Dispersive waves in microstructured solids

On solitary waves in case of amplitude-dependent nonlinearity

“…Usually (like, for example, in the KdV equation or various modifications of Boussinesq equations) higher amplitude solitons travel faster, see for example [1,13,21,16] and references therein. However, systems where smaller amplitude disturbances travel faster do exist, for example in the case of loop solitons where indeed smaller amplitude solitons travel faster [14] and even under some parameter combinations for the classical Boussinesq equation [2] -such an effect was just unexpected here.…”

Negative group velocity in solids

“…Under the used parameter combinations the added small nonlinearity makes the velocity amplitude dependent and introduces following effects: (a) propagating wave profiles evolve into an asymmetric shape (peak tilted slightly in the direction of the propagation), (b) interactions are no longer fully elastic (small additional radiation from interaction events between the pulses) and (c) the maximum amplitude of the emitted oscillatory structure is marginally smaller and the main pulse maintains its amplitude marginally better (less than 1% difference) than in the linear case. It should be added that if the ratio of nonlinearity versus dispersive effects is just right then the nonlinearity can balance the dispersive effects so that solitons can emerge (see [20,29] and references therein). However, this analysis did not reveal any effects that might arise from the interaction of the NGV region in the dispersion curves with the nonlinear effects.…”

“…We start with Lagrangian L = K − W , where K is the kinetic and W is the potential energy and derive the governing equations by using Euler-Lagrange equations after determining the K and W . For two microstructures with different scales one of the simplest potentials W which accounts for nonlinear and dispersive terms can be taken as (see [19,20] and references therein)…”

“…In the following we deal with the case A 2 = 0. Following the Euler-Lagrange formalism (see [14,15,20] for details), the system of governing equations in the dimensionless normalised form is…”

On modelling of wave propagation in microstructured solids

“…the fourth-order derivatives U T T XX and U XXXX govern the dispersion effects. The dispersion analysis for the derived model is represented in several studies [ 18,19,24,27 ]. From the results of the dispersion analysis it is worth to mention the following:…”