Derivation of the Camassa–Holm equations for elastic waves
Introduction
In the present paper we show that, in the long wave limit, small-but-finite waves propagating in a one-dimensional medium made of nonlocally and nonlinearly elastic material satisfy the Camassa–Holm (CH) equation [1] and the fractional CH equation (see Eq. (4.10)) when a proper balance between dispersion and nonlinearity exists.
The CH equation was derived for the propagation of unidirectional small-amplitude shallow-water waves [1], [2], [3], [4], [5], [6] when the nonlinear effects are stronger than the dispersive effects. Due to the fact that, even for smooth initial data, the solution of the CH equation stays bounded as its slope becomes unbounded, it is often used as an appropriate model capturing the essential features of wave-breaking of shallow-water waves [7]. However, recalling that (1.1) is derived under the long wavelength assumption, it follows that the CH equation is valid only when the solutions and their derivatives remain bounded [5]. For a discussion on a different criterion for wave-breaking in long wave models we refer the reader to [8]. It is interesting to note that the CH equation as a model for wave-breaking of water waves is an infinite-dimensional completely integrable Hamiltonian system [9], [10]. Another interesting property of the CH equation is the existence of the so-called peakon solitary wave solutions when [1]. At this point, it is worth pointing out that the derivation in the present study is also based on the long wavelength assumption and that is nonzero for the resulting equation. In addition to the studies about water waves, there are also studies that derive the CH equation as an appropriate model equation for nonlinear dispersive elastic waves. We refer the reader to [11] for the derivation of a CH-type equation governing the propagation of long waves in a compressible hyperelastic rod, and to [12] for the derivation of a two-dimensional CH-type equation governing the propagation of long waves in a compressible hyperelastic plate. However, these studies relied only on the “geometrical” dispersion resulting from the existence of the boundaries, that is, from the existence of a bounded elastic solid, like a rod or a plate. Another type of dispersion for elastic waves is the “physical” dispersion produced by the internal structure of the medium. Therefore, one interesting question is to investigate whether the CH equation can be derived as an asymptotic approximation for physically dispersive nonlinear elastic waves in the absence of the geometrical dispersion. In this study, we consider the one-dimensional wave propagation in an infinite, nonlinearly and nonlocally elastic medium whose constitutive behavior is described by a convolution integral. We then show that, for an exponential-type kernel function, the CH equation can model the propagation of elastic waves even in the absence of the geometrical dispersion. Furthermore, by considering a fractional-type kernel function we are able to derive a fractional-type CH equation, which indicates the possibility of obtaining more general evolution equations for suitable kernel functions. It is well known that the KdV and BBM equations are valid at the same level of approximation while the CH equation is more accurate than the KdV and BBM equations. Therefore, when we neglect the highest order terms in the asymptotic expansion, the KdV and BBM equations and their fractional generalizations are also obtained as a by-product of the present derivation. We underline that the asymptotic derivation of the CH equation needs a double asymptotic expansion in two small parameters characterizing nonlinear and dispersive effects. However, assuming simply that the two parameters are equal, the asymptotic derivations of the KdV and BBM equations can also be based on a single asymptotic expansion in one small parameter resulting from the balance of nonlinear and dispersive effects.
The paper is organized as follows. Section 2 presents the governing equations of one-dimensional nonlocal nonlinear elasticity theory and gives the equation of motion in dimensionless quantities for various forms of the kernel function. In Section 3, using a multiple scale asymptotic expansion, the CH equation is derived from the improved Boussinesq (IBq) equation which is the equation of motion for the exponential kernel function. Section 4 presents the derivation of a fractional CH equation from the equation of motion corresponding to a fractional-type kernel function.
Section snippets
A one-dimensional nonlinear theory of nonlocal elasticity
We consider a one-dimensional, infinite, homogeneous, elastic medium with a nonlinear and nonlocal stress–strain relation (see [13], [14], [15] and the references cited therein for a more detailed discussion of the nonlocal model). In the absence of body forces the equation of motion is where the scalar function represents the displacement of a reference point X at time t, is the mass density of the medium, is the stress and the subscripts denote partial
Derivation of the Camassa–Holm equation in the long wave limit
In this section we provide a formal derivation of the CH equation from (2.5) with in the long wave limit. We restrict our attention to the quadratic nonlinearity since the KdV, BBM and CH equations all contain only quadratic nonlinearities. For a data on a compact support, (2.5) has both right-going and left-going wave solutions that are moving apart. Assuming the two waves no longer overlap for a sufficiently large time, in the rest of this section we consider right-going,
Derivation of a fractional Camassa–Holm equation
In this section we will handle (2.6) with and explore how to extend the asymptotic expansion of the previous section to (2.6). As we did in the previous section, to make the asymptotic behavior of (2.6) more transparent, we use the scaling transformation (3.1) in (2.6) and we get We then seek an asymptotic solution of (4.1) in the form with the assumptions we made for (3.3).
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