710 research outputs found
Bäcklund transformations for noncommutative anti-self-dual Yang-Mills equations
We present Bäcklund transformations for the non-commutative anti-self-dual Yang–Mills equations where the gauge group is G = GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants and belong to a non-commutative version of the Atiyah–Ward ansatz. In the commutative limit, our results coincide with those by Corrigan, Fairlie, Yates and Goddard
Plastic flow in polycrystal states in a binary mixture
Using molecular dynamics simulation we examine dynamics in sheared
polycrystal states in a binary mixture containing 10% larger particles in two
dimensions. We find large stress fluctuations arising from sliding motions of
the particles at the grain boundaries, which occur cooperatively to release the
elastic energy stored. These dynamic processes are visualized with the aid of a
sixfold angle representing the local crystal orientation and a
disorder variable representing a deviation from the hexagonal order
for particle .Comment: 3 pages, 3 figure
Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies
We discuss commuting flows and conservation laws for Lax hierarchies on
noncommutative spaces in the framework of the Sato theory. On commutative
spaces, the Sato theory has revealed essential aspects of the integrability for
wide class of soliton equations which are derived from the Lax hierarchies in
terms of pseudo-differential operators. Noncommutative extension of the Sato
theory has been already studied by the author and Kouichi Toda, and the
existence of various noncommutative Lax hierarchies are guaranteed. In the
present paper, we present conservation laws for the noncommutative Lax
hierarchies with both space-space and space-time noncommutativities and prove
the existence of infinite number of conserved densities. We also give the
explicit representations of them in terms of Lax operators. Our results include
noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera,
modified KdV equations and so on.Comment: 22 pages, LaTeX, v2: typos corrected, references added, version to
appear in JM
On a direct approach to quasideterminant solutions of a noncommutative KP equation
A noncommutative version of the KP equation and two families of its solutions
expressed as quasideterminants are discussed. The origin of these solutions is
explained by means of Darboux and binary Darboux transformations. Additionally,
it is shown that these solutions may also be verified directly. This approach
is reminiscent of the wronskian technique used for the Hirota bilinear form of
the regular, commutative KP equation but, in the noncommutative case, no
bilinearising transformation is available.Comment: 11 page
Transitions among crystal, glass, and liquid in a binary mixture with changing particle size ratio and temperature
Using molecular dynamics simulation we examine changeovers among crystal,
glass, and liquid at high density in a two dimensional binary mixture. We
change the ratio between the diameters of the two components and the
temperature. The transitions from crystal to glass or liquid occur with
proliferation of defects. We visualize the defects in terms of a disorder
variable "D_j(t)" representing a deviation from the hexagonal order for
particle j. The defect structures are heterogeneous and are particularly
extended in polycrystal states. They look similar at the crystal-glass
crossover and at the melting. Taking the average of "D_j(t)" over the
particles, we define a disorder parameter "D(t)", which conveniently measures
the degree of overall disorder. Its relaxation after quenching becomes slow at
low temperature in the presence of size dispersity. Its steady state average is
small in crystal and large in glass and liquid.Comment: 7 pages, 10 figure
Molecular Dynamics Simulation of Heat-Conducting Near-Critical Fluids
Using molecular dynamics simulations, we study supercritical fluids near the
gas-liquid critical point under heat flow in two dimensions. We calculate the
steady-state temperature and density profiles. The resultant thermal
conductivity exhibits critical singularity in agreement with the mode-coupling
theory in two dimensions. We also calculate distributions of the momentum and
heat fluxes at fixed density. They indicate that liquid-like (entropy-poor)
clusters move toward the warmer boundary and gas-like (entropy-rich) regions
move toward the cooler boundary in a temperature gradient. This counterflow
results in critical enhancement of the thermal conductivity
B\"acklund Transformations and the Atiyah-Ward ansatz for Noncommutative Anti-Self-Dual Yang-Mills Equations
We present Backlund transformations for the noncommutative anti-self-dual
Yang-Mills equation where the gauge group is G=GL(2) and use it to generate a
series of exact solutions from a simple seed solution. The solutions generated
by this approach are represented in terms of quasideterminants. We also explain
the origins of all of the ingredients of the Backlund transformations within
the framework of noncommutative twistor theory. In particular we show that the
generated solutions belong to a noncommutative version of the Atiyah-Ward
ansatz.Comment: v2: 21 pages, published versio
Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation
Matrix solutions of a noncommutative KP and a noncommutative mKP equation
which can be expressed as quasideterminants are discussed. In particular, we
investigate interaction properties of two-soliton solutions.Comment: 2 figure
On Non-Commutative Integrable Burgers Equations
We construct the recursion operators for the non-commutative Burgers
equations using their Lax operators. We investigate the existence of any
integrable mixed version of left- and right-handed Burgers equations on higher
symmetry grounds.Comment: 8 page
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