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solv-int/9806007 | Juhi-Lian Julian Ting | S. Lee, Julian J.-L. Ting and S. Kim | Phonon Scattering by Breathers in the Discrete Nonlinear Schroedinger
Equation | 13 pages 11 figures in EPS, RevTeX, Phys. Rev. E accepted | null | null | null | solv-int cond-mat.dis-nn nlin.SI quant-ph | null | Linear theory for phonon scattering by discrete breathers in the discrete
nonlinear Schroedinger equation using the transfer matrix approach is
presented. Transmission and reflection coefficients are obtained as a function
of the wave vector of the input phonon. The occurrence of a nonzero
transmission, which in fact becomes perfect for a symmetric breather, is shown
to be connected with localized eigenmodes thresholds. In the weak-coupling
limit, perfect reflection are shown to exist, which requires two scattering
channels. A necessary condition for a system to have a perfect reflection is
also considered in a general context.
| 2007-05-23 |
solv-int/9806008 | Leonid Bogdanov | L.V. Bogdanov (Landau ITP, Moscow) and B.G. Konopelchenko (Universita'
degli Studi di Lecce) | M\"obius invariant integrable lattice equations associated with KP and
2DTL hierarchies | 13 pages, LaTeX; talk at SIDE III conference, Sabaudia, Italy, May
1998 | null | 10.1016/S0375-9601(99)00199-1 | null | solv-int nlin.SI | null | Integrable lattice equations arising in the context of singular manifold
equations for scalar, multicomponent KP hierarchies and 2D Toda lattice
hierarchy are considered. These equation generate the corresponding continuous
hierarchy of singular manifold equations, its B\"acklund transformations and
different forms of superposition principles. They possess rather special form
of compatibility representation. The distinctive feature of these equations is
invariance under the action of M\"obius transformation. Geometric
interpretation of these discrete equations is given.
| 2009-10-31 |
solv-int/9806009 | null | G.F. Helminck, J.W. van de Leur | Geometric B\"acklund--Darboux transformations for the KP hierarchy | 44 pages Latex2e | null | null | null | solv-int hep-th math.QA nlin.SI | null | We shown that, if you have two planes in the Segal-Wilson Grassmannian that
have an intersection of finite codimension, then the corresponding solutions of
the KP hierarchy are linked by B\"acklund-Darboux transformations (BDT). The
pseudodifferential operator that performs this transformation is shown to be
built up in a geometric way from elementary BDT's and is given here in a closed
form. The geometric description of elementary DBT's requires that one has a
geometric interpretation of the dual wavefunctions involved. This is done here
with the help of a suitable algebraic characterization of the wavefunction. The
BDT's also induce transformations of the tau-function associated to a plane in
the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric
characterization of the BDT'ss that preserves these subsystems of the KP
hierarchy. This generalizes the classical Darboux-transformations. we also
determine an explicit expression for the squared eigenfunction potentials. Next
a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy.
It is shown that infinite flags in the Grassmannian yield solutions of the
latter hierarchy. these flags can be constructed by means of BDT's, starting
from some plane. Other applications of these BDT's are a geometric way to
characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy
and the construction of a vast collection of orthogonal polynomials, playing a
role in matrix models.
| 2007-05-23 |
solv-int/9807001 | Juhi-Lian Julian Ting | Julian Juhi-Lian Ting | DNA Transcription Mechanism with a Moving Enzyme | paper published long time ago. 11 pages RevTeX 4 EPS files | Int. J. Bifurcat. Chaos.7:5, 1125-1132 (1997) | null | null | solv-int nlin.SI q-bio | null | Previous numerical investigations of an one-dimensional DNA model with an
extended modified coupling constant by transcripting enzyme are integrated to
longer time and demonstrated explicitly the trapping of breathers by DNA chains
with realistic parameters obtained from experiments. Furthermore, collective
coordinate method is used to explain a previously observed numerical evidence
that breathers placed far from defects are difficult to trap, and the motional
effect of RNA-polymerase is investigated.
| 2021-01-21 |
solv-int/9807002 | Wen-Xiu Ma | Wen-Xiu Ma and Maxim Pavlov | Extending Hamiltonian Operators to Get Bi-Hamiltonian Coupled KdV
Systems | 13 pages, latex | null | 10.1016/S0375-9601(98)00555-6 | null | solv-int nlin.SI | null | An analysis of extension of Hamiltonian operators from lower order to higher
order of matrix paves a way for constructing Hamiltonian pairs which may result
in hereditary operators. Based on a specific choice of Hamiltonian operators of
lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a
consequence of bi-Hamiltonian structure, they all possess infinitely many
symmetries and infinitely many conserved densities.
| 2016-09-08 |
solv-int/9807003 | Lafortune | A. Ramani, B. Grammaticos and S. Lafortune | Schlesinger Transformations for Linearisable Equations | 14 pages, no figures, Tex file | Lett.Math.Phys. 46, 131-145 (1998). | null | Preprint 98 | solv-int nlin.SI | null | We introduce the Schlesinger transformations of the Gambier equation. The
latter can be written, in both the continuous and discrete cases, as a system
of two coupled Riccati equations in cascade involving an integer parameter n.
In the continuous case the parameter appears explicitly in the equation while
in the discrete case it corresponds to the number of steps for singularity
confinement. Two Schlesinger transformations are obtained relating the
solutions for some value $n$ to that corresponding to either n+1 or n+2.
| 2007-05-23 |
solv-int/9807004 | I. A. B. Strachan | I.A.B.Strachan | Degenerate Frobenius manifolds and the bi-Hamiltonian structure of
rational Lax equations | 28 pages, LaTeX | J. Math. Phys. 40, 5058 (1999); | 10.1063/1.533015 | null | solv-int nlin.SI | null | The bi-Hamiltonian structure of certain multi-component integrable systems,
generalizations of the dispersionless Toda hierarchy, is studies for systems
derived from a rational Lax function. One consequence of having a rational
rather than a polynomial Lax function is that the corresponding bi-Hamiltonian
structures are degenerate, i.e. the metric which defines the Hamiltonian
structure has vanishing determinant. Frobenius manifolds provide a natural
setting in which to study the bi-Hamiltonian structure of certain classes of
hydrodynamic systems. Some ideas on how this structure may be extanded to
include degenerate bi-Hamiltonian structures, such as those given in the first
part of the paper, are given.
| 2020-12-16 |
solv-int/9807005 | V. E. Vekslerchik | V.E. Vekslerchik | 'Universality' of the Ablowitz-Ladik hierarchy | 21 pages, LaTeX | null | null | IC/98/52 | solv-int nlin.SI | null | The aim of this paper is to summarize some recently obtained relations
between the Ablowitz-Ladik hierarchy (ALH) and other integrable equations. It
has been shown that solutions of finite subsystems of the ALH can be used to
derive a wide range of solutions for, e.g., the 2D Toda lattice, nonlinear
Schr\"odinger, Davey-Stewartson, Kadomtsev-Petviashvili (KP) and some other
equations. Similar approach has been used to construct new integrable models:
O(3,1) and multi-field sigma models. Such 'universality' of the ALH becomes
more transparent in the framework of the Hirota's bilinear method. The ALH,
which is usually considered as an infinite set of differential-difference
equations, has been presented as a finite system of functional-difference
equations, which can be viewed as a generalization of the famous bilinear
identities for the KP tau-functions.
| 2007-05-23 |
solv-int/9807006 | Helge Holden | F. Gesztesy, H. Holden | Dubrovin equations and integrable systems on hyperelliptic curves | LaTeX2e | null | null | null | solv-int nlin.SI | null | We introduce the most general version of Dubrovin-type equations for divisors
on a hyperelliptic curve of arbitrary genus, and provide a new argument for
linearizing the corresponding completely integrable flows. Detailed
applications to completely integrable systems, including the KdV, AKNS, Toda,
and the combined sine-Gordon and mKdV hierarchies, are made. These
investigations uncover a new principle for 1+1-dimensional integrable soliton
equations in the sense that the Dubrovin equations, combined with appropriate
trace formulas, encode all hierarchies of soliton equations associated with
hyperelliptic curves. In other words, completely integable hierarchies of
soliton equations determine Dubrovin equations and associated trace formulas
and, vice versa, Dubrovin-type equations combined with trace formulas permit
the construction of hierarchies of soliton equations.
| 2007-05-23 |
solv-int/9807007 | H. J. S. Dorren | H.J.S. Dorren | On the integrability of nonlinear partial differential equations | 12 pages Latex | null | 10.1063/1.532843 | null | solv-int nlin.SI | null | We investigate the integrability of Nonlinear Partial Differential Equations
(NPDEs). The concepts are developed by firstly discussing the integrability of
the KdV equation. We proceed by generalizing the ideas introduced for the KdV
equation to other NPDEs. The method is based upon a linearization principle
which can be applied on nonlinearities which have a polynomial form. We
illustrate the potential of the method by finding solutions of the (coupled)
nonlinear Schr\"{o}dinger equation and the Manakov equation which play an
important role in optical fiber communication. Finally, it is shown that the
method can also be generalized to higher-dimensions.
| 2015-06-26 |
solv-int/9807008 | Takashi Takebe | Evgueni K. Sklyanin, Takashi Takebe | Separation of Variables in the Elliptic Gaudin Model | 24 pages, Latex; minor corrections | Communications in Mathematical Physics 204:1 (1999) 17-38 | 10.1007/s002200050635 | UTMS 98-28, PDMI 15/98 | solv-int hep-th math.QA nlin.SI | null | For the elliptic Gaudin model (a degenerate case of XYZ integrable spin
chain) a separation of variables is constructed in the classical case. The
corresponding separated coordinates are obtained as the poles of a suitably
normalized Baker-Akhiezer function. The classical results are generalized to
the quantum case where the kernel of separating integral operator is
constructed. The simplest one-degree-of-freedom case is studied in detail.
| 2015-11-13 |
solv-int/9807009 | Pilar G. Estevez | Pilar G. Estevez (Universidad de Salamanca) and Pilar R. Gordoa
(Universidad de Salamanca) | Non-classical symmetries and the singular manifold method: A further two
examples | 9 pages (latex), to appear in Journal of Physics A | null | 10.1088/0305-4470/31/37/011 | AFTUS-98-15 | solv-int nlin.SI | null | This paper discusses two equations with the conditional Painleve property.
