8 research outputs found
Scattering of bulk strain solitary waves in bi-layers with delamination
We study the scattering of longitudinal bulk strain solitary waves in delaminated bi-layers with different types of bonding. The direct numerical modelling of these problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and the Integrability Theory of the Korteweg - de Vries equation by the Inverse Scattering Transform. We show that the semi-analytical approach agrees well with the direct numerical simulations and use it to study the scattering of different types of longitudinal bulk strain solitary waves in a wide range of bi-layers with delamination. In particular, we model the dynamics of a long longitudinal strain solitary wave in a symmetric perfectly bonded bi-layer with delamination. The numerical modelling confirms that delamination causes fission of an incident solitary wave and, thus, can be used to detect the defect. We then extend our approaches to the modelling of the waves in bi-layers with soft ("imperfect") bonding, described by a system of coupled Boussinesq equations and supporting radiating solitary waves. The results may help us to control the integrity of layered structures
Modulational instability of two pairs of counter-propagating waves and energy exchange in two-component media
The dynamics of two pairs of counter-propagating waves in two-component media
is considered within the framework of two generally nonintegrable coupled
Sine-Gordon equations. We consider the dynamics of weakly nonlinear wave
packets, and using an asymptotic multiple-scales expansion we obtain a suite of
evolution equations to describe energy exchange between the two components of
the system. Depending on the wave packet length-scale vis-a-vis the wave
amplitude scale, these evolution equations are either four non-dispersive and
nonlinearly coupled envelope equations, or four non-locally coupled nonlinear
Schroedinger equations. We also consider a set of fully coupled nonlinear
Schroedinger equations, even though this system contains small dispersive terms
which are strictly beyond the leading order of the asymptotic multiple-scales
expansion method. Using both the theoretical predictions following from these
asymptotic models and numerical simulations of the original unapproximated
equations, we investigate the stability of plane-wave solutions, and show that
they may be modulationally unstable. These instabilities can then lead to the
formation of localized structures, and to a modification of the energy exchange
between the components. When the system is close to being integrable, the
time-evolution is distinguished by a remarkable almost periodic sequence of
energy exchange scenarios, with spatial patterns alternating between
approximately uniform wavetrains and localized structures.Comment: 35 pages, 13 figure
On integrability of (2+1)-dimensional quasilinear systems
A (2+1)-dimensional quasilinear system is said to be `integrable' if it can
be decoupled in infinitely many ways into a pair of compatible n-component
one-dimensional systems in Riemann invariants. Exact solutions described by
these reductions, known as nonlinear interactions of planar simple waves, can
be viewed as natural dispersionless analogs of n-gap solutions. It is
demonstrated that the requirement of the existence of 'sufficiently many'
n-component reductions provides the effective classification criterion. As an
example of this approach we classify integrable (2+1)-dimensional systems of
conservation laws possessing a convex quadratic entropy.Comment: 23 page