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Submitter: None
Authors: Jongbae Kim (ETRI), Q-Han Park and H.J. Shin (Kyunghee U.)
Title: Conservation Laws in Higher-Order Nonlinear Optical Effects
Abstract: Conservation laws of the nonlinear Schr\"{o}dinger equation are studied in
the presence of higher-order nonlinear optical effects including the
third-order dispersion and the self-steepening. In a context of group theory,
we derive a general expression for infinitely many conserved currents and
charges of the coupled higher-order nonlinear Schr\"{o}dinger equation. The
first few currents and charges are also presented explicitly. Due to the
higher-order effects, conservation laws of the nonlinear Schr\"{o}dinger
equation are violated in general. The differences between the types of the
conserved currents for the Hirota and the Sasa-Satsuma equations imply that the
higher-order terms determine the inherent types of conserved quantities for
each integrable cases of the higher-order nonlinear Schr\"{o}dinger equation.
Journal: Phys. Rev. E {\bf 58} 6746, 1998 |
Submitter: None
Authors: Q-Han Park and H.J. Shin (Kyunghee U.)
Title: Painlev\'{e} analysis of the coupled nonlinear Schr\"{o}dinger equation
for polarized optical waves in an isotropic medium
Abstract: Using the Painlev\'{e} analysis, we investigate the integrability properties
of a system of two coupled nonlinear Schr\"{o}dinger equations that describe
the propagation of orthogonally polarized optical waves in an isotropic medium.
Besides the well-known integrable vector nonlinear Schr\"{o}dinger equation, we
show that there exist a new set of equations passing the Painlev\'{e} test
where the self and cross phase modulational terms are of different magnitude.
We introduce the Hirota bilinearization and the B\"{a}cklund transformation to
obtain soliton solutions and prove integrability by making a change of
variables. The conditions on the third-order susceptibility tensor $\chi^{(3)}
$ imposed by these new integrable equations are explained.
Journal: Phys. Rev. E {\bf 59} 2373, 1999 |
Submitter: Baryakhtar
Authors: I.V.Baryakhtar, V.G.Baryakhtar, E.N.Economou
Title: Kinetic and Transport Equations for Localized Excitations in Sine-Gordon
Model
Abstract: We analyze the kinetic behavior of localized excitations - solitons,
breathers and phonons - in Sine-Gordon model. Collision integrals for all type
of localized excitation collision processes are constructed, and the kinetic
equations are derived. We analyze the kinetic behavior of localized excitations
- solitons, breathers and phonons - in Sine-Gordon model. Collision integrals
for all type of localized excitation collision processes are constructed, and
the kinetic equations are derived. We prove that the entropy production in the
system of localized excitations takes place only in the case of inhomogeneous
distribution of these excitations in real and phase spaces. We derive transport
equations for soliton and breather densities, temperatures and mean velocities
i.e. show that collisions of localized excitations lead to creation of
diffusion, thermoconductivity and intrinsic friction processes. The diffusion
coefficients for solitons and breathers, describing the diffusion processes in
real and phase spaces, are calculated. It is shown that diffusion processes in
real space are much faster than the diffusion processes in phase space.
Journal: None |
Submitter: Nugmanova G. N.
Authors: R. Myrzakulov
Title: On the M-XX equation
Abstract: The (2+1)-dimensional integrable M-XX equation is considered.
Journal: None |
Submitter: Paul Fendley
Authors: P. Fendley and H. Saleur
Title: Differential equations and duality in massless integrable field theories
at zero temperature
Abstract: Functional relations play a key role in the study of integrable models. We
argue in this paper that for massless field theories at zero temperature, these
relations can in fact be interpreted as monodromy relations. Combined with a
recently discovered duality, this gives a way to bypass the Bethe ansatz, and
compute directly physical quantities as solutions of a linear differential
equation, or as integrals over a hyperelliptic curve. We illustrate these ideas
in details in the case of the $c=1$ theory, and the associated boundary
sine-Gordon model.
Journal: Nucl.Phys.B574:571-586,2000 |
Submitter: Sergei M. Sergeev
Authors: S. Sergeev
Title: Solitons in a 3d integrable model
Abstract: Equations of motion for a classical 3d discrete model, whose auxialiary
system is a linear system, are investigated. The Lagrangian form of the
equations of motion is derived. The Lagrangian variables are a triplet of "tau
functions". The equations of motion for the Triplet of Tau functions are Three
Trilinear equations. Simple solitons for the trilinear equations are given.
Both the dispersion relation and the phase shift reflect the triplet structure
of equations.
Journal: None |
Submitter: Robert Milson
Authors: Niky Kamran, Robert Milson, Peter Olver
Title: Invariant Modules and the Reduction of Nonlinear Partial Differential
Equations to Dynamical Systems
Abstract: We completely characterize all nonlinear partial differential equations
leaving a given finite-dimensional vector space of analytic functions
invariant. Existence of an invariant subspace leads to a re duction of the
associated dynamical partial differential equations to a system of ordinary
differential equations, and provide a nonlinear counterpart to quasi-exactly
solvable quantum Hamiltonians. These results rely on a useful extension of the
classical Wronskian determinant condition for linear independence of functions.
In addition, new approaches to the characterization o f the annihilating
differential operators for spaces of analytic functions are presented.
Journal: None |
Submitter: Nikolay Asenov Kostov
Authors: N.A.Kostov, Z.T. Kostova
Title: Nonlinear waves, differential resultant, computer algebra and completely
integrable dynamical systems
Abstract: The hierarchy of integrable equations are considered. The dynamical approach
to the theory of nonlinear waves is proposed. The special solutions(nonlinear
waves) of considered equations are derived. We use powerful methods of computer
algebra such differential resultant and others.
Journal: None |
Submitter: Nikolay Asenov Kostov
Authors: N.A. Kostov
Title: Korteweg-de Vries hierarchy and related completely integrable systems:
I. Algebro-geometrical approach
Abstract: We consider complementary dynamical systems related to stationary Korteweg-de
Vries hierarchy of equations. A general approach for finding elliptic solutions
is given. The solutions are expressed in terms of Novikov polynomials in
general quais-periodic case. For periodic case these polynomials coincide with
Hermite and Lam\'e polynomials. As byproduct we derive $2\times 2$ matrix Lax
representation for Rosochatius-Wojciechiwski, Rosochatius, second flow of
stationary nonlinear vectro Schr\"{o}dinger equations and complex Neumann
system.
Journal: None |
Submitter: Nikolay Asenov Kostov
Authors: P.L. Christiansen, J.C. Eilbeck, V.Z. Enolskii, and N.A. Kostov
Title: Quasi-Periodic and Periodic Solutions for Systems of Coupled Nonlinear
SCHR\"Odinger Equations
Abstract: We consider travelling periodic and quasiperiodic wave solutions of a set of
coupled nonlinear Schr\"odimger equations. In fibre optics these equations can
be used to model single mode fibers with strong birefringence and two-mode
optical fibres. Recently these equations appear as modes, which describe
pulse-pulse interaction in wavelength-division-multiplexed channels of optical
fiber transmission systems. Two phase quasi-periodic solutions for integrable
Manakov system are given in tems of two-dimensional Kleinian functions. The
reduction of quasi-periodic solutions to elliptic functions is dicussed. New
solutions in terms of generalized Hermite polynomilas, which are associated
with two-gap Treibich-Verdier potentials are found.
Journal: None |
Submitter: Sudipta Nandy
Authors: Sasanka Ghosh, Anjan Kundu, Sudipta Nandy
Title: Soliton solutions, Liouville integrability and gauge equivalence of Sasa
Satsuma equation
Abstract: Exact integrability of the Sasa Satsuma eqation (SSE) in the Liouville sense
is established by showing the existence of an infinite set of conservation
laws. The explicit form of the conserved quantities in term of the fields are
obtained by solving the Riccati equation for the associated 3x3 Lax operator.