The usefulness of the singular manifold method as a tool for determining the
non-classical symmetries that reduce the equations to ordinary differential
equations with the Painleve property is confirmed once more
| 2009-10-31 |
solv-int/9807010 | Wen-Xiu Ma | Benno Fuchssteiner and Wen-Xiu Ma | An Approach to Master Symmetries of Lattice Equations | 14 pages, latex, to appear in Proceedings of SIDEII, UK | null | null | null | solv-int nlin.SI | null | An approach to master symmetries of lattice equations is proposed by the use
of discrete zero curvature equation. Its key is to generate non-isospectral
flows from the discrete spectral problem associated with a given lattice
equation. A Volterra-type lattice hierarchy and the Toda lattice hierarchy are
analyzed as two illustrative examples.
| 2007-05-23 |
solv-int/9808001 | Roman Paunov | H. Belich and R. Paunov | $A_n^{(1)}$ Toda Solitons: a Relation between Dressing transformations
and Vertex Operators | 17 pages, Latex, Talk given at the IV International Conference on Non
Associative Algebra and its Applications, University of Sao Paulo, July
19-24, 1988 | null | null | CBPF/NF/050/98 | solv-int hep-th nlin.SI | null | Affine Toda equations based on simple Lie algebras arise by imposing zero
curvature condition on a Lax connection which belongs to the corresponding loop
Lie algebra in the principal gradation. In the particular case of $A_n^{(1)}$
Toda models, we exploit the symmetry of the underlying linear problem to
calculate the dressing group element which generates arbitrary $N$-soliton
solution from the vacuum. Starting from this result we recover the vertex
operator representation of the soliton tau functions.
| 2016-09-08 |
solv-int/9808002 | Antonio Lima Santos | A. Lima-Santos | Polynomial rings of the chiral $SU(N)_{2}$ models | 10 pages, LaTex (ioplppt.sty) | J. Phys. A: Math. Gen 30 (1997) 8653-8660 | 10.1088/0305-4470/30/24/024 | null | solv-int nlin.SI | null | Via explicit diagonalization of the chiral $SU(N)_{2}$ fusion matrices, we
discuss the possibility of representing the fusion ring of the chiral SU(N)
models, at level K=2, by a polynomial ring in a single variable when $N$ is odd
and by a polynomial ring in two variables when $N$ is even.
| 2009-10-31 |
solv-int/9808003 | Henrik Aratyn | H. Aratyn, E. Nissimov and S. Pacheva | From One-Component KP Hierarchy to Two-Component KP Hierarchy and Back | LaTeX, 9 pgs., contribution to Festschrift for A.H. Zimerman | null | null | null | solv-int hep-th nlin.SI | null | We show that the system of the standard one-component KP hierarchy endowed
with a special infinite set of abelian additional symmetries, generated by
squared eigenfunction potentials, is equivalent to the two-component KP
hierarchy.
| 2007-05-23 |
solv-int/9808004 | Henrik Aratyn | H. Aratyn, E. Nissimov and S. Pacheva | Berezinian Construction of Super-Solitons in Supersymmetric Constrained
KP Hierarchies | LaTeX, 9 pgs., contribution to Festschrift for A.H. Zimerman | null | null | null | solv-int hep-th nlin.SI | null | We consider a broad class of consistently reduced Manin-Radul supersymmetric
KP hierarchies (MR-SKP) which are supersymmetric analogs of the ordinary
bosonic constrained KP models. Compatibility of these reductions with the MR
fermionic isospectral flows is achieved via appropriate modification of the
latter preserving their (anti-)commutation algebra. Unlike the general
unconstrained MR-SKP case, Darboux-Backlund transformations do preserve the
fermionic isospectral flows of the reduced MR-SKP hierarchies. This allows for
a systematic derivation of explicit Berezinian solutions for the
super-tau-functions (super-solitons) for these models.
| 2007-05-23 |
solv-int/9808005 | Pijush Kanti Ghosh | Pijush K. Ghosh and Avinash Khare | Relationship Between the Energy Eigenstates of Calogero-Sutherland
Models With Oscillator and Coulomb-like Potentials | 23 pages, RevTeX, no figure, some clarifications added, version to
appear in Journal of Physics A | Journal of Physics A : Math. & Gen. 32 (1999) 2129-2140 | 10.1088/0305-4470/32/11/008 | IMSC/98/07/47, IP/BBSR/98-25 | solv-int cond-mat hep-th nlin.SI | null | We establish a simple algebraic relationship between the energy eigenstates
of the rational Calogero-Sutherland model with harmonic oscillator and
Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in
both of these models which plays a crucial role in such an identification.
Further, we show that our analysis is in fact valid for any many-particle
system in arbitrary dimensions whose potential term (apart from the oscillator
or the Coulomb-like potential) is a homogeneous function of coordinates of
degree -2. The explicit coordinate transformation which maps the Coulomb-like
problem to the oscillator one has also been determined in some specific cases.
| 2009-10-31 |
solv-int/9808006 | Zixiang Zhou | Zixiang Zhou (Institute of Mathematics, Fudan University, Shanghai,
China) | Darboux transformations for twisted so(p,q) system and local isometric
immersion of space forms | LaTeX, 21 pages, 5 Postscript figures, to appear in Inverse Problems
(1998) | Inverse Problems 14 (1998) 1353-1370 | 10.1088/0266-5611/14/5/018 | null | solv-int nlin.SI | null | For the n-dimensional integrable system with a twisted so(p,q) reduction,
Darboux transformations given by Darboux matrices of degree 2 are constructed
explicitly. These Darboux transformations are applied to the local isometric
immersion of space forms with flat normal bundle and linearly independent
curvature normals to give the explicit expression of the position vector. Some
examples are given from the trivial solutions and standard imbedding T^n\to
R^{2n}.
| 2009-10-31 |
solv-int/9808007 | Vadim V. Varlamov | Vadim V. Varlamov | Equations of Geodesic Deviation and the Inverse Scattering Transform | 32 pages, LaTeX2e, to appear in "Relativity, Gravitation, Cosmology"
(Nova Science Publishers, New York) | null | null | null | solv-int gr-qc nlin.SI | null | Solutions of equations of geodesic deviation in three- and four- dimensional
spaces obtained by the inverse scattering transform are considered. It is shown
that in the case of three-dimensional space solutions of geodesic deviation
equations are reduced to solutions of the well-known Zakharov-Shabat problem.
In four- dimensional space system of geodesic deviation equations is associated
with $3\times 3$ matrix Schr\"{o}dinger equation, and dependence on parameters
defined by the nonlinear equations of three-wave interaction.
| 2007-05-23 |
solv-int/9808008 | J. vandeLeur | J.W. van de Leur and R.Martini | The construction of Frobenius manifolds from KP tau-functions | 29 pages, latex2e, no figures | null | 10.1007/s002200050691 | null | solv-int hep-th math.AG math.QA nlin.SI | null | Frobenius manifolds (solutions of WDVV equations) in canonical coordinates
are determined by the system of Darboux-Egoroff equations. This system of
partial differential equations appears as a specific subset of the
$n$-component KP hierarchy. KP representation theory and the related Sato
infinite Grassmannian are used to construct solutions of this Darboux-Egoroff
system and the related Frobenius manifolds. Finally we show that for these
solutions Dubrovin's isomonodromy tau-function can be expressed in the KP
tau-function.
| 2009-10-31 |
solv-int/9808009 | Boris Lorbeer | Boris Lorbeer | Finite gap integration of a discrete Euler top | 21 pages, 6 figures | null | null | null | solv-int nlin.SI | null | In [1] new discretizations of the Euler top have been found. They can be
discribed with a Lax pair with a spectral parameter on an elliptic curve. This
is used in this paper to perform a finite gap integration.
| 2007-05-23 |
solv-int/9808010 | Wen-Xiu Ma | Yishen Li and Wen-Xiu Ma | Binary Nonlinearization of AKNS Spectral Problem under Higher-Order
Symmetry Constraints | 16 pages, latex, to appear in Chaos, Solitons and Fractals | null | null | null | solv-int nlin.SI | null | Binary nonlinearization of AKNS spectral problem is extended to the cases of
higher-order symmetry constraints. The Hamiltonian structures, Lax
representations, $r$-matrices and integrals of motion in involution are
explicitly proposed for the resulting constrained systems in the cases of the
first four orders. The obtained integrals of motion are proved to be
functionally independent and thus the constrained systems are completely
integrable in the Liouville sense.
| 2007-05-23 |
solv-int/9808011 | Zixiang Zhou | Zixiang Zhou | Localized solitons of hyperbolic su(N) AKNS system | 15 pages, 5 figures, to appear in Inverse Problems | Inverse Problems 14 (1998) 1371-1383 | 10.1088/0266-5611/14/5/019 | null | solv-int nlin.SI | null | Using the nonlinear constraint and Darboux transformation methods, the
(m_1,...,m_N) localized solitons of the hyperbolic su(N) AKNS system are
constructed. Here "hyperbolic su(N)" means that the first part of the Lax pair
is F_y=JF_x+U(x,y,t)F where J is constant real diagonal and U^*=-U. When
different solitons move in different velocities, each component U_{ij} of the
solution U has at most m_i m_j peaks as t tends to infinity. This corresponds
to the (M,N) solitons for the DSI equation. When all the solitons move in the
same velocity, U_{ij} still has at most m_i m_j peaks if the phase differences
are large enough.
