| Abstract: |
| We are considering convolution-type nonlocal equations of the form$\
u_{tt}=\beta \ast g(u)_{xx}$, where $\beta $ is a general $\ $kernel and $g$
represents the nonlinearity. This form can be considered as a model for the
bi-directional propagation of one-dimensional nonlinear waves in a non-local
elastic material. For particular choices of the kernel, the nonlocal
equation reduces to well known examples as the regularized Boussinesq type
equations. Our investigation covers a wide range of questions as
well-posedness of the Cauchy problem, existence of traveling waves and
numerical methods taking advantage of the convolution. We have similar
results for the unidirectional problem $u_{t}=\beta \ast g(u)_{x}$,
generalizing BBM and BBM-KdV type equations. The talk will concentrate on
comparing solutions of two such equations with two different kernels. We
show that in the long wave small amplitude regime, solutions stay close
over long times. In particular, this implies that when $\beta $ approaches
the Dirac measure solutions converge uniformly to solutions of the classical
elasticity equation $u_{tt}=(u+u^{p+1})_{xx}.$ |
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