The soliton solutions in particular, one and two soliton solutions, are
constructed by the Hirota's bilinear method. The one soliton solutions is also
compared with that found through the inverse scattering method. The gauge
equivalence of the SSE with a generalized Landau Lifshitz equation is
established with the explicit construction o
Journal: None |
Submitter: Sudipta Nandy
Authors: Sasanka Ghosh, Sudipta Nandy
Title: Optical solitons in higher order nonlinear Schrodinger equation
Abstract: We show the complete integrability and the existence of optical solitons of
higher order nonlinear Schrodinger equation by inverse scattering method for a
wide range of values of coefficients. This is achieved first by invoking a
novel connection between the integrability of a nonlinear evolution equation
and the dimensions of a family of matrix Lax pairs. It is shown that Lax pairs
of different dimensions lead to the same evolution equation only with the
coefficients of the terms in different integer ratios. Optical solitons, thus
obtained by inverse scattering method, have been found by solving an n
dimensional eigenvalue problem.
Journal: None |
Submitter: Dr P. K. Panigrahi
Authors: C. Nagaraja Kumar and Prasanta K. Panigrahi (School of Physics,
University of Hyderabad, Hyderabad, India)
Title: Compacton-like Solutions for Modified KdV and other Nonlinear Equations
Abstract: We present compacton-like solution of the modified KdV equation and compare
its properties with those of the compactons and solitons. We further show that,
the nonlinear Schr{\"o}dinger equation with a source term and other higher
order KdV-like equations also possess compact solutions of the similar form.
Journal: None |
Submitter: Sudipta Nandy
Authors: Sasanka Ghosh and Sudipta Nandy
Title: Inverse scattering method and vector higher order nonlinear Schrodinger
equation
Abstract: A generalised inverse scattering method has been developed for arbitrary n
dimensional Lax equations. Subsequently, the method has been used to obtain N
soliton solutions of a vector higher order nonlinear Schrodinger equation,
proposed by us. It has been shown that under suitable reduction, vector higher
order nonlinear Schrodinger equation reduces to higher order nonlinear
Schrodinger equation. The infinite number of conserved quantities have been
obtained by solving a set of coupled Riccati equation. A gauge equivalence is
shown between the vector higher order nonlinear Schrodinger equation and the
generalized Landau Lifshitz equation and the Lax pair for the latter equation
has also been constructed in terms of the spin field, establishing direct
integrability of the spin system.
Journal: None |
Submitter: Unal Goktas
Authors: Willy Hereman (Colorado School of Mines), Unal Goktas (Wolfram
Research, Inc.)
Title: Integrability Tests for Nonlinear Evolution Equations
Abstract: Discusses several integrability tests for nonlinear evolution equations.
Journal: None |
Submitter: Andrew Pickering
Authors: Pilar R. Gordoa, Nalini Joshi and Andrew Pickering
Title: Mappings preserving locations of movable poles: a new extension of the
truncation method to ordinary differential equations
Abstract: The truncation method is a collective name for techniques that arise from
truncating a Laurent series expansion (with leading term) of generic solutions
of nonlinear partial differential equations (PDEs). Despite its utility in
finding Backlund transformations and other remarkable properties of integrable
PDEs, it has not been generally extended to ordinary differential equations
(ODEs). Here we give a new general method that provides such an extension and
show how to apply it to the classical nonlinear ODEs called the Painleve
equations. Our main new idea is to consider mappings that preserve the
locations of a natural subset of the movable poles admitted by the equation. In
this way we are able to recover all known fundamental Backlund transformations
for the equations considered. We are also able to derive Backlund
transformations onto other ODEs in the Painleve classification.
Journal: None |
Submitter: Svetlana Pacheva-Nissimov
Authors: Henrik Aratyn, Emil Nissimov and Svetlana Pacheva
Title: Multi-Component Matrix KP Hierarchies as Symmetry-Enhanced Scalar KP
Hierarchies and Their Darboux-B"acklund Solutions
Abstract: We show that any multi-component matrix KP hierarchy is equivalent to the
standard one-component (scalar) KP hierarchy endowed with a special infinite
set of abelian additional symmetries, generated by squared eigenfunction
potentials. This allows to employ a special version of the familiar
Darboux-B"acklund transformation techniques within the ordinary scalar KP
hierarchy in the Sato formulation for a systematic derivation of explicit
multiple-Wronskian tau-function solutions of all multi-component matrix KP
hierarchies.
Journal: None |
Submitter: Polterovich Iosif
Authors: Iosif Polterovich
Title: From Agmon-Kannai expansion to Korteweg-de Vries hierarchy
Abstract: We present a new method for computation of the Korteweg-de Vries hierarchy
via heat invariants of the 1-dimensional Schrodinger operator. As a result new
explicit formulas for the KdV hierarchy are obtained. Our method is based on an
asymptotic expansion of resolvent kernels of elliptic operators due to S.Agmon
and Y.Kannai.
Journal: None |
Submitter: Katrina Elfrieda Hibberd
Authors: Katrina Hibberd, Itzhak Roditi, Jon Links and Angela Foerster
Title: Bethe ansatz solution of the closed anisotropic supersymmetric U model
with quantum supersymmetry
Abstract: The nested algebraic Bethe ansatz is presented for the anisotropic
supersymmetric $U$ model maintaining quantum supersymmetry. The Bethe ansatz
equations of the model are obtained on a one-dimensional closed lattice and an
expression for the energy is given.
Journal: None |
Submitter: Alexander Turbiner
Authors: Alexander Turbiner, Pavel Winternitz
Title: Solutions of Non-linear Differential and Difference Equations with
Superposition Formulas
Abstract: Matrix Riccati equations and other nonlinear ordinary differential equations
with superposition formulas are, in the case of constant coefficients, shown to
have the same exact solutions as their group theoretical discretizations.
Explicit solutions of certain classes of scalar and matrix Riccati equations
are presented as an illustration of the general results.
Journal: None |
Submitter: Takayuki Tsuchida
Authors: T. Tsuchida, M. Wadati
Title: New integrable systems of derivative nonlinear Schr\"{o}dinger equations
with multiple components
Abstract: The Lax pair for a derivative nonlinear Schr\"{o}dinger equation proposed by
Chen-Lee-Liu is generalized into matrix form. This gives new types of
integrable coupled derivative nonlinear Schr\"{o}dinger equations. By virtue of
a gauge transformation, a new multi-component extension of a derivative
nonlinear Schr\"{o}dinger equation proposed by Kaup-Newell is also obtained.
Journal: Phys. Lett. A 257 (1999) 53-64 |
Submitter: Luis Martinez Alonso
Authors: Boris Konopelchenko and Luis Martinez Alonso
Title: The KP Hierarchy in Miwa Coordinates
Abstract: A systematic reformulation of the KP hierarchy by using continuous Miwa
variables is presented. Basic quantities and relations are defined and
determinantal expressions for Fay's identities are obtained. It is shown that
in terms of these variables the KP hierarchy gives rise to a Darboux system
describing an infinite-dimensional conjugate net.
Journal: None |
Submitter: R. Radhakrishnan
Authors: R. Radhakrishnan, A. Kundu and M. Lakshmanan
Title: Coupled nonlinear Schrodinger equations with cubic-quintic nonlinearity:
Integrability and soliton interaction in non-Kerr media
Abstract: We propose an integrable system of coupled nonlinear Schrodinger equations
with cubic-quintic terms describing the effects of quintic nonlinearity on the
ultra-short optical soliton pulse propagation in non-Kerr media. Lax pair,
conserved quantities and exact soliton solutions for the proposed integrable
model are given. Explicit form of two-solitons are used to study soliton
interaction showing many intriguing features including inelastic (shape
changing) scattering. Another novel system of coupled equations with
fifth-degree nonlinearity is derived, which represents vector generalization of
the known chiral-soliton bearing system.
Journal: None |
Submitter: Hubert Saleur
Authors: H. Saleur
Title: The continuum limit of sl(N/K) integrable super spin chains
Abstract: I discuss in this paper the continuum limit of integrable spin chains based
on the superalgebras sl(N/K). The general conclusion is that, with the full
``supersymmetry'', none of these models is relativistic. When the supersymmetry
is broken by the generator of the sub u(1), Gross Neveu models of various types
are obtained. For instance, in the case of sl(N/K) with a typical fermionic
representation on every site, the continuum limit is the GN model with N colors
and K flavors. In the case of sl(N/1) and atypical representations of spin j, a
close cousin of the GN model with N colors, j flavors and flavor anisotropy is
obtained. The Dynkin parameter associated with the fermionic root, while
providing solutions to the Yang Baxter equation with a continuous parameter,
thus does not give rise to any new physics in the field theory limit.
These features are generalized to the case where an impurity is embedded in
the system.