| 2009-10-31 |
solv-int/9808012 | Antonio Lima Santos | A. Lima-Santos and Wagner Utiel | On The KMS Condition for the critical Ising model | 8 pages, TcilaTex | Physics Letter A 226 (1997) 65-68 | 10.1016/S0375-9601(96)00919-X | null | solv-int hep-th nlin.SI | null | Using the KMS condition and exchange algebras we discuss the monodromy and
modular properties of two-point KMS states of the critical Ising model.
| 2009-10-31 |
solv-int/9808013 | Antonio Lima Santos | A. Lima-Santos | On fusion algebra of chiral $SU(N)_{k}$ models | 11 pages, ioplppt | J.Phys.A: Math. Gen. 30 (1997) 5123-5131 | 10.1088/0305-4470/30/14/021 | null | solv-int nlin.SI | null | We discuss some algebraic setting of chiral $SU(N)_{k}$ models in terms of
the statistical dimensions of their fields. In particular, the conformal
dimensions and the central charge of the chiral $SU(N)_{k}$ models are
calculated from their braid matrices. Futhermore, at level K=2, we present the
characteristic polynomials of their fusion matrices in a factored form.
| 2009-10-31 |
solv-int/9808014 | David H. Sattinger | M. Haragus-Courcelle and D.H. Sattinger | Inversion of the linearized Korteweg-deVries equation at the
multi-soliton solutions | 39 pages, 1 figure | Zeit fur Angew. Math. und Physik (ZAMP), vol 49, (1998), pp.
436-469 | 10.1007/s000000050101 | null | solv-int nlin.SI | null | Uniform estimates for the decay structure of the $n$-soliton solution of the
Korteweg-deVries equation are obtained. The KdV equation, linearized at the
$n$-soliton solution is investigated in a class $\WW$ consisting of sums of
travelling waves plus an exponentially decaying residual term. An analog of the
kernel of the time-independent equation is proposed, leading to solvability
conditions on the inhomogeneous term. Estimates on the inversion of the
linearized KdV equation at the $n$-soliton are obtained.
| 2018-08-29 |
solv-int/9808015 | Harold Widom | Harold Widom (University of California, Santa Cruz) | On the Solution of a Painlev\'e III Equation | LaTeX file. 9 pages | null | null | null | solv-int math.FA nlin.SI | null | In a 1977 paper of McCoy, Tracy and Wu there appeared for the first time the
solution of a Painlev\'e equation in terms of Fredholm determinants of integral
operators. This equation is $\psi''(t)+t^{-1}\psi'(t)=(1/2) \sinh 2\psi+2\alpha
t^{-1} \sinh\psi$, a special case of the Painlev\'e III equation. The proof in
the cited paper is complicated, and the purpose of this note is to give a more
straightforward one. First we give an equivalent formulation of the solution in
terms of the kernel ${e^{-t (x+x^{-1})/2}\over x+y}\Big|{x-1\over
x+1}\Big|^{2\alpha}$. There are already in the literature relatively simple
proofs of the fact that when $\alpha=0$ Fredholm determinants of this kernel
give solutions to the equation. We extend this result here to general $\alpha$.
| 2007-05-23 |
solv-int/9808016 | Vsevolod Adler | V.E. Adler (Ufa Institute of Mathematics, Ufa, Russia) | Legendre transformations on the triangular lattice | 13 pages, latex | Functional Analysis and Its Applications, 2000, Volume 34, Issue
1, pp 1-9 | 10.1007/BF02467062 | null | solv-int nlin.SI | null | The main purpose of the paper is to demonstrate that condition of invariance
with respect to the Legendre transformations allows effectively isolate the
class of integrable difference equations on the triangular lattice, which can
be considered as discrete analogues of relativistic Toda type lattices. Some of
obtained equations are new, up to the author knowledge. As an example, one of
them is studied in more details, in particular, its higher continuous
symmetries and zero curvature representation are found.
| 2014-08-27 |
solv-int/9808017 | Christiane Quesne | Avinash Khare, C. Quesne | On Some One-Parameter Families of Three-Body Problems in One Dimension:
Exchange Operator Formalism in Polar Coordinates and Scattering Properties | 10 pages, LaTeX, no figure | Phys. Lett. A 250 (1998) 33-38 | 10.1016/S0375-9601(98)00752-X | ULB/229/CQ/98/7, IOP-BBSR/98-26 | solv-int hep-th nlin.SI quant-ph | null | We apply the exchange operator formalism in polar coordinates to a
one-parameter family of three-body problems in one dimension and prove the
integrability of the model both with and without the oscillator potential. We
also present exact scattering solution of a new family of three-body problems
in one dimension.
| 2009-10-31 |
solv-int/9808018 | Kirill Vaninsky | K. L. Vaninsky | On Explicit Parametrisation of Spectral Curves for Moser-Calogero
Particles and its Applications | 20 pages, 6 pictures | Int.Math.Res.Not. 10 (1999) 509-529 | null | null | solv-int hep-ph hep-th nlin.SI | null | The system of $N$ classical particles on the line with the Weierstrass $\wp$
function as potential is known to be completely integrable. Recently D'Hoker
and Phong found a beautiful parameterization by the polynomial of degree $N$ of
the space of Riemann surfaces associated with this system. In the trigonometric
limit of the elliptic potential these Riemann surfaces degenerate into rational
curves. The D'Hoker-Phong polynomial in the limit describes the intersection
points of the rational curves. We found an explicit determinant representation
of the polynomial in the trigonometric case. We consider applications of this
result to the theory of Toeplitz determinants and to geometry of the spectral
curves. We also prove our earlier conjecture on the asymptotic behavior of the
ratio of two symplectic volumes when the number of particles tends to infinity.
| 2007-05-23 |
solv-int/9808019 | James D. E. Grant | J.D.E. Grant and I.A.B. Strachan | Hypercomplex Integrable Systems | Latex file, 19 pages | Nonlinearity 12 (1999) 1247 | 10.1088/0951-7715/12/5/302 | null | solv-int hep-th nlin.SI | null | In this paper we study hypercomplex manifolds in four dimensions. Rather than
using an approach based on differential forms, we develop a dual approach using
vector fields. The condition on these vector fields may then be interpreted as
Lax equations, exhibiting the integrability properties of such manifolds. A
number of different field equations for such hypercomplex manifolds are
derived, one of which is in Cauchy-Kovaleskaya form which enables a formal
general solution to be given. Various other properties of the field equations
and their solutions are studied, such as their symmetry properties and the
associated hierarchy of conservation laws.
| 2020-12-16 |
solv-int/9809001 | Angela Foerster | Jon Links, Angela Foerster and Michael Karowski | Bethe ansatz solution of a closed spin 1 XXZ Heisenberg chain with
quantum algebra symmetry | 13 pages, LaTeX, to appear in J. Math. Phys | null | 10.1063/1.532701 | null | solv-int nlin.SI | null | A quantum algebra invariant integrable closed spin 1 chain is introduced and
analysed in detail. The Bethe ansatz equations as well as the energy
eigenvalues of the model are obtained. The highest weight property of the Bethe
vectors with respect to U_q(sl(2)) is proved.
| 2015-06-26 |
solv-int/9809002 | Antonio Lima Santos | A. Lima-Santos and R.C.T. Ghiotto | A Bethe ansatz solution for the closed $U_{q}[sl(2)]$ Temperley-Lieb
quantum spin chains | 12 pages, tcilatex | J.Phys.A:Math.Gen.31 (1998)505-512 | 10.1088/0305-4470/31/2/011 | null | solv-int cond-mat.str-el hep-th nlin.SI | null | We solve the spectrum pf the closed Temperley-Lieb quantum spin chains using
the coordinate Bethe ansatz. These Hamiltonians are invariante under the
quantum group $U_{q}[sl(2)]$
| 2009-10-31 |
solv-int/9809003 | Antonio Lima Santos | A. Lima-Santos | Osp(1|2) Off-shell Bethe Ansatz Equations | 21 pages, LaTex, no significant changes | Nucl. Phys. B543 (1999)499-517 | 10.1016/S0550-3213(98)00861-X | null | solv-int hep-th nlin.SI | null | The semiclassical limit of the algebraic quantum inverse scattering method is
used to solve the theory of the Gaudin model. Via Off-shell Bethe ansatz
equations of an integrable representation of the graded osp(1|2) vertex model
we find the spectrum of N-1 independent Hamiltonians of Gaudin. Integral
representations of the N-point correlators are presented as solutions of the
Knizhnik-Zamolodchikov equation. These results are extended for highest
representations of the osp(1|2) Gaudin algebra.
| 2009-10-31 |
solv-int/9809004 | Fritz Gesztesy | Ronnie Dickson, Fritz Gesztesy, Karl Unterkofler | Algebro-Geometric Solutions of the Boussinesq Hierarchy | LaTeX, 48 pages | null | 10.1142/S0129055X9900026X | null | solv-int nlin.SI | null | We continue a recently developed systematic approach to the Bousinesq (Bsq)
hierarchy and its algebro-geometric solutions. Our formalism includes a
recursive construction of Lax pairs and establishes associated
Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations
for analogs of Dirichlet and Neumann divisors. The principal aim of this paper
is a detailed theta function representation of all algebro-geometric
quasi-periodic solutions and related quantities of the Bsq hierarchy.
