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solv-int/9812019 | David Gomez-Ullate | D. Gomez-Ullate, S. Lafortune and P. Winternitz | Symmetries of Discrete Dynamical Systems Involving Two Species | 40 pages, no figures, typed in AMS-LaTeX | J. Math. Phys. 40 (1999) 2782-2804 | 10.1063/1.532728 | CRM-2567 | solv-int nlin.SI | null | The Lie point symmetries of a coupled system of two nonlinear
differential-difference equations are investigated. It is shown that in special
cases the symmetry group can be infinite dimensional, in other cases up to 10
dimensional. The equations can describe the interaction of two long molecular
chains, each involving one type of atoms.
| 2009-10-31 |
solv-int/9812020 | V. E. Vekslerchik | V.E. Vekslerchik | Functional representation of the Ablowitz-Ladik hierarchy. II | arxiv version is already official | J. Nonlinear Math. Phys. 9, no. 2 (2002) 157-180 | 10.2991/jnmp.2002.9.2.3 | null | solv-int nlin.SI | null | In this paper I continue studies of the functional representation of the
Ablowitz-Ladik hierarchy (ALH). Using formal series solutions of the
zero-curvature condition I rederive the functional equations for the
tau-functions of the ALH and obtain some new equations which provide more
straightforward description of the ALH and which were absent in the previous
paper. These results are used to establish relations between the ALH and the
discrete-time nonlinear Schrodinger equations, to deduce the superposition
formulae (Fay's identities) for the tau-functions of the hierarchy and to
obtain some new results related to the Lax representation of the ALH and its
conservation laws. Using the previously found connections between the ALH and
other integrable systems I derive functional equations which are equivalent to
the AKNS, derivative nonlinear Schrodinger and Davey-Stewartson hierarchies.
| 2012-11-09 |
solv-int/9812021 | Maillet Jean Michel | A.G. Izergin, N. Kitanine, J. M. Maillet, V. Terras | Spontaneous magnetization of the XXZ Heisenberg spin-1/2 chain | 18 pages, Latex2e | Nucl. Phys. B 554 (1999) 679-696 | 10.1016/S0550-3213(99)00273-4 | LPENSL-TH-13/98 | solv-int nlin.SI | null | Determinant representations of form factors are used to represent the
spontaneous magnetization of the Heisenberg XXZ chain (Delta >1) on the finite
lattice as the ratio of two determinants. In the thermodynamic limit (the
lattice of infinite length), the Baxter formula is reproduced in the framework
of Algebraic Bethe Ansatz. It is shown that the finite size corrections to the
Baxter formula are exponentially small.
| 2018-08-30 |
solv-int/9812022 | Adrian-Stefan Carstea | A.S.Carstea | Extension of the bilinear formalism to supersymmetric KdV-type equations | 11 pages, revtex, no figures, some corrected typos | null | null | null | solv-int nlin.SI | null | Extending the gauge-invariance principle for \tau functions of the standard
bilinear formalism to the supersymmetric case, we define N=1 supersymmetric
Hirota operators. Using them, we bilinearize SUSY KdV-type equations (KdV,
Sawada-Kotera, Hirota-Satsuma). The solutions for multiple collisions of
super-solitons and extension to SUSY sine-Gordon are also discussed.
| 2007-05-23 |
solv-int/9812023 | Yavuz Nutku | Y. Nutku | Hamiltonian structure of real Monge-Amp\`ere equations | published in J. Phys. A 29 (1996) 3257 | null | 10.1088/0305-4470/29/12/029 | null | solv-int nlin.SI | null | The real homogeneous Monge-Amp\`{e}re equation in one space and one time
dimensions admits infinitely many Hamiltonian operators and is completely
integrable by Magri's theorem. This remarkable property holds in arbitrary
number of dimensions as well, so that among all integrable nonlinear evolution
equations the real homogeneous Monge-Amp\`{e}re equation is distinguished as
one that retains its character as an integrable system in multi-dimensions.
