78 research outputs found
Soliton Resonances for MKP-II
Using the second flow - the Derivative Reaction-Diffusion system, and the
third one of the dissipative SL(2,R) Kaup-Newell hierarchy, we show that the
product of two functions, satisfying those systems is a solution of the
modified Kadomtsev-Petviashvili equation in 2+1 dimension with negative
dispersion (MKP-II). We construct Hirota's bilinear representation for both
flows and combine them together as the bilinear system for MKP-II. Using this
bilinear form we find one and two soliton solutions for the MKP-II. For special
values of parameters our solution shows resonance behaviour with creation of
four virtual solitons. Our approach allows one to interpret the resonance
soliton as a composite object of two dissipative solitons in 1+1 dimensions.Comment: 11 pages, 2 figures, Talk on International Conference "Nonlinear
Physics. Theory and Experiment. III", 24 June-3 July, 2004, Gallipoli(Lecce),
Ital
Topological Field Theory and Nonlinear -Models on Symmetric Spaces
We show that the classical non-abelian pure Chern-Simons action is related to
nonrelativistic models in (2+1)-dimensions, via reductions of the gauge
connection in Hermitian symmetric spaces. In such models the matter fields are
coupled to gauge Chern-Simons fields, which are associated with the isotropy
subgroup of the considered symmetric space. Moreover, they can be related to
certain (integrable and non-integrable) evolution systems, as the Ishimori and
the Heisenberg model. The main classical and quantum properties of these
systems are discussed in connection with the topological field theory and the
condensed matter physics.Comment: LaTeX format, 31 page
Dissipation and Topologically Massive Gauge Theories in Pseudoeuclidean Plane
In the pseudo-euclidean metrics Chern-Simons gauge theory in the infrared
region is found to be associated with dissipative dynamics. In the infrared
limit the Lagrangian of 2+1 dimensional pseudo-euclidean topologically massive
electrodynamics has indeed the same form of the Lagrangian of the damped
harmonic oscillator. On the hyperbolic plane a set of two damped harmonic
oscillators, each other time-reversed, is shown to be equivalent to a single
undamped harmonic oscillator. The equations for the damped oscillators are
proven to be the same as the ones for the Lorentz force acting on two particles
carrying opposite charge in a constant magnetic field and in the electric
harmonic potential. This provides an immediate link with Chern-Simons-like
dynamics of Bloch electrons in solids propagating along the lattice plane with
hyperbolic energy surface. The symplectic structure of the reduced theory is
finally discussed in the Dirac constrained canonical formalism.Comment: 22 pages, LaTe
Degenerate Four Virtual Soliton Resonance for KP-II
By using disipative version of the second and the third members of AKNS
hierarchy, a new method to solve 2+1 dimensional Kadomtsev-Petviashvili (KP-II)
equation is proposed. We show that dissipative solitons (dissipatons) of those
members give rise to the real solitons of KP-II. From the Hirota bilinear form
of the SL(2,R) AKNS flows, we formulate a new bilinear representation for
KP-II, by which, one and two soliton solutions are constructed and the
resonance character of their mutual interactions is studied. By our bilinear
form, we first time created four virtual soliton resonance solution for KP-II
and established relations of it with degenerate four-soliton solution in the
Hirota-Satsuma bilinear form for KP-II.Comment: 10 pages, 5 figures, Talk on International Conference Nonlinear
Physics. Theory and Experiment. III, 24 June-3 July, 2004, Gallipoli(Lecce),
Ital
Integrable Hierarchies and Information Measures
In this paper we investigate integrable models from the perspective of
information theory, exhibiting various connections. We begin by showing that
compressible hydrodynamics for a one-dimesional isentropic fluid, with an
appropriately motivated information theoretic extension, is described by a
general nonlinear Schrodinger (NLS) equation. Depending on the choice of the
enthalpy function, one obtains the cubic NLS or other modified NLS equations
that have applications in various fields. Next, by considering the integrable