| 2015-06-26 |
solv-int/9809005 | Fritz Gesztesy | Fritz Gesztesy and Rudi Weikard | Elliptic Algebro-Geometric Solutions of the KdV and AKNS Hierarchies -
An Analytic Approach | LaTeX, 46 pages, to appear in Bull. A.M.S | null | null | null | solv-int nlin.SI | null | We provide an overview of elliptic algebro-geometric solutions of the KdV and
AKNS hierarchies, with special emphasis on Floquet theoretic and spectral
theoretic methods. Our treatment includes an effective characterization of all
stationary elliptic KdV and AKNS solutions based on a theory developed by
Hermite and Picard.
| 2007-05-23 |
solv-int/9809006 | Maciej Dunajski | M. Dunajski, L.J. Mason and N.M.J. Woodhouse | From 2D Integrable Systems to Self-Dual Gravity | 9 pages, LaTex, no figures | J.Phys.A: Math.Gen 31 (1998) 6019-6028 | 10.1088/0305-4470/31/28/015 | null | solv-int nlin.SI | null | We explain how to construct solutions to the self-dual Einstein vacuum
equations from solutions of various two-dimensional integrable systems by
exploiting the fact that the Lax formulations of both systems can be embedded
in that of the self-dual Yang--Mills equations. We illustrate this by
constructing explicit self-dual vacuum metrics on $\R^2\times \Sigma$, where
$\Sigma$ is a homogeneous space for a real subgroup of $SL(2, \C)$ associated
with the two-dimensional system.
| 2009-10-31 |
solv-int/9809007 | C. Chandre | C. Chandre | A comparison of two discrete mKdV equations | 2 pages, REVTeX | Physica Scripta 55, 129 (1997) | 10.1088/0031-8949/55/2/001 | null | solv-int nlin.SI | null | We consider here two discrete versions of the modified KdV equation. In one
case, some solitary wave solutions, B\"acklund transformations and integrals of
motion are known. In the other one, only solitary wave solutions were given,
and we supply the corresponding results for this equation. We also derive the
integrability of the second equation and give a transformation which links the
two models.
| 2009-10-31 |
solv-int/9809008 | Victor Enolskii | J C Eilbeck, V Z Enol'skii, V B Kuznetsov, D V Leykin | Linear r-Matrix Algebra for a Hierarchy of One-Dimensional Particle
Systems Separable in Parabolic Coordinates | plain LaTeX, 28 pages | null | null | null | solv-int nlin.SI | null | We consider a hierarchy of many-particle systems on the line with polynomial
potentials separable in parabolic coordinates. The first non-trivial member of
this hierarchy is a generalization of an integrable case of the H\'enon-Heiles
system. We give a Lax representation in terms of $2\times 2$ matrices for the
whole hierarchy and construct the associated linear r-matrix algebra with the
r-matrix dependent on the dynamical variables. A Yang-Baxter equation of
dynamical type is proposed. Classical integration in a particular case is
carried out and quantization of the system is discussed with the help of
separation variables. This paper was published in the rary issues: Sfb 288
Preprint No. 110, Berlin and Nonlinear Mathematical Physics, {\bf 1(3)},
275-294 (1994)
| 2019-08-17 |
solv-int/9809009 | Wen-Xiu Ma | Wen-Xiu Ma and Benno Fuchssteiner | Algebraic Structure of Discrete Zero Curvature Equations and Master
Symmetries of Discrete Evolution Equations | 24 pages, LaTex, revised | null | 10.1063/1.532872 | null | solv-int nlin.SI | null | An algebraic structure related to discrete zero curvature equations is
established. It is used to give an approach for generating master symmetries of
first degree for systems of discrete evolution equations and an answer to why
there exist such master symmetries. The key of the theory is to generate
nonisospectral flows $(\lambda_t=\lambda ^l, l\ge0)$ from the discrete spectral
problem associated with a given system of discrete evolution equations. Three
examples are given.
| 2015-06-26 |
solv-int/9809010 | Bireswar Basu-Mallick | B. Basu-Mallick | Symmetries and exact solutions of some integrable Haldane-Shastry like
spin chains | 35 pages, latex, no figures, minor type errors are corrected, version
to appear in Nucl. Phys. B | null | 10.1016/S0550-3213(98)00784-6 | null | solv-int cond-mat.stat-mech hep-th nlin.SI | null | By using a class of `anyon like' representations of permutation algebra,
which pick up nontrivial phase factors while interchanging the spins of two
lattice sites, we construct some integrable variants of $SU(M)$ Haldane-Shastry
(HS) spin chain. Lax pairs and conserved quantities for these spin chains are
also found and it is established that these models exhibit multi-parameter
deformed or nonstandard variants of $Y(gl_M)$ Yangian symmetry. Moreover, by
projecting the eigenstates of Dunkl operators in a suitable way, we derive a
class of exact eigenfunctions for such HS like spin chain and subsequently
conjecture that these exact eigenfunctions would lead to the highest weight
states associated with a multi-parameter deformed or nonstandard variant of
$Y(gl_M)$ Yangian algebra. By using this conjecture, and acting descendent
operator on the highest weight states associated with a nonstandard $Y(gl_2)$
Yangian algebra, we are able to find out the complete set of eigenvalues and
eigenfunctions for the related HS like spin-${1\over 2}$ chain. It turns out
that some additional energy levels, which are forbidden due to a selection rule
in the case of SU(2) HS model, interestingly appear in the spectrum of above
mentioned HS like spin chain having nonstandard $Y(gl_2)$ Yangian symmetry.
| 2009-10-31 |
solv-int/9809011 | Antoine Balan | A.Balan (Ecole Polytechnique) | The Generalised mKdV Equations for Level -3 of $\hat{sl}_2$ | null | null | null | null | solv-int nlin.SI | null | A certain generalisation of the hierarchy of mKdV equations (modified KdV),
which forms an integrable system, is studied here. This generalisation is based
on a Lax operator associated to the equations, with principal components of
degrees between -3 and 0. The results are the following ones: 1) an isomorphism
between the space of jets of the system and a quotient of ${Sl}_2({\CC}((t)))$;
2) the fact that the monodromy matrixes of the Lax operators have, morover,
Poisson brackets given by the trigonometric r-matrix; 3) a definition of the
action of screening operators on the densities; 4) an identification of the
intersection of the kernel with the integrals of motion.
| 2007-05-23 |
solv-int/9809012 | Gavrilov | Lubomir Gavrilov and Angel Zhivkov | The complex geometry of Lagrange top | LaTex, 26 pages | L'Enseignement Mathematique, tome 44 (1998) p.133-170 | null | null | solv-int nlin.SI | null | We prove that the heavy symmetric top (Lagrange, 1788) linearizes on a
two-dimensional non-compact algebraic group -- the generalized Jacobian of an
elliptic curve with two points identified. This leads to a transparent
description of its complex and real invariant level sets. We also deduce, by
making use of a Baker-Akhiezer function, simple explicit formulae for the
general solution of Lagrange top.
| 2007-05-23 |
solv-int/9809013 | R. S. Ward | R. S. Ward | Lax Pairs for Integrable Lattice Systems | 15 pages, plainTeX, to appear in J Math Phys | null | 10.1063/1.532772 | DTP 98/59 | solv-int nlin.SI | null | This paper studies the structure of Lax pairs associated with integrable
lattice systems (where space is a one-dimensional lattice, and time is
continuous). It describes a procedure for generating examples of such systems,
and emphasizes the features that are needed to obtain equations which are local
on the spatial lattice.
| 2015-06-26 |
solv-int/9809014 | Nalini Joshi | N. Joshi, A. Ramani, B. Grammaticos | A Bilinear Approach to Discrete Miura Transformations | 7 pages in TeX, to appear in Phys. Letts. A | null | 10.1016/S0375-9601(98)00624-0 | null | solv-int nlin.SI | null | We present a systematic approach to the construction of Miura transformations
for discrete Painlev\'e equations. Our method is based on the bilinear
formalism and we start with the expression of the nonlinear discrete equation
in terms of $\tau$-functions. Elimination of $\tau$-functions from the
resulting system leads to another nonlinear equation, which is a ``modified''
version of the original equation. The procedure therefore yields Miura
transformations. In this letter, we illustrate this approach by reproducing
previously known Miura transformations and constructing new ones.
| 2009-10-31 |
solv-int/9810001 | Gavrilov | Lubomir Gavrilov | Generalized Jacobians of spectral curves and completely integrable
systems | 20 pages, LaTex2e, to appear in Math. Zeitschrift | Math. Zeitschrift, 230, 487-508 (1999) | null | null | solv-int nlin.SI | null | Consider an ordinary differential equation which has a Lax pair
representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a
fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such
an equation can be considered as a completely integrable complex Hamiltonian
system. We show that the generic complex invariant manifold {A(x): det(A(x)-y
I)= P(x,y)} of this Lax pair is an affine part of a non-compact commutative
algebraic group---the generalized Jacobian of the spectral curve {(x,y):
P(x,y)=0} with its points at "infinity" identified. Moreover, for suitable
B(x), the Hamiltonian vector field defined by the Lax pairon the generalized
Jacobian is translation--invariant. We provide two examples in which the above
result applies.
| 2010-05-04 |
solv-int/9810002 | Daniel Arnaudon | D. Arnaudon and Z. Maassarani | Integrable open boundary conditions for XXC models | Latex2e, 10 pages | JHEP 10 (1998) 024 | 10.1088/1126-6708/1998/10/024 | LAPTH-695/98, LAVAL-PHY-22/98 | solv-int cond-mat.str-el math-ph math.MP nlin.SI | null | The XXC models are multistate generalizations of the well known spin 1/2 XXZ
model. These integrable models share a common underlying su(2) structure. We
derive integrable open boundary conditions for the hierarchy of conserved
quantities of the XXC models . Due to lack of crossing unitarity of the
R-matrix, we develop specific methods to prove integrability. The symmetry of
the spectrum is determined.
| 2009-10-31 |
solv-int/9810003 | Anton Zabrodin | A.Zabrodin | Tau-function for discrete sine-Gordon equation and quantum R-matrix | 14 pages, latex | null | null | ITEP-TH-55/98 | solv-int hep-th nlin.SI | null | We prove that the tau-function of the integrable discrete sine-Gordon model
apart from the "standard" bilinar identities obeys a number of "non-standard"
ones. They can be combined into a bivector 3-dimensional difference equation
which is shown to contain Hirota's difference analogue of the sine-Gordon
equation and both auxiliary linear problems for it. We observe that this
equation is most naturally written in terms of the quantum R-matrix for the XXZ
spin chain and looks then like a relation of the "vertex-face correspondence"
type.
| 2007-05-23 |
solv-int/9810004 | John Palmer | John Palmer | Zeros of the Jimbo, Miwa, Ueno tau function | 59 pages | null | 10.1063/1.533112 | null | solv-int nlin.SI | null | We introduce a family of local deformations for meromorphic connections on
the Riemann sphere in the neighborhood of a higher rank (simple) singularity.