Journal: Nucl.Phys. B578 (2000) 552-576 |
Submitter: Ruslan Sharipov
Authors: R. F. Bikbaev, R. A. Sharipov
Title: Magnetization waves in Landau-Lifshitz Model
Abstract: The solutions of the Landau-Lifshitz equation with finite-gap behavior at
infinity are considered. By means of the inverse scattering method the
large-time asymptotics is obtained.
Journal: Phys. Lett. 134A (1988), no. 2, 105-107. |
Submitter: Takeo Kojima
Authors: H. Furutsu and T. Kojima (Nihon Univ.)
Title: $U_q(\hat{sl}_n)$-analog of the XXZ chain with a boundary
Abstract: We study $U_q(\hat{sl}_n)$ analog of the XXZ spin chain with a boundary
magnetic field h. We construct explicit bosonic formulas of the vacuum vector
and the dual vacuum vector with a boundary magnetic field. We derive integral
formulas of the correlation functions.
Journal: J.Math.Phys. 41 (2000) 4413-4436 |
Submitter: Nalini Joshi
Authors: Nalini Joshi, Johannes A. Petersen, and Luke M. Schubert
Title: Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili
equations
Abstract: We study characteristic Cauchy problems for the Korteweg-deVries (KdV)
equation $u_t=uu_x+u_{xxx}$, and the Kadomtsev-Petviashvili (KP) equation
$u_{yy}=\bigl(u_{xxx}+uu_x+u_t\bigr)_x$ with holomorphic initial data
possessing nonnegative Taylor coefficients around the origin. For the KdV
equation with initial value $u(0,x)=u_0(x)$, we show that there is no solution
holomorphic in any neighbourhood of $(t,x)=(0,0)$ in ${\mathbb C}^2$ unless
$u_0(x)=a_0+a_1x$. This also furnishes a nonexistence result for a class of
$y$-independent solutions of the KP equation. We extend this to $y$-dependent
cases by considering initial values given at $y=0$, $u(t,x,0)=u_0(x,t)$,
$u_y(t,x,0)=u_1(x,t)$, where the Taylor coefficients of $u_0$ and $u_1$ around
$t=0$, $x=0$ are assumed nonnegative. We prove that there is no holomorphic
solution around the origin in ${\mathbb C}^3$ unless $u_0$ and $u_1$ are
polynomials of degree 2 or lower.
Journal: None |
Submitter: Askold Perelomov
Authors: L. Gavrilov (U. Paul Sabatier, Toulouse), A. Perelomov (MPIM Bonn)
Title: On the explicit solutions of the elliptic Calogero system
Abstract: Let $q_1,q_2,...,q_N$ be the coordinates of $N$ particles on the circle,
interacting with the integrable potential $\sum_{j<k}^N\wp(q_j-q_k)$, where
$\wp$ is the Weierstrass elliptic function. We show that every symmetric
elliptic function in $q_1,q_2,...,q_N$ is a meromorphic function in time. We
give explicit formulae for these functions in terms of genus $N-1$ theta
functions.
Journal: None |
Submitter: Doc. Dr. Ayse Humeyra Bilge
Authors: Ayse Humeyra Bilge
Title: A System with a Recursion Operator but One Higher Local Symmetry of the
Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$
Abstract: We construct a recursion operator for the system $(u_t,v_t)=(u_4+v^2,1/5
v_4)$, for which only one local symmetry is known and we show that the action
of the recursion operator on $(u_t,v_t)$ is a local function.
Journal: Lie Groups and Their Applications, Vol.1, No 2, pp.132-139, (1994) |
Submitter: H. J. S. Dorren
Authors: H.J.S. Dorren and J.J.B. van den Heuvel
Title: On pulse broadening for optical solitons
Abstract: Pulse broadening for optical solitons due to birefringence is investigated.
We present an analytical solution which describes the propagation of solitons
in birefringent optical fibers. The special solutions consist of a combination
of purely solitonic terms propagating along the principal birefringence axes
and soliton-soliton interaction terms. The solitonic part of the solutions
indicates that the decay of initially localized pulses could be due to
different propagation velocities along the birefringence axes. We show that the
disintegration of solitonic pulses in birefringent optical fibers can be caused
by two effects. The first effect is similar as in linear birefringence and is
related to the unequal propagation velocities of the modes along the
birefringence axes. The second effect is related to the nonlinear
soliton-soliton interaction between the modes, which makes the solitonic
pulse-shape blurred.
Journal: None |
Submitter: David H. Sattinger
Authors: Richard Beals, D.H. Sattinger, and J. Szmigielski
Title: Multipeakons and the Classical Moment Problem
Abstract: Classical results of Stieltjes are used to obtain explicit formulas for the
peakon-antipeakon solutions of the Camassa-Holm equation. The closed form
solution is expressed in terms of the orthogonal polynomials of the related
classical moment problem. It is shown that collisions occur only in
peakon-antipeakon pairs, and the details of the collisions are analyzed using
results {}from the moment problem. A sharp result on the steepening of the
slope at the time of collision is given. Asymptotic formulas are given, and the
scattering shifts are calculated explicitly
Journal: None |
Submitter: David H. Sattinger
Authors: Yi Li and D.H. Sattinger
Title: Soliton Collisions in the Ion Acoustic Plasma Equations
Abstract: Numerical experiments involving the interaction of two solitary waves of the
ion acoustic plasma equations are described. An exact 2-soliton solution of the
relevant KdV equation was fitted to the initial data, and good agreement was
maintained throughout the entire interaction. The data demonstrates that the
soliton interactions are virtually elastic
Journal: Journal of Mathematical Fluid Mechanics, volume 1, (1999), pp.
117-130 |
Submitter: Antonio Lima Santos
Authors: A. Lima-Santos
Title: Reflection K-Matrices for 19-Vertex Models
Abstract: We derive and classify all regular solutions of the boundary Yang-Baxter
equation for 19-vertex models known as Zamolodchikov-Fateev or $A_{1}^{(1)}$
model, Izergin-Korepin or $A_{2}^{(2)}$ model, sl(2|1) model and osp(2|1)
model. We find that there is a general solution for $A_{1}^{(1)}$ and sl(2|1)
models. In both models it is a complete K-matrix with three free parameters.
For the $A_{2}^{(2)}$ and osp(2|1) models we find three general solutions,
being two complete reflection K-matrices solutions and one incomplete
reflection K-matrix solution with some null entries. In both models these
solutions have two free parameters. Integrable spin-1 Hamiltonians with general
boundary interactions are also presented. Several reduced solutions from these
general solutions are presented in the appendices.
Journal: Nucl. Phys. B 558 [PM] 637-667 |
Submitter: John Harnad
Authors: J. Harnad (C.R.M., U. de Montreal and Concordia U.)
Title: On the bilinear equations for Fredholm determinants appearing in random
matrices
Abstract: It is shown how the bilinear differential equations satisfied by Fredholm
determinants of integral operators appearing as spectral distribution functions
for random matrices may be deduced from the associated systems of nonautonomous
Hamiltonian equations satisfied by auxiliary canonical phase space variables
introduced by Tracy and Widom. The essential step is to recast the latter as
isomonodromic deformation equations for families of rational covariant
derivative operators on the Riemann sphere and interpret the Fredholm
determinants as isomonodromic $\tau$-functions.
Journal: J. Nonlinear Math. Phys., volume 9, no. 4 (2002) 530-550 |
Submitter: Runliang Lin
Authors: Yunbo Zeng (1), Wen-Xiu Ma (2) ((1)Tsinghua University, Beijing,
China, (2) City University of Hong Kong, China)
Title: Families of quai-bi-Hamiltonian systems and separability
Abstract: It is shown how to construct an infinite number of families of
quasi-bi-Hamiltonian (QBH) systems by means of the constrained flows of soliton
equations. Three explicit QBH structures are presented for the first three
families of the constrained flows. The Nijenhuis coordinates defined by the
Nijenhuis tensor for the corresponding families of QBH systems are proved to be
exactly the same as the separated variables introduced by means of the Lax
matrices for the constrained flows.
