1,304 research outputs found
Nonlinear ring waves in a two-layer fluid
Surface and interfacial weakly-nonlinear ring waves in a two-layer fluid are
modelled numerically, within the framework of the recently derived
2+1-dimensional cKdV-type equation. In a case study, we consider concentric
waves from a localised initial condition and waves in a 2D version of the
dam-break problem, as well as discussing the effect of a piecewise-constant
shear flow. The modelling shows, in particular, the formation of 2D dispersive
shock waves (DSWs) and oscillatory wave trains. The surface and interfacial
DSWs generated in our numerical experiments look distinctively different.Comment: 16 pages, 21 figure
Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability
A (d+1)-dimensional dispersionless PDE is said to be integrable if its
n-component hydrodynamic reductions are locally parametrized by (d-1)n
arbitrary functions of one variable. Given a PDE which does not pass the
integrability test, the method of hydrodynamic reductions allows one to
effectively reconstruct additional differential constraints which, when added
to the equation, make it an integrable system in fewer dimensions (if
consistent).Comment: 16 page
Stasheff polytopes and the coordinate ring of the cluster X-variety of type A_n
We define Stasheff polytopes in the space of tropical points of cluster
A-varieties. We study the supports of products of elements of canonical bases
for cluster X-varieties. We prove that, for the cluster X-variety of type A_n,
such supports are Stasheff polytopes.Comment: 30 pages, 8 figure
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
Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian
We investigate integrable second order equations of the form
F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0.
Familiar examples include the Boyer-Finley equation, the potential form of
the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota
equation, etc. The integrability is understood as the existence of infinitely
many hydrodynamic reductions. We demonstrate that the natural equivalence group
of the problem is isomorphic to Sp(6), revealing a remarkable correspondence
between differential equations of the above type and hypersurfaces of the
Lagrangian Grassmannian. We prove that the moduli space of integrable equations
of the dispersionless Hirota type is 21-dimensional, and the action of the
equivalence group Sp(6) on the moduli space has an open orbit.Comment: 32 page
Coupled Ostrovsky equations for internal waves in a shear flow
In the context of fluid flows, the coupled Ostrovsky equations arise when two
distinct linear long wave modes have nearly coincident phase speeds in the
presence of background rotation. In this paper, nonlinear waves in a stratified
fluid in the presence of shear flow are investigated both analytically, using
techniques from asymptotic perturbation theory, and through numerical
simulations. The dispersion relation of the system, based on a three-layer
model of a stratified shear flow, reveals various dynamical behaviours,
including the existence of unsteady and steady envelope wave packets.Comment: 47 pages, 39 figures, accepted to Physics of Fluid
On Boussinesq-type models for long longitudinal waves in elastic rods
In this paper we revisit the derivations of model equations describing long
nonlinear longitudinal bulk strain waves in elastic rods within the scope of
the Murnaghan model in order to derive a Boussinesq-type model, and extend
these derivations to include axially symmetric loading on the lateral boundary
surface, and longitudinal pre-stretch. We systematically derive two forced
Boussinesq-type models from the full equations of motion and non-zero surface
boundary conditions, utilising the presence of two small parameters
characterising the smallness of the wave amplitude and the long wavelength
compared to the radius of the waveguide. We compare the basic dynamical
properties of both models (linear dispersion curves and solitary wave
solutions). We also briefly describe the laboratory experiments on generation
of bulk strain solitary waves in the Ioffe Institute, and suggest that this
generation process can be modelled using the derived equations.Comment: 19 pages, 5 figures, submitted to the Special Issue of Wave Motion,
"Nonlinear Waves in Solids", in Memory of Professor Alexander M. Samsono
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