Following a scheme introduced by Malgrange we use these local models to prove
that the zeros of the tau function introduced by Jimbo, Miwa and Ueno occur
precisely at those points in the deformation space at which a certain
Birkhoff-Riemann- Hilbert problem fails to have a solution.
| 2009-10-31 |
solv-int/9810005 | Anca Visinescu | D. Grecu, A.S. C\^arstea, Anca Visinescu | Long range interaction corrections on the quantum vibronic soliton | Proceedings of the Conference "Path-Integral from pev to TeV. 50
years from Feynman's paper", Florence, August 1998, 4 pages, latex,
sprocl.sty | null | null | IFA-FT-437(1998) | solv-int cond-mat nlin.SI | null | Self-localized modes in a quantum vibronic system, with long range
interaction of Kac-Baker type and interacting nonlinearly with an acoustical
phonon bath, is studied. One works in the coherent state approximation.
Following a procedure of Sarker and Krumhansl, the problem is reduced to a
nearest neighbours one. In the continuum limit the localized state satisfy a
mKdV equation. An approximate expression for its frequency is found.
| 2007-05-23 |
solv-int/9810006 | Alexandr Andrianov | A.A.Andrianov, M.V.Ioffe, D.N.Nishnianidze (St.Petersburg Univ.) | Classical Integrable 2-dim Models Inspired by SUSY Quantum Mechanics | 19 pages, LaTeX, final version to be published in J.Phys.A | J.Phys.A32:4641,1999 | 10.1088/0305-4470/32/25/307 | null | solv-int hep-th math-ph math.MP nlin.SI quant-ph | null | A class of integrable 2-dim classical systems with integrals of motion of
fourth order in momenta is obtained from the quantum analogues with the help of
deformed SUSY algebra. With similar technique a new class of potentials
connected with Lax method is found which provides the integrability of
corresponding 2-dim hamiltonian systems. In addition, some integrable 2-dim
systems with potentials expressed in elliptic functions are explored.
| 2008-11-26 |
solv-int/9810007 | Paul Zinn-Justin | P. Zinn-Justin | Quelques applications de l'Ansatz de Bethe (Some applications of the
Bethe Ansatz) | PhD dissertation. In French. the articles are not included (they're
already on the archive) | null | null | null | solv-int nlin.SI | null | The Bethe Ansatz is a method that is used in quantum integrable models in
order to solve them explicitly. This method is explained here in a general
framework, which applies to 1D quantum spin chains, 2D statistical lattice
models (vertex models) and relativistic field theories with 1 space dimension
and 1 time dimension. The connection with quantum groups is expounded. Several
applications are then presented. Finite size corrections are calculated via two
methods: The Non-Linear Integral Equations, which are applied to the study of
the states of the affine Toda model with imaginary coupling, and their
interpolation between the high energy (ultra-violet) and low energy (infra-red)
regions; and the Thermodynamic Bethe Ansatz Equations, along with the
associated Fusion Equations, which are used to determine the thermodynamic
properties of the generalized multi-channel Kondo model. The latter is then
studied in more detail, still using the Bethe Ansatz and quantum groups, so as
to characterize the spectrum of the low energy excitations.
| 2007-05-23 |
solv-int/9810008 | Doc. Dr. Ayse Humeyra Bilge | Ayse Humeyra Bilge and Fatma Ozdemir | Miura Transformations for Integrable Evolution Equations of the Form
$u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ | The paper is withdrawn | null | null | null | solv-int nlin.SI | null | The paper is withdrawn due to an error in Section 3.2. The remaining of the
results are included in the preprint solv-int/9605004.
| 2007-05-23 |
solv-int/9810009 | Olaf Lechtenfeld | Olaf Lechtenfeld and Alexander Sorin | Fermionic flows and tau function of the N=(1|1) superconformal Toda
lattice hierarchy | 11 pages, no figures, revised version published in Nucl. Phys. B | Nucl.Phys. B557 (1999) 535-547 | 10.1016/S0550-3213(99)00063-2 | ITP-UH-23/98, JINR E2-98-285 | solv-int hep-th math-ph math.MP nlin.SI | null | An infinite class of fermionic flows of the N=(1|1) superconformal Toda
lattice hierarchy is constructed and their algebraic structure is studied. We
completely solve the semi-infinite N=(1|1) Toda lattice and chain hierarchies
and derive their tau functions, which may be relevant for building
supersymmetric matrix models. Their bosonic limit is also discussed.
| 2009-10-31 |
solv-int/9810010 | Juan A. Calzada | J.A. Calzada, M.A. del Olmo, M.A. Rodriguez | Pseudo-orthogonal groups and integrable dynamical systems in two
dimensions | 32 pages,revtex | null | 10.1063/1.532768 | null | solv-int nlin.SI | null | Integrable systems in low dimensions, constructed through the symmetry
reduction method, are studied using phase portrait and variable separation
techniques. In particular, invariant quantities and explicit periodic solutions
are determined. Widely applied models in Physics are shown to appear as
particular cases of the method.
| 2009-10-31 |
solv-int/9810011 | Martin Goliath | Martin Goliath, Max Karlovini, and Kjell Rosquist | Lax pair tensors in arbitrary dimensions | 8 pages, uses IOP style files. Minor correction. Submitted to J. Phys
A | null | 10.1088/0305-4470/32/18/311 | null | solv-int gr-qc nlin.SI | null | A recipe is presented for obtaining Lax tensors for any n-dimensional
Hamiltonian system admitting a Lax representation of dimension n. Our approach
is to use the Jacobi geometry and coupling-constant metamorphosis to obtain a
geometric Lax formulation. We also exploit the results to construct integrable
spacetimes, satisfying the weak energy condition.
| 2009-10-31 |
solv-int/9810012 | Manuel Manas | Adam Doliwa, Manuel Manas and Luis Martinez Alonso | Generating Quadrilateral and Circular Lattices in KP Theory | 20 pages, 1 figure, LaTeX2e with AMSLaTeX, Babel, graphicx and psfrag
packages | null | 10.1016/S0375-9601(99)00579-4 | null | solv-int nlin.SI | null | The bilinear equations of the $N$-component KP and BKP hierarchies and a
corresponding extended Miwa transformation allow us to generate quadrilateral
and circular lattices from conjugate and orthogonal nets, respectively. The
main geometrical objects are expressed in terms of Baker functions.
| 2009-10-31 |
solv-int/9810013 | Gennady El | Gennady A. El, Alexander L.Krylov | Stochastic Soliton Lattices | 11 pages. To be published in Proceedings of the International
Conference `Solitons, Geometry and Topology: on the Crossroads', Moscow, 1998 | null | null | null | solv-int nlin.SI | null | We introduce a new concept, Stochastic Soliton Lattice, as a random process
generated by a finite-gap potential of the Shroedinger operator. We study the
basic properties of this stochastic process and consider its KdV evolution
| 2007-05-23 |
solv-int/9810014 | A. David Trubatch | M. J. Ablowitz, Y. Ohta, A. D. Trubatch | On Discretizations of the Vector Nonlinear Schrodinger Equation | 16 pages, 1 figure, 1 table | null | 10.1016/S0375-9601(99)00048-1 | APPM 349 | solv-int nlin.SI | null | Two discretizations of the vector nonlinear Schrodinger (NLS) equation are
studied. One of these discretizations, referred to as the symmetric system, is
a natural vector extension of the scalar integrable discrete NLS equation. The
other discretization, referred to as the asymmetric system, has an associated
linear scattering pair. General formulae for soliton solutions of the
asymmetric system are presented. Formulae for a constrained class of solutions
of the symmetric system may be obtained. Numerical studies support the
hypothesis that the symmetric system has general soliton solutions.
| 2009-10-31 |
solv-int/9810015 | Antoine Balan | Antoine Balan (Ecole Polytechnique) | The Lax operators $\cal L$ of the Benney type equations bound with the
circle | 5 pages, LaTex, no figure | null | null | null | solv-int nlin.SI | null | The Lax operators of the Benney type equations are studied on the circle. The
vectors fields of the Lax operators are showed to commute with each other
| 2007-05-23 |
solv-int/9810016 | Nikita A. Slavnov | N. A. Slavnov (Steklov Mathematical Institute, Moscow, Russia) | Asymptotics of the Fredholm determinant associated with the correlation
functions of the quantum Nonlinear Schrodinger equation | 14 pages, Latex, no figures | null | null | MI-98-06 | solv-int nlin.SI | null | The correlation functions of the quantum nonlinear Schrodinger equation can
be presented in terms of a Fredholm determinant. The explicit expression for
this determinant is found for the large time and long distance.