Journal: None |
Submitter: F. Nijhoff
Authors: F.W. Nijhoff, N. Joshi, A. Hone
Title: On the discrete and continuous Miura Chain associated with the Sixth
Painlev\'e Equation
Abstract: A Miura chain is a (closed) sequence of differential (or difference)
equations that are related by Miura or B\"acklund transformations. We describe
such a chain for the sixth Painlev\'e equation (\pvi), containing, apart from
\pvi itself, a Schwarzian version as well as a second-order second-degree
ordinary differential equation (ODE). As a byproduct we derive an
auto-B\"acklund transformation, relating two copies of \pvi with different
parameters. We also establish the analogous ordinary difference equations in
the discrete counterpart of the chain. Such difference equations govern
iterations of solutions of \pvi under B\"acklund transformations. Both discrete
and continuous equations constitute a larger system which include partial
difference equations, differential-difference equations and partial
differential equations, all associated with the lattice Korteweg-de Vries
equation subject to similarity constraints.
Journal: None |
Submitter: Sergei M. Sergeev
Authors: Sergei M. Sergeev
Title: On exact solution of a classical 3D integrable model
Abstract: We investigate some classical evolution model in the discrete 2+1 space-time.
A map, giving an one-step time evolution, may be derived as the compatibility
condition for some systems of linear equations for a set of auxiliary linear
variables. Dynamical variables for the evolution model are the coefficients of
these systems of linear equations. Determinant of any system of linear
equations is a polynomial of two numerical quasimomenta of the auxiliary linear
variables. For one, this determinant is the generating functions of all
integrals of motion for the evolution, and on the other hand it defines a high
genus algebraic curve. The dependence of the dynamical variables on the
space-time point (exact solution) may be expressed in terms of theta functions
on the jacobian of this curve. This is the main result of our paper.
Journal: J. Nonlinear Math. Phys. 7 (2000), no. 1, 57-72 |
Submitter: James D. E. Grant
Authors: James D.E. Grant
Title: Paraconformal Structures and Integrable Systems
Abstract: We consider some natural connections which arise between right-flat (p, q)
paraconformal structures and integrable systems. We find that such systems may
be formulated in Lax form, with a "Lax p-tuple" of linear differential
operators, depending a spectral parameter which lives in (q-1)-dimensional
complex projective space. Generally, the differential operators contain partial
derivatives with respect to the spectral parameter.
Journal: None |
Submitter: G. Tondo
Authors: G. Falqui, F. Magri, G. Tondo
Title: Reduction of bihamiltonian systems and separation of variables: an
example from the Boussinesq hierarchy
Abstract: We discuss the Boussinesq system with $t_5$ stationary, within a general
framework for the analysis of stationary flows of n-Gel'fand-Dickey
hierarchies. We show how a careful use of its bihamiltonian structure can be
used to provide a set of separation coordinates for the corresponding
Hamilton--Jacobi equations.
Journal: None |
Submitter: Yuri B. Suris
Authors: Yuri B. Suris
Title: r-matrices for relativistic deformations of integrable systems
Abstract: We include the relativistic lattice KP hierarchy, introduced by Gibbons and
Kupershmidt, into the $r$-matrix framework. An $r$-matrix account of the
nonrelativistic lattice KP hierarchy is also provided for the reader's
convenience. All relativistic constructions are regular one-parameter
perturbations of the nonrelativistic ones. We derive in a simple way the linear
Hamiltonian structure of the relativistic lattice KP, and find for the first
time its quadratic Hamiltonian structure. Amasingly, the latter turns out to
coincide with its nonrelativistic counterpart (a phenomenon, known previously
only for the simplest case of the relativistic Toda lattice).
Journal: J. Nonlinear Math. Phys. 6 (1999), no. 4, 411-447 |
Submitter: None
Authors: Shigeki Matsutani
Title: p-adic Difference-Difference Lotka-Volterra Equation and Ultra-Discrete
Limit
Abstract: In this article, we have studied the difference-difference Lotka-Volterra
equations in p-adic number space and its p-adic valuation version. We pointed
out that the structure of the space given by taking the ultra-discrete limit is
the same as that of the $p$-adic valuation space.
Journal: None |
Submitter: S. Yu. Sakovich
Authors: S. Yu. Sakovich
Title: Integrability of the higher-order nonlinear Schroedinger equation
revisited
Abstract: Only the known integrable cases of the Kodama-Hasegawa higher-order nonlinear
Schroedinger equation pass the Painleve test. Recent results of Ghosh and Nandy
add no new integrable cases of this equation.
Journal: None |
Submitter: Nadja Kutz
Authors: Tim Hoffmann, Johannes Kellendonk, Nadja Kutz and Nicolai Reshetikhin
Title: Factorization dynamics and Coxeter-Toda lattices
Abstract: It is shown that the factorization relation on simple Lie groups with
standard Poisson Lie structure restricted to Coxeter symplectic leaves gives an
integrable dynamical system. This system can be regarded as a discretization of
the Toda flow. In case of $SL_n$ the integrals of the factorization dynamics
are integrals of the relativistic Toda system. A substantial part of the paper
is devoted to the study of symplectic leaves in simple complex Lie groups, its
Borel subgroups and their doubles.
Journal: Comm. Math. Phys. 212, Issue 2, 297-321 (2000) |
Submitter: Peter Forrester
Authors: M. Adler, P.J. Forrester, T. Nagao and P. van Moerbeke
Title: Classical skew orthogonal polynomials and random matrices
Abstract: Skew orthogonal polynomials arise in the calculation of the $n$-point
distribution function for the eigenvalues of ensembles of random matrices with
orthogonal or symplectic symmetry. In particular, the distribution functions
are completely determined by a certain sum involving the skew orthogonal
polynomials. In the cases that the eigenvalue probability density function
involves a classical weight function, explicit formulas for the skew orthogonal
polynomials are given in terms of related orthogonal polynomials, and the
structure is used to give a closed form expression for the sum. This theory
treates all classical cases on an equal footing, giving formulas applicable at
once to the Hermite, Laguerre and Jacobi cases.
Journal: None |
Submitter: Tim Hoffmann
Authors: Tim Hoffmann
Title: On the equivalence of the discrete nonlinear Schr\"odinger equation and
the discrete isotropic Heisenberg magnet
Abstract: The equivalence of the discrete isotropic Heisenberg magnet (IHM) model and
the discrete nonlinear Schr\"odinger equation (NLSE) given by Ablowitz and
Ladik is shown. This is used to derive the equivalence of their discretization
with the one by Izergin and Korepin. Moreover a doubly discrete IHM is
presented that is equivalent to Ablowitz' and Ladiks doubly discrete NLSE.
Journal: None |
Submitter: Sudipta Nandy
Authors: Sasanka Ghosh and Sudipta Nandy
Title: A New Class of Optical Solitons
Abstract: Existence of a new class of soliton solutions is shown for higher order
nonlinear Schrodinger equation, describing thrid order dispersion, Kerr effect
and stimulated Raman scattering. These new solutions have been obtaiened by
invoking a group of nonlinear transformations acting on localised stable
solutions. Stability of these solutions has been studied for different values
of the arbitrary coefficients, involved in the recursion relation and
consequently, different values of coefficient lead to different transmission
rates for almost same input power. Another series solution containing even
powers of localised stable solution is shown to exist for higher order
nonlinear Schrodinger equation.
Journal: None |
Submitter: Alexander Sorin
Authors: F. Delduc, L. Gallot and A. Sorin
Title: N=2 local and N=4 nonlocal reductions of supersymmetric KP hierarchy in
N=2 superspace
Abstract: A N=4 supersymmetric matrix KP hierarchy is proposed and a wide class of its
reductions which are characterized by a finite number of fields are described.
This class includes the one-dimensional reduction of the two-dimensional
N=(2|2) superconformal Toda lattice hierarchy possessing the N=4 supersymmetry
-- the N=4 Toda chain hierarchy -- which may be relevant in the construction of
supersymmetric matrix models. The Lax pair representations of the bosonic and
fermionic flows, corresponding local and nonlocal Hamiltonians, finite and
infinite discrete symmetries, the first two Hamiltonian structures and the
recursion operator connecting all evolution equations and the Hamiltonian
structures of the N=4 Toda chain hierarchy are constructed in explicit form.
Its secondary reduction to the N=2 supersymmetric alpha=-2 KdV hierarchy is
discussed.
Journal: Nucl.Phys. B558 (1999) 545-572 |
Submitter: Atalay Karasu
Authors: Atalay Karasu
Title: On A Recently Proposed Relation Between oHS and Ito Systems
Abstract: The bi-Hamiltonian structure of original Hirota-Satsuma system proposed by
Roy based on a modification of the bi-Hamiltonian structure of Ito system is
incorrect.
