622 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
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
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
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