| 2007-05-23 |
solv-int/9810017 | Robert Milson | N. Kamran and R. Milson | Algebraic Exact Solvability of trigonometric-type Hamiltonians
associated to root systems | 14 pages | null | 10.1063/1.533012 | null | solv-int math-ph math.MP math.SP nlin.SI | null | In this article, we study and settle several structural questions concerning
the exact solvability of the Olshanetsky-Perelomov quantum Hamiltonians
corresponding to an arbitrary root system. We show that these operators can be
written as linear combinations of certain basic operators admitting infinite
flags of invariant subspaces, namely the Laplacian and the logarithmic gradient
of invariant factors of the Weyl denominator. The coefficients of the
constituent linear combination become the coupling constants of the final
model. We also demonstr ate the $L^2$ completeness of the eigenfunctions
obtained by this procedure, and describe a straight-forward recursive procedure
based on the Freudenthal multiplicity formula for constructing the
eigenfunctions explicitly.
| 2015-06-26 |
solv-int/9810018 | Yuri B. Suris | A.I.Bobenko, Yu.B.Suris (Technische Universitaet Berlin) | Discrete time Lagrangian mechanics on Lie groups, with an application to
the Lagrange top | LaTeX 2e, 44 pages, 1 figure | Commun. Math. Phys., 1999, V. 204, p. 147-188 | 10.1007/s002200050642 | SFB288 preprint No. 345 | solv-int nlin.SI | null | We develop the theory of discrete time Lagrangian mechanics on Lie groups,
originated in the work of Veselov and Moser, and the theory of Lagrangian
reduction in the discrete time setting. The results thus obtained are applied
to the investigation of an integrable time discretization of a famous
integrable system of classical mechanics, -- the Lagrange top. We recall the
derivation of the Euler--Poinsot equations of motion both in the frame moving
with the body and in the rest frame (the latter ones being less widely known).
We find a discrete time Lagrange function turning into the known continuous
time Lagrangian in the continuous limit, and elaborate both descriptions of the
resulting discrete time system, namely in the body frame and in the rest frame.
This system naturally inherits Poisson properties of the continuous time
system, the integrals of motion being deformed. The discrete time Lax
representations are also found. Kirchhoff's kinetic analogy between elastic
curves and motions of the Lagrange top is also generalised to the discrete
context.
| 2009-10-31 |
solv-int/9810019 | Faruk Gungor | F. Gungor (Istanbul Technical University) | Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation | 16 pages, no figures, revised version | Physica Scripta, Vol. 61, 385-390, 2000 | 10.1238/Physica.Regular.061a00385 | null | solv-int nlin.SI | null | The purpose of this paper is to present a class of particular solutions of a
C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry
reduction. Using the subgroups of similitude group reduced ordinary
differential equations of second order and their solutions by a singularity
analysis are classified. In particular, it has been shown that whenever they
have the Painlev\'e property, they can be transformed to standard forms by
Moebius transformations of dependent variable and arbitrary smooth
transformations of independent variable whose solutions, depending on the
values of parameters, are expressible in terms of either elementary functions
or Jacobi elliptic functions.
| 2009-10-31 |
solv-int/9810020 | A. David Trubatch | M. J. Ablowitz, Y. Ohta, A. D. Trubatch | On Integrability and Chaos in Discrete Systems | 14 pages, 1 figure, 2 tables | null | null | APPM 350 | solv-int nlin.SI | null | The scalar nonlinear Schrodinger (NLS) equation and a suitable discretization
are well known integrable systems which exhibit the phenomena of ``effective''
chaos. Vector generalizations of both the continuous and discrete system are
discussed. Some attention is directed upon the issue of the integrability of a
discrete version of the vector NLS equation.
| 2007-05-23 |
solv-int/9811001 | Minoru Wakimoto | K. Iohara (Kyoto Univ.), Y. Saito (Hiroshima Univ.), M. Wakimoto
(Kyushu Univ.) | Hirota bilinear forms with 2-toroidal symmetry | 11 pages, AMS-latex file | null | 10.1016/S0375-9601(99)00093-6 | null | solv-int nlin.SI | null | In this note, we compute Hirota bilinear forms arising from both homogeneous
and principal realization of vertex representations of 2-toroidal Lie algebras
of type $A_l, D_l, E_l$.
| 2009-10-31 |
solv-int/9811002 | Angela Foerster | A. Lima-Santos, I. Roditi and A. Foerster | Bethe ansatz solution of the anisotropic correlated electron model
associated with the Temperley-Lieb algebra | 21 pages | Int. Jour. of Mod. Phys. A, Vol. 13, No. 25 (1998) 4309-4324 | 10.1142/S0217751X98002080 | null | solv-int nlin.SI | null | A recently proposed strongly correlated electron system associated with the
Temperley-Lieb algebra is solved by means of the coordinate Bethe ansatz for
periodic and closed boundary conditions.
| 2009-10-31 |
solv-int/9811003 | Sergei M. Sergeev | S. M. Sergeev | Quantum 2+1 evolution model | LaTeX, 37pages | null | 10.1088/0305-4470/32/30/313 | Branch of INP preprint No. 98-02 | solv-int nlin.SI | null | A quantum evolution model in 2+1 discrete space - time, connected with 3D
fundamental map R, is investigated. Map R is derived as a map providing a zero
curvature of a two dimensional lattice system called "the current system". In a
special case of the local Weyl algebra for dynamical variables the map appears
to be canonical one and it corresponds to known operator-valued R-matrix. The
current system is a kind of the linear problem for 2+1 evolution model. A
generating function for the integrals of motion for the evolution is derived
with a help of the current system. The subject of the paper is rather new, and
so the perspectives of further investigations are widely discussed.
| 2009-10-31 |
solv-int/9811004 | Ming-Hsien Tu | Ming-Hsien Tu, Jiin-Chang Shaw and Chin-Rong Lee | On Darboux-B\"acklund Transformations for the Q-Deformed Korteweg-de
Vries Hierarchy | 12 pages, Revtex, no figures | Lett. Math. Phys., 49(1): 33-45, July 1999 | null | null | solv-int nlin.SI | null | We study Darboux-B\"acklund transformations (DBTs) for the $q$-deformed
Korteweg-de Vries hierarchy by using the $q$-deformed pseudodifferential
operators. We identify the elementary DBTs which are triggered by the gauge
operators constructed from the (adjoint) wave functions of the hierarchy.
Iterating these elementary DBTs we obtain not only $q$-deformed Wronskian-type
but also binary-type representations of the tau-function to the hierarchy.
| 2007-05-23 |
solv-int/9811005 | Max Karlovini | Max Karlovini and Kjell Rosquist | A unified treatment of cubic invariants at fixed and arbitrary energy | 16 pages, LaTeX2e, slightly revised version. To appear in J. Math.
Phys. vol 41, pp 370-384 (2000) | null | 10.1063/1.533137 | SUITP 98-21 | solv-int nlin.SI | null | Cubic invariants for two-dimensional Hamiltonian systems are investigated
using the Jacobi geometrization procedure. This approach allows for a unified
treatment of invariants at both fixed and arbitrary energy. In the geometric
picture the invariant generally corresponds to a third rank Killing tensor,
whose existence at a fixed energy value forces the metric to satisfy a
nonlinear integrability condition expressed in terms of a Kahler potential.
Further conditions, leading to a system of equations which is overdetermined
except for singular cases, are added when the energy is arbitrary. As solutions
to these equations we obtain several new superintegrable cases in addition to
the previously known cases. We also discover a superintegrable case where the
cubic invariant is of a new type which can be represented by an energy
dependent linear invariant. A complete list of all known systems which admit a
cubic invariant at arbitrary energy is given.
| 2009-10-31 |
solv-int/9811006 | Bernard Deconinck | Bernard Deconinck | Canonical variables for multiphase solutions of the KP equation | 52 papes, 3 figures, uses psfig, latexsym | null | null | null | solv-int nlin.SI | null | The KP equation has a large family of quasiperiodic multiphase solutions.
These solutions can be expressed in terms of Riemann-theta functions. In this
paper, a finite-dimensional canonical Hamiltonian system depending on a finite
number of parameters is given for the description of each such solution. The
Hamiltonian systems are completely integrable in the sense of Liouville. In
effect, this provides a solution of the initial-value problem for the
theta-function solutions. Some consequences of this approach are discussed.
| 2007-05-23 |
solv-int/9811007 | G. Tondo | C. Morosi, G. Tondo | On a class of dynamical systems both quasi-bi-Hamiltonian and
bi-Hamiltonian | 11 pages, AMS-LaTex 1.1 | Phys. Lett. A 247 (1998), 59-64 | 10.1016/S0375-9601(98)00543-X | null | solv-int nlin.SI | null | It is shown that a class of dynamical systems (encompassing the one recently
considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both
quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the
separability of these systems; the second one is obtained trough a non
canonical map whose form is directly suggested by the associated Nijenhuis
tensor.
| 2009-10-31 |
solv-int/9811008 | G. Tondo | G. Tondo, C. Morosi | Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation
variables | 12 pages, no figures, LaTeX 2.09, to be published in Report on
Mathematical Physics | null | 10.1016/S0034-4877(99)80167-0 | null | solv-int nlin.SI | null | We discuss from a bi-Hamiltonian point of view the Hamilton-Jacobi
separability of a few dynamical systems. They are shown to admit, in their
natural phase space, a quasi-bi-Hamiltonian formulation of Pfaffian type. This
property allows us to straightforwardly recover a set of separation variables
for the corresponding Hamilton-Jacobi equation.