Journal: None |
Submitter: Marcio J. Martins
Authors: M.J. Martins and X.W. Guan
Title: Integrable supersymmetric correlated electron chain with open boundaries
Abstract: We construct an extended Hubbard model with open boundaries from a $R$-matrix
based on the $U_q[Osp(2|2)]$ superalgebra. We study the reflection equation and
find two classes of diagonal solutions. The corresponding one-dimensional open
Hamiltonians are diagonalized by means of the Bethe ansatz approach.
Journal: Nucl. Phys. B 562 (1999) 433-444 |
Submitter: Hendry Izaac Elim
Authors: Hendry I. Elim
Title: New Integrable Coupled Nonlinear Schrodinger Equations
Abstract: Two types of integrable coupled nonlinear Schrodinger (NLS) equations are
derived by using Zakharov-Shabat (ZS) dressing method.The Lax pairs for the
coupled NLS equations are also investigated using the ZS dressing method. These
give new types of the integrable coupled NLS equations with certain additional
terms. Then, the exact solutions of the new types are obtained. We find that
the solution of these new types do not always produce a soliton solution even
they are the kind of the integrable NLS equations.
Journal: None |
Submitter: Peter Forrester
Authors: P.J. Forrester and E.M. Rains
Title: Inter-relationships between orthogonal, unitary and symplectic matrix
ensembles
Abstract: We consider the following problem: When do alternate eigenvalues taken from a
matrix ensemble themselves form a matrix ensemble? More precisely, we classify
all weight functions for which alternate eigenvalues from the corresponding
orthogonal ensemble form a symplectic ensemble, and similarly classify those
weights for which alternate eigenvalues from a union of two orthogonal
ensembles forms a unitary ensemble. Also considered are the $k$-point
distributions for the decimated orthogonal ensembles.
Journal: None |
Submitter: Fritz Gesztesy
Authors: Fritz Gesztesy
Title: Integrable Systems in the Infinite Genus Limit
Abstract: We provide an elementary approach to integrable systems associated with
hyperelliptic curves of infinite genus. In particular, we explore the extent to
which the classical Burchnall-Chaundy theory generalizes in the infinite genus
limit, and systematically study the effect of Darboux transformations for the
KdV hierarchy on such infinite genus curves. Our approach applies to
complex-valued periodic solutions of the KdV hierarchy and naturally identifies
the Riemann surface familiar from standard Floquet theoretic considerations
with a limit of Burchnall-Chaundy curves.
Journal: None |
Submitter: Jon Links
Authors: Jon Links (U. of Queensland)
Title: A construction for R-matrices without difference property in the
spectral parameter
Abstract: A new construction is given for obtaining R-matrices which solve the
McGuire-Yang-Baxter equation in such a way that the spectral parameters do not
possess the difference property. A discussion of the derivation of the
supersymmetric U model is given in this context such that applied chemical
potential and magnetic field terms can be coupled arbitrarily. As a limiting
case the Bariev model is obtained.
Journal: Phys. Lett. A 265 (2000) 194-206 |
Submitter: Dmitry Pelinovsky
Authors: Dmitry E. Pelinovsky and Catherine Sulem
Title: Spectral decomposition for the Dirac system associated to the DSII
equation
Abstract: A new (scalar) spectral decomposition is found for the Dirac system in two
dimensions associated to the focusing Davey--Stewartson II (DSII) equation.
Discrete spectrum in the spectral problem corresponds to eigenvalues embedded
into a two-dimensional essential spectrum. We show that these embedded
eigenvalues are structurally unstable under small variations of the initial
data. This instability leads to the decay of localized initial data into
continuous wave packets prescribed by the nonlinear dynamics of the DSII
equation.
Journal: None |
Submitter: Adam Doliwa
Authors: Adam Doliwa and Paolo Maria Santini
Title: The symmetric, D-invariant and Egorov reductions of the quadrilateral
lattice
Abstract: We present a detailed study of the geometric and algebraic properties of the
multidimensional quadrilateral lattice (a lattice whose elementary
quadrilaterals are planar; the discrete analogue of a conjugate net) and of its
basic reductions. To make this study, we introduce the notions of forward and
backward data, which allow us to give a geometric meaning to the tau-function
of the lattice, defined as the potential connecting these data. Together with
the known circular lattice (a lattice whose elementary quadrilaterals can be
inscribed in circles; the discrete analogue of an orthogonal conjugate net) we
introduce and study two other basic reductions of the quadrilateral lattice:
the symmetric lattice, for which the forward and backward data coincide, and
the D-invariant lattice, characterized by the invariance of a certain natural
frame along the main diagonal. We finally discuss the Egorov lattice, which is,
at the same time, symmetric, circular and D-invariant. The integrability
properties of all these lattices are established using geometric, algebraic and
analytic means; in particular we present a D-bar formalism to construct large
classes of such lattices. We also discuss quadrilateral hyperplane lattices and
the interplay between quadrilateral point and hyperplane lattices in all the
above reductions.
Journal: None |
Submitter: Adam Doliwa
Authors: Adam Doliwa
Title: Lattice geometry of the Hirota equation
Abstract: Geometric interpretation of the Hirota equation is presented as equation
describing the Laplace sequence of two-dimensional quadrilateral lattices.
Different forms of the equation are given together with their geometric
interpretation: (i) the discrete coupled Volterra system for the coefficients
of the Laplace equation, (ii) the gauge invariant form of the Hirota equation
for projective invariants of the Laplace sequence, (iii) the discrete Toda
system for the rotation coefficients, (iv) the original form of the Hirota
equation for the tau-function of the quadrilateral lattice.
Journal: None |
Submitter: Adam Doliwa
Authors: Adam Doliwa and Paolo Maria Santini
Title: Integrable Discrete Geometry: the Quadrilateral Lattice, its
Transformations and Reductions
Abstract: We review recent results on Integrable Discrete Geometry. It turns out that
most of the known (continuous and/or discrete) integrable systems are
particular symmetries of the quadrilateral lattice, a multidimensional lattice
characterized by the planarity of its elementary quadrilaterals. Therefore the
linear property of planarity seems to be a basic geometric property underlying
integrability. We present the geometric meaning of its tau-function, as the
potential connecting its forward and backward data. We present the theory of
transformations of the quadrilateral lattice, which is based on the discrete
analogue of the theory of rectilinear congruences. In particular, we discuss
the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental
transformations and their interrelations. We also show how the sequence of
Laplace transformations of a quadrilateral surface is described by the discrete
Toda system. We finally show that these classical transformations are strictly
related to the basic operators associated with the quantum field theoretical
formulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review
the properties of quadrilateral hyperplane lattices, which play an interesting
role in the reduction theory, when the introduction of additional geometric
structures allows to establish a connection between point and hyperplane
lattices. We present and fully characterize some geometrically distinguished
reductions of the quadrilateral lattice, like the symmetric, circular and
Egorov lattices; we review also basic geometric results of the theory of
quadrilateral lattices in quadrics, and the corresponding analogue of the
Ribaucour reduction of the fundamental transformation.
Journal: None |
Submitter: F. Nijhoff
Authors: F.W. Nijhoff (University of Leeds)
Title: Discrete Dubrovin Equations and Separation of Variables for Discrete
Systems
Abstract: A universal system of difference equations associated with a hyperelliptic
curve is derived constituting the discrete analogue of the Dubrovin equations
arising in the theory of finite-gap integration. The parametrisation of the
solutions in terms of Abelian functions of Kleinian type (i.e. the higher-genus
analogues of the Weierstrass elliptic functions) is discussed as well as the
connections with the method of separation of variables.
Journal: None |
Submitter: Nobuhiko Shinzawa
Authors: Nobuhiko Shinzawa
Title: Symmetric Linear Backlund Transformation for Discrete BKP and DKP
equation
Abstract: Proper lattices for the discrete BKP and the discrete DKP equaitons are
determined. Linear B\"acklund transformation equations for the discrete BKP and
the DKP equations are constructed, which possesses the lattice symmetries and
generate auto-B\"acklund transformations
Journal: None |
Submitter: Fritz Gesztesy
Authors: F. Gesztesy, C. K. R. T. Jones, Y. Latushkin, and M. Stanislavova
Title: A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear
Schr\"odinger Equations
Abstract: A spectral mapping theorem is proved that resolves a key problem in applying
invariant manifold theorems to nonlinear Schr\" odinger type equations. The
theorem is applied to the operator that arises as the linearization of the
equation around a standing wave solution. We cast the problem in the context of
space-dependent nonlinearities that arise in optical waveguide problems. The
result is, however, more generally applicable including to equations in higher
dimensions and even systems. The consequence is that stable, unstable, and
center manifolds exist in the neighborhood of a (stable or unstable) standing
wave, such as a waveguide mode, under simple and commonly verifiable spectral
conditions.
