A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

Abstract

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.

Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: the KdV, BBM, and CH equations

“…In this section we briefly state the nonlinear and nonlocal model discussed in [5,6]. In the absence of body forces the equation of motion for a one-dimensional, homogeneous, elastic medium is…”

“…As in [5,6], we consider a nonlinear and nonlocal elastic medium whose constitutive equation is given by…”

“…In this section we briefly review the unidirectional wave equations derived in [9] for small-but-finite amplitude long waves propagating in the one-dimensional nonlocal media governed by (4) and (5). From a wave propagation point of view, both (4) and (5) are dispersive wave equations.…”

“…Nonlocal elastic models have been successfully used to obtain solutions for linear problems arising in various applications (see for instance [2][3][4] and the references cited therein). Recently, a nonlinear model of nonlocal elasticity was proposed in [5,6] in which the nonlocal stress is expressed as the weighted average of the local stress over space and consequently the nonlinear constitutive relation is based on a convolution integral over space. In subsequent papers [7,8], global existence and nonexistence of solutions of the initial-value problems posed for various generalizations of the model were investigated.…”

Some remarks on the stability and instability properties of solitary waves for the double dispersion equation

“…To see this, notice that when [7] and the improved Boussinesq [8] belong to this class, and they can be considered as the limiting cases of the double dispersion equation. One example for higher-order differential operators is the higher-order improved Boussinesq equation u tt − u xx − u xxtt + bu xxxxtt = (g(u)) xx (for which K = 0, M = −∂ 2 x + b∂ 4 x ) [9]. To consider a truly nonlocal case, taking the operator (I + M) −1 as a convolution integral with kernel β and K = 0 we get the nonlinear nonlocal wave equation u tt = [β * (u + g(u))] xx of nonlocal elasticity [10].…”