| 2009-10-31 |
solv-int/9811009 | Alex Kasman | Alex Kasman | Spectral Difference Equations Satisfied by KP Soliton Wavefunctions | to appear in "Inverse Problems" | null | 10.1088/0266-5611/14/6/008 | null | solv-int nlin.SI | null | The Baker-Akhiezer (wave) functions corresponding to soliton solutions of the
KP hierarchy are shown to satisfy eigenvalue equations for a commutative ring
of translational operators in the spectral parameter. In the rational limit,
these translational operators converge to the differential operators in the
spectral parameter previously discussed as part of the theory of
"bispectrality". Consequently, these translational operators can be seen as
demonstrating a form of bispectrality for the non-rational solitons as well.
| 2009-10-31 |
solv-int/9811010 | Ming-Hsien Tu | Ming-Hsien Tu | Q-deformed KP hierarchy: Its additional symmetries and infinitesimal
B\"acklund transformations | 9 pages, Revtex, no figures | Lett. Math. Phys., 49(2): 95-103, July 1999 | null | null | solv-int nlin.SI | null | We study the additional symmetries associated with the $q$-deformed
Kadomtsev-Petviashvili ($q$-KP) hierarchy. After identifying the resolvent
operator as the generator of the additional symmetries, the $q$-KP hierarchy
can be consistently reduced to the so-called $q$-deformed constrained KP
($q$-cKP) hierarchy. We then show that the additional symmetries acting on the
wave function can be viewed as infinitesimal B\"acklund transformations by
acting the vertex operator on the tau-function of the $q$-KP hierarchy. This
establishes the Adler-Shiota-van Moerbeke formula for the $q$-KP hierarchy.
| 2007-05-23 |
solv-int/9811011 | Pilar G. Estevez | Pilar Garcia Estevez | Darboux Transformations and solutions for an equation in 2+1 dimensions | LaTeX 16 pages with 6 figures. Journal of Mathematical Physics (to
appear) | null | null | University of Salamanca (SPAIN) preprint NLPG 25/98 | solv-int nlin.SI | null | Painleve analysis and the singular manifold method are the tools used in this
paper to perform a complete study of an equation in 2+1 dimensions. This
procedure has allowed us to obtain the Lax pair, Darboux transformation and tau
functions in such a way that a plethora of different solutions with solitonic
behavior can be constructed iteratively
| 2007-05-23 |
solv-int/9811012 | R. S. Ward | R S Ward | Two Integrable Systems Related to Hyperbolic Monopoles | 12 pages, plainTeX | null | null | DTP-98/77 | solv-int nlin.SI | null | Monopoles on hyperbolic 3-space were introduced by Atiyah in 1984. This
article describes two integrable systems which are closely related to
hyperbolic monopoles: a one-dimensional lattice equation (the Braam-Austin or
discrete Nahm equation), and a soliton system in (2+1)-dimensional
anti-deSitter space-time.
| 2007-05-23 |
solv-int/9811013 | Metin Gurses | Metin Gurses and Atalay Karasu | Integrable KdV Systems: Recursion Operators of Degree Four | Latex File, to be published in Physucs Letters A | null | 10.1016/S0375-9601(98)00910-4 | null | solv-int nlin.SI | null | The recursion operator and bi-Hamiltonian formulation of the Drinfeld-
Sokolov system are given
| 2009-10-31 |
solv-int/9811014 | Nalini Joshi | Peter A. Clarkson, Nalini Joshi and Andrew Pickering | B\"acklund transformations for the second Painlev\'e hierarchy: a
modified truncation approach | 12 pages in LaTeX 2.09 (uses ioplppt.sty), to appear in Inverse
Problems | null | 10.1088/0266-5611/15/1/019 | null | solv-int nlin.SI | null | The second Painlev\'e hierarchy is defined as the hierarchy of ordinary
differential equations obtained by similarity reduction from the modified
Korteweg-de Vries hierarchy. Its first member is the well-known second
Painlev\'e equation, P2.
In this paper we use this hierarchy in order to illustrate our application of
the truncation procedure in Painlev\'e analysis to ordinary differential
equations. We extend these techniques in order to derive auto-B\"acklund
transformations for the second Painlev\'e hierarchy. We also derive a number of
other B\"acklund transformations, including a B\"acklund transformation onto a
hierarchy of P34 equations, and a little known B\"acklund transformation for P2
itself.
We then use our results on B\"acklund transformations to obtain, for each
member of the P2 hierarchy, a sequence of special integrals.
| 2009-10-31 |
solv-int/9811015 | Faruk Gungor | Faruk Gungor | Nonlinear Evolution Equations Invariant Under Schroedinger Group in
three-dimensional Space-time | null | J. Phys. A: Math. and Gen. 32 (1999) 977-988 | 10.1088/0305-4470/32/6/010 | null | solv-int nlin.SI | null | A classification of all possible realizations of the Galilei,
Galilei-similitude and Schroedinger Lie algebras in three-dimensional
space-time in terms of vector fields under the action of the group of local
diffeomorphisms of the space $\R^3\times\C$ is presented. Using this result a
variety of general second order evolution equations invariant under the
corresponding groups are constructed and their physical significance are
discussed.
| 2009-10-31 |
solv-int/9811016 | Chand Devchand | Chandrashekar Devchand and Jeremy Schiff | The supersymmetric Camassa-Holm equation and geodesic flow on the
superconformal group | 14 pages, latex file | J.Math.Phys. 42 (2001) 260-273 | 10.1063/1.1330196 | null | solv-int hep-th math-ph math.DG math.MP nlin.SI | null | We study a family of fermionic extensions of the Camassa-Holm equation.
Within this family we identify three interesting classes: (a) equations, which
are inherently hamiltonian, describing geodesic flow with respect to an H^1
metric on the group of superconformal transformations in two dimensions, (b)
equations which are hamiltonian with respect to a different hamiltonian
structure and (c) supersymmetric flow equations. Classes (a) and (b) have no
intersection, but the intersection of classes (a) and (c) gives a candidate for
a new supersymmetric integrable system. We demonstrate the Painlev\'e property
for some simple but nontrivial reductions of this system.
| 2009-10-31 |
solv-int/9812001 | Red Hat Linux User | Andrey Tsiganov | Duality between integrable Stackel systems | LaTeX2e, 18 pages | J. Phys. A, Math. Gen. 32, No.45, 7965-7982, (1999) | 10.1088/0305-4470/32/45/311 | null | solv-int nlin.SI | null | For the Stackel family of the integrable systems a non-canonical
transformation of the time variable is considered. This transformation may be
associated to the ambiguity of the Abel map on the corresponding hyperelliptic
curve. For some Stackel's systems with two degrees of freedom the 2x2 Lax
representations and the dynamical r-matrix algebras are constructed. As an
examples the Henon-Heiles systems, integrable Holt potentials and the
integrable deformations of the Kepler problem are discussed in detail.
| 2009-10-31 |
solv-int/9812002 | Ihabib | I.T.Habibullin and A.N.Vil'danov | Integrable boundary conditions for nonlinear lattices | 11 pages | null | null | null | solv-int nlin.SI | null | Integrable boundary conditions in 1+1 and 2+1 dimensions are discussed from
the higher symmetries point of view. Boundary conditions consistent with the
discrete Landau-Lifshitz model and infinite 2D Toda lattice are represented.
| 2007-05-23 |
solv-int/9812003 | Takayuki Tsuchida | T. Tsuchida, M. Wadati | The Coupled Modified Korteweg-de Vries Equations | 26 pages, LaTex209 file, uses jpsj.sty | J. Phys. Soc. Jpn. 67 (1998) 1175-1187 | 10.1143/JPSJ.67.1175 | null | solv-int nlin.SI | null | Generalization of the modified KdV equation to a multi-component system, that
is expressed by $(\partial u_i)/(\partial t) + 6 (\sum_{j,k=0}^{M-1} C_{jk} u_j
u_k) (\partial u_i)/(\partial x) + (\partial^3 u_{i})/(\partial x^3) = 0, i=0,
1, ..., M-1 $, is studied. We apply a new extended version of the inverse
scattering method to this system. It is shown that this system has an infinite
number of conservation laws and multi-soliton solutions. Further, the initial
value problem of the model is solved.
| 2009-10-31 |
solv-int/9812004 | Pawel Maslanka | C.Gonera, P.Kosi\'nski, M.Majewski, P.Ma\'slanka | On Calogero wave functions | 10 pages LaTeX2e file | null | null | null | solv-int nlin.SI | null | Two properties of Calogero wave functions for rational Calogero models are
studied: (i) the representation of the wave functions in terms of the
exponential of Lassalle operators, (ii) the $sL(2,\rr)$ structure of the
Calogero--Moser wave functions.
| 2007-05-23 |
solv-int/9812005 | Pawel Maslanka | C.Gonera, M.Majewski, P.Ma\'slanka | On the Calogero model with negative harmonic term | 6 pages LaTeX2e file | null | null | null | solv-int nlin.SI | null | The Calogero model with negative harmonic term is shown to be equivalent to
the set of negative harmonic oscillators. Two time-independent canonical
transformations relating both models are constructed: one based on the recent
results concerning quantum Calogero model and one obtained from dynamical
$sL(2,\rr)$ algebra. The two-particle case is discussed in some detail.