Journal: None |
Submitter: Takayuki Tsuchida
Authors: Takayuki Tsuchida, Miki Wadati (University of Tokyo)
Title: Multi-Field Integrable Systems Related to WKI-Type Eigenvalue Problems
Abstract: Higher flows of the Heisenberg ferromagnet equation and the
Wadati-Konno-Ichikawa equation are generalized into multi-component systems on
the basis of the Lax formulation. It is shown that there is a correspondence
between the multi-component systems through a gauge transformation. An
integrable semi-discretization of the multi-component higher Heisenberg
ferromagnet system is obtained.
Journal: J. Phys. Soc. Jpn. 68 (1999) 2241-2245 |
Submitter: Liu Qing Ping
Authors: Q.P. Liu
Title: Miura Map between Lattice KP and its Modification is Canonical
Abstract: We consider the Miura map between the lattice KP hierarchy and the lattice
modified KP hierarchy and prove that the map is canonical not only between the
first Hamiltonian structures, but also between the second Hamiltonian
structures.
Journal: None |
Submitter: Goro Hatayama
Authors: Goro Hatayama, Atsuo Kuniba, and Taichiro Takagi
Title: Soliton Cellular Automata Associated With Crystal Bases
Abstract: We introduce a class of cellular automata associated with crystals of
irreducible finite dimensional representations of quantum affine algebras
U'_q(\hat{\geh}_n). They have solitons labeled by crystals of the smaller
algebra U'_q(\hat{\geh}_{n-1}). We prove stable propagation of one soliton for
\hat{\geh}_n = A^{(2)}_{2n-1}, A^{(2)}_{2n}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n
and D^{(2)}_{n+1}. For \gh_n = C^{(1)}_n, we also prove that the scattering
matrices of two solitons coincide with the combinatorial R matrices of
U'_q(C^{(1)}_{n-1})-crystals.
Journal: Nuclear Physics B577[PM](2000) 619-645 |
Submitter: Olaf Lechtenfeld
Authors: Olaf Lechtenfeld and Alexander Sorin
Title: Supersymmetric KP hierarchy in N=1 superspace and its N=2 reductions
Abstract: A wide class of N=2 reductions of the supersymmetric KP hierarchy in N=1
superspace is described. This class includes a new N=2 supersymmetric
generalization of the Toda chain hierarchy. The Lax pair representations of the
bosonic and fermionic flows, local and nonlocal Hamiltonians, finite and
infinite discrete symmetries, first two Hamiltonian structures and the
recursion operator of this hierarchy are constructed. Its secondary reduction
to new N=2 supersymmetric modified KdV hierarchy is discussed.
Journal: Nucl.Phys. B566 (2000) 489-510 |
Submitter: Ziad Maassarani
Authors: Z. Maassarani (University of Virginia)
Title: Non-additive fusion, Hubbard models and non-locality
Abstract: In the framework of quantum groups and additive R-matrices, the fusion
procedure allows to construct higher-dimensional solutions of the Yang-Baxter
equation. These solutions lead to integrable one-dimensional spin-chain
Hamiltonians. Here fusion is shown to generalize naturally to non-additive
R-matrices, which therefore do not have a quantum group symmetry. This method
is then applied to the generalized Hubbard models. Although the resulting
integrable models are not as simple as the starting ones, the general structure
is that of two spin-(s times s') sl(2) models coupled at the free-fermion
point. An important issue is the probable lack of regular points which give
local Hamiltonians. This problem is related to the existence of second order
zeroes in the unitarity equation, and arises for the XX models of higher spins,
the building blocks of the Hubbard models. A possible connection between some
Lax operators L and R-matrices is noted.
Journal: J. Phys. A 32 (1999) 8691-8703 |
Submitter: V. Kuznetsov
Authors: V.B.Kuznetsov, M.Salerno and E.K.Sklyanin
Title: Quantum Backlund transformation for the integrable DST model
Abstract: For the integrable case of the discrete self-trapping (DST) model we
construct a Backlund transformation. The dual Lax matrix and the corresponding
dual Backlund transformation are also found and studied. The quantum analog of
the Backlund transformation (Q-operator) is constructed as the trace of a
monodromy matrix with an infinite-dimensional auxiliary space. We present the
Q-operator as an explicit integral operator as well as describe its action on
the monomial basis. As a result we obtain a family of integral equations for
multivariable polynomial eigenfunctions of the quantum integrable DST model.
These eigenfunctions are special functions of the Heun class which is beyond
the hypergeometric class. The found integral equations are new and they shall
provide a basis for efficient analytical and numerical studies of such
complicated functions.
Journal: J.Phys.A33:171-189,2000 |
Submitter: Konstantin Selivanov
Authors: K.G. Selivanov (ITEP, Moscow)
Title: Classical Solutions Generating Tree Form-Factors in Yang-Mills,
Sin(h)-Gordon and Gravity
Abstract: Classical solutions generating tree form-factors are defined and constructed
in various models.
Journal: None |
Submitter: Alexander Mikhailov
Authors: A. V. Mikhailov, V. V. Sokolov
Title: Integrable ODEs on Associative Algebras
Abstract: In this paper we give definitions of basic concepts such as symmetries, first
integrals, Hamiltonian and recursion operators suitable for ordinary
differential equations on associative algebras, and in particular for matrix
differential equations. We choose existence of hierarchies of first integrals
and/or symmetries as a criterion for integrability and justify it by examples.
Using our componentless approach we have solved a number of classification
problems for integrable equations on free associative algebras. Also, in the
simplest case, we have listed all possible Hamiltonian operators of low order.
Journal: None |
Submitter: Wen-Xiu Ma
Authors: Wen-Xiu Ma and Ruguang Zhou
Title: A Coupled AKNS-Kaup-Newell Soliton Hierarchy
Abstract: A coupled AKNS-Kaup-Newell hierarchy of systems of soliton equations is
proposed in terms of hereditary symmetry operators resulted from Hamiltonian
pairs. Zero curvature representations and tri-Hamiltonian structures are
established for all coupled AKNS-Kaup-Newell systems in the hierarchy.
Therefore all systems have infinitely many commuting symmetries and
conservation laws. Two reductions of the systems lead to the AKNS hierarchy and
the Kaup-Newell hierarchy, and thus those two soliton hierarchies also possess
tri-Hamiltonian structures.
Journal: None |
Submitter: Takayuki Tsuchida
Authors: Takayuki Tsuchida, Miki Wadati (University of Tokyo)
Title: Complete integrability of derivative nonlinear Schr\"{o}dinger-type
equations
Abstract: We study matrix generalizations of derivative nonlinear Schr\"{o}dinger-type
equations, which were shown by Olver and Sokolov to possess a higher symmetry.
We prove that two of them are `C-integrable' and the rest of them are
`S-integrable' in Calogero's terminology.
Journal: Inverse Problems 15 (1999) 1363-1373 |
Submitter: Kenji Kajiwara
Authors: Kenji Kajiwara(Doshisha Univ.), Tetsu Masuda(Doshisha Univ.),
Masatoshi Noumi(Kobe Univ.), Yasuhiro Ohta(Hiroshima Univ.), Yasuhiko
Yamada(Kobe Univ.)
Title: Determinant Formulas for the Toda and Discrete Toda Equations
Abstract: Determinant formulas for the general solutions of the Toda and discrete Toda
equations are presented. Application to the $\tau$ functions for the Painlev\'e
equations is also discussed.
Journal: None |
Submitter: Ladislav Hlavaty
Authors: L. Hlavaty
Title: Towards the Lax formulation of SU(2) principal models with nonconstant
metric
Abstract: The equations that define the Lax pairs for generalized principal chiral
models can be solved for any constant nondegenerate bilinear form on SU(2).
Necessary conditions for the nonconstant metric on SU(2) that define the
integrable models are given.