hierarchy associated with the NLS model, we propose higher order information
measures which include the Fisher measure as their first member. The lowest
members of the hiearchy are shown to be included in the expansion of a
regularized Kullback-Leibler measure while, on the other hand, a suitable
combination of the NLS hierarchy leads to a Wootters type measure related to a
NLS equation with a relativistic dispersion relation. Finally, through our
approach, we are led to construct an integrable semi-relativistic NLS equation.Comment: 11 page
Resonance NLS Solitons as Black Holes in Madelung Fluid
A new resonance version of NLS equation is found and embedded to the
reaction-diffusion system, equivalent to the anti-de Sitter valued Heisenberg
model, realizing a particular gauge fixing condition of the Jackiw-Teitelboim
gravity. The space-time points where dispersion change the sign correspond to
the event horizon, and the soliton solutions to the AdS black holes. The
soliton with velocity bounded above describes evolution on the hyperboloid with
nontrivial winding number and create under collisions the resonance states with
a specific life time.Comment: Plain Tex, 12 pages, 6 figure
Solitons of the Resonant Nonlinear Schrodinger Equation with Nontrivial Boundary Conditions and Hirota Bilinear Method
Physically relevant soliton solutions of the resonant nonlinear Schrodinger
(RNLS) equation with nontrivial boundary conditions, recently proposed for
description of uniaxial waves in a cold collisionless plasma, are considered in
the Hirota bilinear approach. By the Madelung representation, the model is
transformed to the reaction-diffusion analog of the NLS equation for which the
bilinear representation, soliton solutions and their mutual interactions are
studied.Comment: 15 pages, 1 figure, talk presented in Workshop `Nonlinear Physics IV:
Theory and Experiment`, 22-30 June 2006, Gallipoli, Ital
Young diagrams and N-soliton solutions of the KP equation
We consider -soliton solutions of the KP equation,
(-4u_t+u_{xxx}+6uu_x)_x+3u_{yy}=0 . An -soliton solution is a solution
which has the same set of line soliton solutions in both
asymptotics and . The -soliton solutions include
all possible resonant interactions among those line solitons. We then classify
those -soliton solutions by defining a pair of -numbers with , which labels line solitons in the solution. The
classification is related to the Schubert decomposition of the Grassmann
manifolds Gr, where the solution of the KP equation is defined as a
torus orbit. Then the interaction pattern of -soliton solution can be
described by the pair of Young diagrams associated with . We also show that -soliton solutions of the KdV equation obtained by
the constraint cannot have resonant interaction.Comment: 22 pages, 5 figures, some minor corrections and added one section on
the KdV N-soliton solution
Abelian Chern-Simons Vortices and Holomorphic Burgers' Hierarchy
The Abelian Chern-Simons Gauge Field Theory in 2+1 dimensions and its
relation with holomorphic Burgers' Hierarchy is considered. It is shown that
the relation between complex potential and the complex gauge field as in
incompressible and irrotational hydrodynamics, has meaning of the analytic
Cole-Hopf transformation, linearizing the Burgers Hierarchy in terms of the
holomorphic Schr\"odinger Hierarchy. Then the motion of planar vortices in
Chern-Simons theory, appearing as pole singularities of the gauge field,
corresponds to motion of zeroes of the hierarchy. Using boost transformations
of the complex Galilean group of the hierarchy, a rich set of exact solutions,
describing integrable dynamics of planar vortices and vortex lattices in terms
of the generalized Kampe de Feriet and Hermite polynomials is constructed. The
results are applied to the holomorphic reduction of the Ishimori model and the
corresponding hierarchy, describing dynamics of magnetic vortices and
corresponding lattices in terms of complexified Calogero-Moser models.
Corrections on two vortex dynamics from the Moyal space-time non-commutativity
in terms of Airy functions are found.Comment: 15 pages, talk presented in Workshop `Nonlinear Physics IV: Theory
and Experiment`, 22-30 June 2006, Gallipoli, Ital
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