the Camassa-Holm equations for elastic waves
    (Elsevier, 2015-06-05) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 that 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.
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    Comparison of nonlocal nonlinear wave equations in the long-wave limit
    (Taylor & Francis, 2020-11-17) Erbay, Hüsnü Ata; Erbay, Saadet; Erkip, A.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata; ERBAY, Saadet
    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 and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular, our results show that, in the long-wave limit, solutions of such nonlocal equations can be well approximated by those of improved Boussinesq-type equations.