On Boussinesq's paradigm in nonlinear wave propagation

Abstract

Boussinesq's original derivation of his celebrated equation for surface waves on a fluid layer opened up new horizons that were to yield the concept of the soliton. The present contribution concerns the set of Boussinesq-like equations under the general title of 'Boussinesq's paradigm'. These are true bi-directional wave equations occurring in many physical instances and sharing analogous properties. The emphasis is placed: (i) on generalized Boussinesq systems that involve higher-order linear dispersion through either additional space derivatives or additional wave operators (so-called double-dispersion equations); and (ii) on the 'mechanics' of the most representative localized nonlinear wave solutions. Dissipative cases and two-dimensional generalizations are also considered.

A Note About Waves in Dissipative and Dispersive Solids

“…Then the assumption Q = µ 0 + µ 1 Ω 2 (where µ 0 , µ 1 are positive constants) is the simplest one we can make to uncover nonlinear governing equations; it corresponds to fourth-order elasticity theory 10 . With these assumptions, equation (19) reduces to…”

“…A thorough review on several mechanical aspects and applications of this equation may be found in the recent paper by Christov et al 19 . When we study travelling waves of (30) in the linearly polarized case, we obtain…”

“…(14); however, the equations were the result of a small parameter expansion, see Section 3. We now remark that it is also possible to study travelling waves for the exact equation (19), both in the case of linearly-polarized waves and in the non-linearly polarized case, when a given constitutive behaviour is chosen. For instance, we make the constitutive assumptions that α = const., ν = const., which are the simplest assumptions we can make for the modelling of dispersion and dissipation.…”

Phragmén–Lindelöf alternative of exponential type for the solutions of a fourth order dispersive equation

“…It has been pointed out recently [1] that, on studying solitary waves propagating on the free surface of a constant depth fluid, Boussinesq derived initially a wellposed equation which contained a mixed fourth order derivative for the dispersion alongside with the purely spatial fourth order derivative. Then, in an attempt to find an analytical solution he went on to replace the time derivatives in the mixed derivative term by purely spatial ones using the ansatz u ;t ¼ Àcu ;x which is true for steady linear waves propagating with phase speed c. If this unnecessary ansatz was not applied by Boussinesq the form of his celebrated equation in one spatial dimension would it be in the linear regime u ;tt À c The equation we usually denote as the Boussinesq equation (BE) is obtained from (1.1) considering b 1 ¼ 0 and b 2 < 0.…”

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

“…The double dispersion equation (1) was introduced as a mathematical model of nonlinear dispersive waves in various contexts (see for instance [3][4][5][6] and the references therein). It belongs to the following general class of nonlocal nonlinear wave equations:…”