Journal: None |
Submitter: Takeo Kojima
Authors: H. Furutsu, T. Kojima, and Y.-H. Quano
Title: Type II vertex operators for the $A_{n-1}^{(1)}$ face model
Abstract: Presented is a free boson representation of the type II vertex operators for
the $A_{n-1}^{(1)}$ face model. Using the bosonization, we derive some
properties of the type II vertex operators, such as commutation, inversion and
duality relations.
Journal: Int.J.Mod.Phys. A15 (2000) 1533-1556 |
Submitter: Wen-Xiu Ma
Authors: Wen-Xiu Ma and Si-Ming Zhu
Title: Non-symmetry constraints of the AKNS system yielding integrable
Hamiltonian systems
Abstract: This paper aims to show that there exist non-symmetry constraints which yield
integrable Hamiltonian systems through nonlinearization of spectral problems of
soliton systems, like symmetry constraints. Taking the AKNS spectral problem as
an illustrative example, a class of such non-symmetry constraints is introduced
for the AKNS system, along with two-dimensional integrable Hamiltonian systems
generated from the AKNS spectral problem.
Journal: None |
Submitter: Alexander I. Bobenko
Authors: Sergey I. Agafonov, Alexander I. Bobenko
Title: Discrete Z^a and Painleve equations
Abstract: A discrete analogue of the holomorphic map z^a is studied. It is given by a
Schramm's circle pattern with the combinatorics of the square grid. It is shown
that the corresponding immersed circle patterns lead to special separatrix
solutions of a discrete Painleve equation. Global properties of these
solutions, as well as of the discrete $z^a$ are established.
Journal: International Math. Research Notices 2000:4 165-193 |
Submitter: Metin Gurses
Authors: Metin Gurses (Bilkent University), Atalay Karasu (METU), and Vladimir
Sokolov (Landau Institue)
Title: On Construction of Recursion Operators From Lax Representation
Abstract: In this work we develop a general procedure for constructing the recursion
operators fro non-linear integrable equations admitting Lax representation.
Svereal new examples are given. In particular we find the recursion operators
for some KdV-type of integrable equations.
Journal: None |
Submitter: Andrey Tsyganov User
Authors: Andrey Tsiganov
Title: On integrable deformations of the spherical top
Abstract: The motion on the sphere $S^2$ with the potential $V= (x_1x_2x_3)^{-2/3}$ is
considered. The Lax representation and the linearisation procedure for this
two-dimensional integrable system are discussed.
Journal: J. Phys. A, Math. Gen. 32, No.47, 8355-8363, (1999) |
Submitter: Dmitry Demskoy
Authors: D.K.Demskoy, A.G.Meshkov
Title: New integrable string-like fields in 1+1 dimensions
Abstract: The symmetry classification method is applied to the string-like scalar
fields in two-dimensional space-time. When the configurational space is
three-dimensional and reducible we present the complete list of the systems
admiting higher polynomial symmetries of the 3rd, 4th and 5th-order.
Journal: None |
Submitter: Andrey V. Tsiganov
Authors: Andrey Tsiganov
Title: Canonical transformations of the extended phase space, Toda lattices and
Stackel family of integrable systems
Abstract: We consider compositions of the transformations of the time variable and
canonical transformations of the other coordinates, which map completely
integrable system into other completely integrable system. Change of the time
gives rise to transformations of the integrals of motion and the Lax pairs,
transformations of the corresponding spectral curves and R-matrices. As an
example, we consider canonical transformations of the extended phase space for
the Toda lattices and the Stackel systems.
Journal: J. Phys. A, Math. Gen. 33, No.22, 4169-4182, (2000) |
Submitter: Hans Jacobus Wospakrik
Authors: Hans J. Wospakrik and Freddy P. Zen
Title: CPT Symmetries and the Backlund Transformations
Abstract: We show that the auto-Backlund transformations of the sine-Gordon,
Korteweg-deVries, nonlinear Schrodinger, and Ernst equations are related to
their respective CPT symmetries. This is shown by applying the CPT symmetries
of these equations to the Riccati equations of the corresponding
pseudopotential functions where the fields are allowed to transform into new
solutions while the pseudopotential functions and the Backlund parameter are
held fixed.
Journal: None |
Submitter: Andrey K. Svinin
Authors: A.K. Svinin
Title: Lie point symmetries of integrable evolution equations and invariant
solutions
Abstract: An integrable hierarchies connected with linear stationary Schr\"odinger
equation with energy dependent potentials (in general case) are considered.
Galilei-like and scaling invariance transformations are constructed. A symmetry
method is applied to construct invariant solutions.
Journal: None |
Submitter: Yuri B. Suris
Authors: Yuri B. Suris (TU Berlin)
Title: The motion of a rigid body in a quadratic potential: an integrable
discretization
Abstract: The motion of a rigid body in a quadratic potential is an important example
of an integrable Hamiltonian system on a dual to a semidirect product Lie
algebra so(n) x Symm(n). We give a Lagrangian derivation of the corresponding
equations of motion, and introduce a discrete time analog of this system. The
construction is based on the discrete time Lagrangian mechanics on Lie groups,
accompanied with the discrete time Lagrangian reduction. The resulting
multi-valued map (correspondence) on the dual to so(n) x Symm(n) is Poisson
with respect to the Lie-Poisson bracket, and is also completely integrable. We
find a Lax representation based on matrix factorisations, in the spirit of
Veselov-Moser.
Journal: Intern. Math. Research Notices, 2000, No 12, p.643-663. |
Submitter: Pierre van Moerbeke
Authors: M. Adler, T. Shiota and P. van Moerbeke
Title: Pfaff tau-functions
Abstract: Consider the evolution $$ \frac{\pl m_\iy}{\pl t_n}=\Lb^n m_\iy, \frac{\pl
m_\iy}{\pl s_n}=-m_\iy(\Lb^\top)^n, $$ on bi- or semi-infinite matrices
$m_\iy=m_\iy(t,s)$, with skew-symmetric initial data $m_{\iy}(0,0)$. Then,
$m_\iy(t,-t)$ is skew-symmetric, and so the determinants of the successive
"upper-left corners" vanish or are squares of Pfaffians. In this paper, we
investigate the rich nature of these Pfaffians, as functions of t. This problem
is motivated by questions concerning the spectrum of symmetric and symplectic
random matrix ensembles.
Journal: None |
Submitter: David Fairlie
Authors: D.B. Fairlie and A.N. Leznov
Title: The Complex Bateman Equation
Abstract: The general solution to the Complex Bateman equation is constructed. It is
given in implicit form in terms of a functional relationship for the unknown
function. The known solution of the usual Bateman equation is recovered as a
special case.
Journal: None |
Submitter: David Fairlie
Authors: D.B. Fairlie and A.N. Leznov
Title: The General Solution of the Complex Monge-Amp\`ere Equation in two
dimensional space
Abstract: The general solution to the Complex Monge-Amp\`ere equation in a two
dimensional space is constructed.
Journal: None |
Submitter: David Fairlie
Authors: D.B. Fairlie and A.N. Leznov
Title: The Complex Bateman Equation in a space of arbitrary dimension
Abstract: A general solution to the Complex Bateman equation in a space of arbitrary
dimensions is constructed.
Journal: None |
Submitter: David Fairlie
Authors: D.B. Fairlie and A.N. Leznov
Title: The General Solution of the Complex Monge-Amp\`ere Equation in a space
of arbitrary dimension
Abstract: A general solution to the Complex Monge-Amp\`ere equation in a space of
arbitrary dimensions is constructed.
Journal: None |
Submitter: Adam Doliwa
Authors: Adam Doliwa (Warsaw University)
Title: Discrete asymptotic nets and W-congruences in Plucker line geometry
Abstract: The asymptotic lattices and their transformations are studied within the line
geometry approach. It is shown that the discrete asymptotic nets are
represented by isotropic congruences in the Plucker quadric. On the basis of
the Lelieuvre-type representation of asymptotic lattices and of the discrete
analog of the Moutard transformation, it is constructed the discrete analog of
the W-congruences, which provide the Darboux-Backlund type transformation of
asymptotic lattices.The permutability theorems for the discrete Moutard
transformation and for the corresponding transformation of asymptotic lattices
are established as well. Moreover, it is proven that the discrete W-congruences
are represented by quadrilateral lattices in the quadric of Plucker. These
results generalize to a discrete level the classical line-geometric approach to
asymptotic nets and W-congruences, and incorporate the theory of asymptotic
lattices into more general theory of quadrilateral lattices and their
reductions.
