3,746 research outputs found
Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems
Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the
requirement of closedness of the differential N-1 forms of rank
N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these
flows, given by the systems of the N-1 quasi-linear differential equations,
describe coisotropic deformations of (N-1)-dimensional linear subspaces. For
the class of solutions which are Laurent polynomials in one variable these
systems coincide with N-dimensional integrable systems such as Liouville
equation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3),
dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4)
and others. Gauge invariant part of the forms provides us with
the compact form of the corresponding hierarchies. Dual quasi-linear systems
associated with the projectively dual Grassmannians Gr(2,N+1) are defined via
the requirement of the closedness of the dual forms . It
is shown that at N=3 the self-dual quasi-linear system, which is associated
with the harmonic (closed and co-closed) form , coincides with the
Maxwell equations for orthogonal electric and magnetic fields.Comment: 26 pages, references adde
Spin-flop transition in uniaxial antiferromagnets: magnetic phases, reorientation effects, multidomain states
The classical spin-flop is the field-driven first-order reorientation
transition in easy-axis antiferromagnets. A comprehensive phenomenological
theory of easy-axis antiferromagnets displaying spin-flops is developed. It is
shown how the hierarchy of magnetic coupling strengths in these
antiferromagnets causes a strongly pronounced two-scale character in their
magnetic phase structure. In contrast to the major part of the magnetic phase
diagram, these antiferromagnets near the spin-flop region are described by an
effective model akin to uniaxial ferromagnets. For a consistent theoretical
description both higher-order anisotropy contributions and dipolar stray-fields
have to be taken into account near the spin-flop. In particular,
thermodynamically stable multidomain states exist in the spin-flop region,
owing to the phase coexistence at this first-order transition. For this region,
equilibrium spin-configurations and parameters of the multidomain states are
derived as functions of the external magnetic field. The components of the
magnetic susceptibility tensor are calculated for homogeneous and multidomain
states in the vicinity of the spin-flop. The remarkable anomalies in these
measurable quantities provide an efficient method to investigate magnetic
states and to determine materials parameters in bulk and confined
antiferromagnets, as well as in nanoscale synthetic antiferromagnets. The
method is demonstrated for experimental data on the magnetic properties near
the spin-flop region in the orthorhombic layered antiferromagnet
(C_2H_5NH_3)_2CuCl_4.Comment: (15 pages, 12 figures; 2nd version: improved notation and figures,
correction of various typos
On the heavenly equation hierarchy and its reductions
Second heavenly equation hierarchy is considered using the framework of
hyper-K\"ahler hierarchy developed by Takasaki. Generating equations for the
hierarchy are introduced, they are used to construct generating equations for
reduced hierarchies. General -reductions, logarithmic reduction and rational
reduction for one of the Lax-Sato functions are discussed. It is demonstrated
that rational reduction is equivalent to the symmetry constraint.Comment: 13 pages, LaTeX, minor misprints corrected, references adde
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