This property can be traced back to the appearance of arbitrary functions in
the Lagrangian formulation of the real homogeneous Monge-Amp\`ere equation
which is degenerate and requires use of Dirac's theory of constraints for its
Hamiltonian formulation. As in the case of most completely integrable systems
the constraints are second class and Dirac brackets directly yield the
Hamiltonian operators. The simplest Hamiltonian operator results in the
Kac-Moody algebra of vector fields and functions on the unit circle.
| 2009-10-31 |
solv-int/9812024 | Lafortune | S. Lafortune, B. Grammaticos and A. Ramani | Discrete and Continuous Linearizable Equations | Plain Tex file, 14 pages, no figure | Physica A, 268, 129-141 (1999) | 10.1016/S0378-4371(99)00026-6 | null | solv-int nlin.SI | null | We study the projective systems in both continuous and discrete settings.
These systems are linearizable by construction and thus, obviously, integrable.
We show that in the continuous case it is possible to eliminate all variables
but one and reduce the system to a single differential equation. This equation
is of the form of those singled-out by Painlev\'e in his quest for integrable
forms. In the discrete case, we extend previous results of ours showing that,
again by elimination of variables, the general projective system can be written
as a mapping for a single variable. We show that this mapping is a member of
the family of multilinear systems (which is not integrable in general). The
continuous limit of multilinear mappings is also discussed.
| 2015-06-26 |
solv-int/9812025 | Fritz Gesztesy | Fritz Gesztesy and Helge Holden | The Cole-Hopf and Miura transformations revisited | LaTeX, 11 pages | null | null | null | solv-int nlin.SI | null | An elementary yet remarkable similarity between the Cole-Hopf transformation
relating the Burgers and heat equation and Miura's transformation connecting
the KdV and mKdV equations is studied in detail.
| 2007-05-23 |
solv-int/9812026 | Fritz Gesztesy | Fritz Gesztesy and Helge Holden | The classical Boussinesq hierarchy revisited | LaTeX, 17 pages | null | null | null | solv-int nlin.SI | null | We develop a systematic approach to the classical Boussinesq (cBsq) hierarchy
based on an elementary polynomial recursion formalism. Moreover, the gauge
equivalence between the cBsq and AKNS hierarchies is studied in detail and used
to provide an effortless derivation of algebro-geometric solutions and their
theta function representations of the cBsq hierarchy.
| 2007-05-23 |
solv-int/9812027 | Nugmanova G. N. | F.B.Altynbaeva, A.K.Danlybaeva, G.N.Nugmanova and R.N.Syzdykova | On some soliton equations in 2+1 dimensions and their 1+1 and/or 2+0
dimensional integrable reductions | 18 pages, Latex, no figures | null | null | null | solv-int nlin.SI | null | Some soliton equation in 2+1 dimensions and their 1+1 and/or dimensional
integrable reductions are considered.
| 2007-05-23 |
solv-int/9812028 | Nikita A. Slavnov | N. A. Slavnov (Steklov Mathematical Institute, Moscow, Russia) | Integral equations for the correlation functions of the quantum
one-dimensional Bose gas | 22 pages, Latex, no figures | null | 10.1007/BF02557233 | MI-98-91 | solv-int nlin.SI | null | The large time and long distance behavior of the temperature correlation
functions of the quantum one-dimensional Bose gas is considered. We obtain
integral equations, which solutions describe the asymptotics. These equations
are closely related to the thermodynamic Bethe Ansatz equations. In the low
temperature limit the solutions of these equations are given in terms of
observables of the model.
| 2009-10-31 |
solv-int/9812029 | Lin Runliang | Yunbo Zeng, Runliang Lin and Xin Cao (Tsinghua University, Beijing,
P.R. China) | The relation between the Toda hierarchy and the KdV hierarchy | 11 pages, Tex, no figures, to be published in Physics Letters A | null | 10.1016/S0375-9601(98)00886-X | null | solv-int nlin.SI | null | Under three relations connecting the field variables of Toda flows and that
of KdV flows, we present three new sequences of combination of the equations in
the Toda hierarchy which have the KdV hierarchy as a continuous limit. The
relation between the Poisson structures of the KdV hierarchy and the Toda
hierarchy in continuous limit is also studied.
| 2009-10-31 |
solv-int/9812030 | Andrey V. Tsiganov | A.V. Tsiganov | The Lax pairs for the Holt system | 7 pages, LaTeX2e, a4.sty | J. Phys. A, Math. Gen. 32, No.45, 7983-7987, (1999) | 10.1088/0305-4470/32/45/312 | null | solv-int nlin.SI | null | By using non-canonical transformation between the Holt system and the
Henon-Heiles system the Lax pairs for all the integrable cases of the Holt
system are constructed from the known Lax representations for the Henon-Heiles
system.