Journal: J. Geom. Phys. 39 (2001) 9-29 |
Submitter: Liu Qing Ping
Authors: Q. P. Liu, Manuel Manas
Title: Darboux Transformation for Supersymmetric KP Hierarchies
Abstract: We construct Darboux transformations for the super-symmetric KP hierarchies
of Manin--Radul and Jacobian types. We also consider the binary Darboux
transformation for the hierarchies. The iterations of both type of Darboux
transformations are briefly discussed.
Journal: Phys.Lett.B485:293-300,2000 |
Submitter: Pilar G. Estevez
Authors: P. G. Estevez and G. A. Hernaez (Universidad de Salamanca, Spain)
Title: Darboux transformations for a Bogoyavlenskii equation in 2+1 dimensions
Abstract: We use the singular manifold method to obtain the Lax pair, Darboux
transformations and soliton solutions for a (2+1) dimensional integrable
equation.
Journal: None |
Submitter: Roman Paunov
Authors: H. Belich, G. Cuba and R. Paunov
Title: Surfaces of Constant negative Scalar Curvature and the Correpondence
between the Liouvulle and the sine-Gordon Equations
Abstract: By studying the {\it internal} Riemannian geometry of the surfaces of
constant negative scalar curvature, we obtain a natural map between the
Liouville, and the sine-Gordon equations. First, considering isometric
immersions into the Lobachevskian plane, we obtain an uniform expression for
the general (locally defined) solution of both the equations. Second, we prove
that there is a Lie-B\"acklund transformation interpolating between Liouville
and sine-Gordon. Third, we use isometric immersions into the Lobachevskian
plane to describe sine-Gordon N-solitons explicitly.
Journal: None |
Submitter: Kazuyasu Shigemoto
Authors: M.Horibe and K.Shigemoto
Title: The Structure of the Bazhanov-Baxter Model and a New Solution of the
Tetrahedron Equation
Abstract: We clarify the structure of the Bazhanov-Baxter model of the 3-dim N-state
integrable model. There are two essential points, i) the cubic symmetries, and
ii) the spherical trigonometry parametrization, to understand the structure of
this model. We propose two approaches to find a candidate as a solution of the
tetrahedron equation, and we find a new solution.
Journal: Progr. Theor. Phys. 102 (1999), 221-236 |
Submitter: Vladimir Gerdjikov
Authors: V. S. Gerdjikov (Institute for Nuclear Research and Nuclear Energy,
Bulg. Acad. of Sci., Sofia,Bulgaria), E. G. Evstatiev (Department of Physics,
University of Texas at Austin, Austin, Texas, USA), R. I. Ivanov (Department
of Mathematical Physics National University of Ireland - Galway, Galway,
Ireland)
Title: The complex Toda chains and the simple Lie algebras - solutions and
large time asymptotics -- II
Abstract: We propose a compact and explicit expression for the solutions of the complex
Toda chains related to the classical series of simple Lie algebras g. The
solutions are parametrized by a minimal set of scattering data for the
corresponding Lax matrix. They are expressed as sums over the weight systems of
the fundamental representations of g and are explicitly covariant under the
corresponding Weyl group action. In deriving these results we start from the
Moser formula for the A_r series and obtain the results for the other classical
series of Lie algebras by imposing appropriate involutions on the scattering
data. Thus we also show how Moser's solution goes into the one of Olshanetsky
and Perelomov. The results for the large-time asymptotics of the A_r -CTC
solutions are extended to the other classical series B_r - D_r. We exhibit also
some `irregular' solutions for the D_{2n+1} algebras whose asymptotic regimes
at t ->\pm\infty are qualitatively different. Interesting examples of bounded
and periodic solutions are presented and the relations between the solutions
for the algebras D_4, B_3 and G_2 $ are analyzed.
Journal: None |
Submitter: Vladimir Marikhin
Authors: M. Boiti, V.G. Marikhin, F. Pempinelli, A.B. Shabat
Title: Self-similar solutions of NLS-type dynamical systems
Abstract: We study self-similar solutions of NLS-type dynamical systems. Lagrangian
approach is used to show that they can be reduced to three canonical forms,
which are related by Miura transformations. The fourth Painleve equation (PIV)
is central in our consideration - it connects Heisenberg model, Volterra model
and Toda model to each other. The connection between the rational solutions of
PIV and Coulomb gas in a parabolic potential is established. We discuss also
the possibility to obtain an exact solution for optical soliton i.e. of the NLS
equation with time-dependent dispersion.
Journal: None |
Submitter: Igor Loutsenko
Authors: I.Loutsenko, V.Spiridonov
Title: Self-similarity in Spectral Problems and q-special Functions
Abstract: Similarity symmetries of the factorization chains for one-dimensional
differential and finite-difference Schr\"odinger equations are discussed.
Properties of the potentials defined by self-similar reductions of these chains
are reviewed. In particular, their algebraic structure, relations to
$q$-special functions, infinite soliton systems, supersymmetry, coherent
states, orthogonal polynomials, one-dimensional Ising chains and random
matrices are outlined.
Journal: None |
Submitter: Saburo Kakei
Authors: Saburo Kakei
Title: Orthogonal and symplectic matrix integrals and coupled KP hierarchy
Abstract: Orthogonal and symplectic matrix integrals are investigated. It is shown that
the matrix integrals can be considered as a $\tau$-function of the coupled KP
hierarchy, whose solution can be expressed in terms of pfaffians.
Journal: J.Phys.Soc.Jap. 68 (1999) 2875-2877 |
Submitter: Saburo Kakei
Authors: Saburo Kakei
Title: Dressing method and the coupled KP hierarchy
Abstract: The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by
using the dressing method. It is shown that the coupled KP hierarchy can be
reformulated as a reduced case of the 2-component KP hierarchy.
Journal: None |
Submitter: Krzysztof Marciniak
Authors: Stefan Rauch-Wojciechowski, Krzysztof Marciniak, Hans Lundmark
Title: Quasi-Lagrangian Systems of Newton Equations
Abstract: Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$
with an integral of motion quadratic in velocities are studied. These equations
generalize the potential case (when A=I, the identity matrix) and they admit a
curious quasi-Lagrangian formulation which differs from the standard Lagrange
equations by the plus sign between terms. A theory of such quasi-Lagrangian
Newton (qLN) systems having two functionally independent integrals of motion is
developed with focus on two-dimensional systems. Such systems admit a
bi-Hamiltonian formulation and are proved to be completely integrable by
embedding into five-dimensional integrable systems. They are characterized by a
linear, second-order PDE which we call the fundamental equation. Fundamental
equations are classified through linear pencils of matrices associated with qLN
systems. The theory is illustrated by two classes of systems: separable
potential systems and driven systems. New separation variables for driven
systems are found. These variables are based on sets of non-confocal conics. An
effective criterion for existence of a qLN formulation of a given system is
formulated and applied to dynamical systems of the Henon-Heiles type.
Journal: None |
Submitter: Andrew Hone
Authors: Frank Nijhoff (University of Leeds), Andrew Hone and Nalini Joshi
(University of Adelaide)
Title: On a Schwarzian PDE associated with the KdV Hierarchy
Abstract: We present a novel integrable non-autonomous partial differential equation of
the Schwarzian type, i.e. invariant under M\"obius transformations, that is
related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered
as the generating equation for the entire hierarchy of Schwarzian KdV
equations. We present its Lax pair, establish its connection with the SKdV
hierarchy, its Miura relations to similar generating PDEs for the modified and
regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate
that its similarity reductions lead to the {\it full} Painlev\'e VI equation,
i.e. with four arbitary parameters.
Journal: None |
Submitter: Sergei Sakovich
Authors: Sergei Sakovich
Title: On two aspects of the Painleve analysis
Abstract: We use the Calogero equation to illustrate the following two aspects of the
Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the
Painleve test for integrability, the singular expansions of its solutions
around characteristic hypersurfaces can be neither single-valued functions of
independent variables nor single-valued functionals of data. Second, if the
truncation of singular expansions of solutions is consistent, the truncation
not necessarily leads to the simplest, or elementary, auto-Backlund
transformation related to the Lax pair.
Journal: Int. J. Analysis 2013 (2013) 172813 (5 pages) |