| 2007-05-23 |
solv-int/9812006 | Pierre van Moerbeke | M. Adler, T. Shiota, and P. van Moerbeke | Random matrices, Virasoro algebras, and noncommutative KP | 56 pages | Duke Math Journal, 94, pp. 379-431, 1998 | null | null | solv-int nlin.SI | null | What is the connection of random matrices with integrable systems? Is this
connection really useful? The answer to these questions leads to a new and
unifying approach to the theory of random matrices. Introducing an appropriate
time t-dependence in the probability distribution of the matrix ensemble, leads
to vertex operator expressions for the n-point correlation functions
(probabilities of n eigenvalues in infinitesimal intervals) and the
corresponding Fredholm determinants (probabilities of no eigenvalue in a Borel
subset E); the latter probability is a ratio of tau-functions for the
KP-equation, whose numerator satisfy partial differential equations, which
decouple into the sum of two parts: a Virasoro-like part depending on time only
and a Vect(S^1)-part depending on the boundary points A_i of E. Upon setting
t=0, and using the KP-hierarchy to eliminate t-derivatives, these PDE's lead to
a hierarchy of non-linear PDE's, purely in terms of the A_i. These PDE's are
nothing else but the KP hierarchy for which the t-partials, viewed as commuting
operators, are replaced by non-commuting operators in the endpoints A_i of the
E under consideration. When the boundary of E consists of one point and for the
known kernels, one recovers the Painleve equations, found in prior work on the
subject.
| 2007-05-23 |
solv-int/9812007 | Robert Conte | R. Conte (CEA Saclay) | Perturbative methods for the Painlev\'e test | 11 pages, no figure, standard Latex, to appear in the proceedings of
``Nonlinear dynamics: integrability and chaos'', Tiruchirapalli, 12--16 Feb
1998, ed. S. Daniel | null | null | S98/048 | solv-int nlin.SI | null | There exist many situations where an ordinary differential equation admits a
movable critical singularity which the test of Kowalevski and Gambier fails to
detect. Some possible reasons are: existence of negative Fuchs indices,
insufficient number of Fuchs indices, multiple family, absence of an algebraic
leading order. Mainly giving examples, we present the methods which answer all
these questions. They are all based on the theorem of perturbations of
Poincar\'e and computerizable.
| 2007-05-23 |
solv-int/9812008 | Robert Conte | R. Conte (CEA Saclay) | Various truncations in Painlev\'e analysis of PDEs | 16 pages, no figure, standard Latex, to appear in the proceedings of
``Nonlinear dynamics: integrability and chaos'', Tiruchirapalli, 12--16 Feb
1998, ed. S. Daniel | null | null | S98/047 | solv-int nlin.SI | null | The ``truncation procedure'' initiated by Weiss et al. is best understood as
a Darboux transformation. If it leads to the Lax pair of the PDE under study,
the B\"acklund transformation follows by an elimination, thus proving the
integrability. We present the state of the art of this powerful technique. The
easy situations were all handled by the WTC one-family truncation and its
homographically invariant version. An updated version of this method has been
recently developed, which is now able to handle the Kaup-Kupershmidt and
Tzitz\'eica equations. It incorporates a new feature, namely the distinction
between two entire functions usually mingled, which are shown to be linked by
formulae established by Gambier for his classification.
| 2007-05-23 |
solv-int/9812009 | Michel Talon | O. Babelon, M. Talon | The symplectic structure of rational Lax pair systems | 8 pages, no figure, Latex | null | 10.1016/S0375-9601(99)00298-4 | PAR LPTHE 98-52 | solv-int nlin.SI | null | We consider dynamical systems associated to Lax pairs depending rationnally
on a spectral parameter. We show that we can express the symplectic form in
terms of algebro--geometric data provided that the symplectic structure on L is
of Kirillov type. In particular, in this case the dynamical system is
integrable.
| 2009-10-31 |
solv-int/9812010 | Michal Marvan | Joseph Krasil'shchik and Michal Marvan | Coverings and integrability of the Gauss-Mainardi-Codazzi equations | 15 pages, LaTeX 2e | null | null | DIPS-8/98, ESI 639 (1998), GA 10/1998 | solv-int nlin.SI | null | Using covering theory approach (zero-curvature representations with the gauge
group SL2), we insert the spectral parameter into the Gauss-Mainardi-Codazzi
equations in Tchebycheff and geodesic coordinates. For each choice, four
integrable systems are obtained.
| 2007-05-23 |
solv-int/9812011 | F. Nijhoff | F. W. Nijhoff, A. Ramani, B. Grammaticos and Y. Ohta | On Discrete Painleve Equations Associated with the Lattice KdV Systems
and the Painleve VI Equation | 60+2 pages, Latex2.09, uses equations.sty | null | null | null | solv-int nlin.SI | null | A new integrable nonautonomous nonlinear ordinary difference equation is
presented which can be considered to be a discrete analogue of the Painleve V
equation. Its derivation is based on the similarity reduction on the
two-dimensional lattice of integrable partial difference equations of KdV type.
The new equation which is referred to as GDP (generalised discrete Painleve
equation) contains various ``discrete Painleve equations'' as subcases for
special values/limits of the parameters, some of which were already given
before in the literature. The general solution of the GDP can be expressed in
terms of Painleve VI (PVI) transcendents. In fact, continuous PVI emerges as
the equation obeyed by the solutions of the discrete equation in terms of the
lattice parameters rather than the lattice variables that label the lattice
sites. We show that the bilinear form of PVI is embedded naturally in the
lattice systems leading to the GDP. Further results include the establishment
of Baecklund and Schlesinger transformations for the GDP, the corresponding
isomonodromic deformation problem, and the self-duality of its bilinear scheme.
| 2007-05-23 |
solv-int/9812012 | Zoran Rajilic | S. Lekic, S. Galamic, Z. Rajilic | Optical Fiber Communications:Group of the Nonlinear Transformations | LaTex2e, eps figure, Presented at the conference "Physics-21",St.
Petersburg 1998 | null | null | null | solv-int nlin.SI | null | A new method for finding solutions of the nonlinear Shr\"{o}dinger equation
is proposed. Comutative multiplicative group of the nonlinear transformations,
which operate on stationary localized solutions, enables a consideration of
fractal subspaces in the solution space, stability and deterministic chaos. An
increase of the transmission rate at the optical fiber communications can be
based on new forms of localized stationary solutions, without significant
change of input power. The estimated transmission rate is 50 Gbit/s, for
certain available soliton transmission systems.
| 2009-09-25 |
solv-int/9812013 | Lafortune | S.Lafortune, P.Winternitz and C.R.Menyuk | Solutions to the Optical Cascading Equations | 21 pages | Physical Review E 58, 2518-2825 (1998) | 10.1103/PhysRevE.58.2518 | null | solv-int nlin.SI physics.optics | null | Group theoretical methods are used to study the equations describing
\chi^{(2)}:\chi^{(2)} cascading. The equations are shown not to be integrable
by inverse scattering techniques. On the other hand, these equations do share
some of the nice properties of soliton equations. Large families of explicit
analytical solutions are obtained in terms of elliptic functions. In special
cases, these periodic solutions reduce to localized ones, i.e., solitary waves.
All previously known explicit solutions are recovered, and many additional ones
are obtained
| 2009-10-31 |
solv-int/9812014 | Hendry Izaac Elim | Hans J. Wospakrik and Freddy P. Zen | Inhomogeneous Burgers Equation and the Feynman-Kac Path Integral | 12 pages | null | null | null | solv-int hep-th nlin.SI | null | By linearizing the inhomogeneous Burgers equation through the Hopf-Cole
transformation, we formulate the solution of the initial value problem of the
corresponding linear heat type equation using the Feynman-Kac path integral
formalism. For illustration, we present the exact solution for the forcing term
of the form: $F(x,t)=\omega ^2x+f(t).$ We also present the initial value
problem solution for the case with a constant forcing term to compare with the
known result.
| 2007-05-23 |
solv-int/9812015 | Satoru Saito | Satoru Saito | A Realization of Discrete Geometry by String Model | LaTeX, 9 pages | null | null | null | solv-int nlin.SI | null | A realization of discrete conjugate net is presented by using correlation
functions of strings in a gauge covariant form.
| 2007-05-23 |
solv-int/9812016 | Igor | I.G. Korepanov and S. Saito | Finite-dimensional analogs of string s <-> t duality and pentagon
equation | LaTeX, 12 pages, 6 eps figures | Theor.Math.Phys. 120 (1999) 862-869; Teor.Mat.Fiz. 120 (1999)
54-63 | 10.1007/BF02557395 | null | solv-int hep-th math-ph math.MP nlin.SI | null | We put forward one of the forms of functional pentagon equation (FPE), known
from the theory of integrable models, as an algebraic explanation to the
phenomenon known in physics as s<->t duality. We present two simple geometrical
examples of FPE solutions, one of them yielding in a particular case the
well-known Veneziano expression for 4-particle amplitude. Finally, we interpret
our solutions of FPE in terms of relations in Lie groups.
| 2009-10-31 |
solv-int/9812017 | Ming-Hsien Tu | Ming-Hsien Tu and Jiin-Chang Shaw | Hamiltonian Structures of Generalized Manin-Radul Super KdV and
Constrained Super KP Hierarchies | 16 pages, Revtex, no figures | J. Math. Phys. 40 (1999) 3021 | 10.1063/1.532741 | null | solv-int nlin.SI | null | A study of Hamiltonian structures associated with supersymmetric Lax
operators is presented. Following a constructive approach, the Hamiltonian
structures of Inami-Kanno super KdV hierarchy and constrained modified super KP
hierarchy are investigated from the reduced supersymmetric Gelfand-Dickey
brackets. By applying a gauge transformation on the Hamiltonian structures
associated with these two nonstandard super Lax hierarchies, we obtain the
Hamiltonian structures of generalized Manin-Radul super KdV and constrained
super KP hierarchies. We also work out a few examples and compare them with the
known results.
| 2015-06-26 |
solv-int/9812018 | Antoine Balan | A. Balan, (Ecole Polytechnique) | The periodic Lax operators $\cL$ of the equations of Benney type II | 14 pages, nofigure. WITHDRAWN by the author | null | null | null | solv-int nlin.SI | null | This text has been withdrawn by the author.
| 2007-05-23 |