| 2009-10-31 |
solv-int/9812031 | O. B. Zaslavskii | O.B.Zaslavskii (Department of Physics, Kharkov State University) | Two- and Many-Dimensional Quasi-Exactly Solvable Models With An
Inhomogeneous Magnetic Field | 7 pages, ReVTeX. Talk given at the 22nd International Colloqium for
Group Theoretical Methods in Physics | `Group22: Proceedings of the XXII International Colloquium on
Group Theoretical Methods in Physics', Eds S P Corney, R Delbourgo and P D
Jarvis (Cambridge, MA: International Press) 1998, pp.234-238 | null | null | solv-int hep-th math-ph math.MP math.SP nlin.SI quant-ph | null | Let group generators having finite-dimensional representation be realized as
Hermitian linear differential operators without nhomogeneous terms as takes
place, for example, for the SO(n) group. Then orresponding group Hamiltonians
containing terms linear in generators (along with quadratic ones) give rise to
quasi-exactly solvable models with a magnetic field in a curved space. In
particular, in the two-dimensional case such models are generated by quantum
tops. In the three-dimensional one for the SO(4) Hamiltonian with an isotropic
quadratic part the manifold within which a quantum particle moves has the
geometry of the Einstein universe.
| 2007-05-23 |
solv-int/9901001 | Jeremy Schiff | Michael Fisher, Jeremy Schiff | The Camassa-Holm Equation: Conserved Quantities and the Initial Value
Problem | 8 pages, LaTeX | null | 10.1016/S0375-9601(99)00466-1 | null | solv-int nlin.SI | null | Using a Miura-Gardner-Kruskal type construction, we show that the
Camassa-Holm equation has an infinite number of local conserved quantities. We
explore the implications of these conserved quantities for global
well-posedness.
| 2009-10-31 |
solv-int/9901002 | Marcio J. Martins | M.J. Martins | Unified algebraic Bethe ansatz for two-dimensional lattice models | plain latex, 9 pages | null | 10.1103/PhysRevE.59.7220 | UFSCARF-TH-98-33 | solv-int nlin.SI | null | We develop a unified formulation of the quantum inverse scattering method for
lattice vertex models associated to the non-exceptional $A^{(2)}_{2r}$,
$A^{(2)}_{2r-1}$, $B^{(1)}_r$, $C^{(1)}_r$, $D^{(1)}_{r+1}$ and $D^{(2)}_{r+1}$
Lie algebras. We recast the Yang-Baxter algebra in terms of novel commutation
relations between creation, annihilation and diagonal fields. The solution of
the $D^{(2)}_{r+1}$ model is based on an interesting sixteen-vertex model which
is solvable without recourse to a Bethe ansatz.
| 2009-10-31 |
solv-int/9901003 | Craig A. Tracy | Craig A. Tracy and Harold Widom | Universality of the distribution functions of random matrix theory | 11 pages, 3 figures | Statistical Physics on the Eve of the 21st Century: In Honour of J
B McGuire on the Occasion of His 65th Birthday, eds. M. T. Batchelor and L.
T. Wille, World Scientific Pub., 1999, pgs. 230-239. | null | null | solv-int math-ph math.MP nlin.SI | null | This paper first surveys the connection of integrable systems of the Painleve
type to various distribution functions appearing in Wigner-Dyson random matrix
theory. A short discussion is then given of the appearance of these same
distributions in other areas of mathematics.
| 2007-05-23 |
solv-int/9901004 | Craig A. Tracy | Craig A. Tracy and Harold Widom | Airy Kernel and Painleve II | 14 pages, 1 figure. References updated in second version | in "Isomonodromic Deformations and Applications in Physics," eds.
A. Its and J. Harnad, CRM Proceedings & Lecture Notes, Vol. 31, Amer. Math.
Soc., Providence, 2002, pp. 85-98. | null | null | solv-int math-ph math.MP nlin.SI | null | We prove that the distribution function of the largest eigenvalue in the
Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in
terms of Painlev\'e II. Our goal is to concentrate on this important example of
the connection between random matrix theory and integrable systems, and in so
doing to introduce the newcomer to the subject as a whole. We also give
sketches of the results for the limiting distribution of the largest eigenvalue
in the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble
(GSE). This work we did some years ago in a more general setting. These notes,
therefore, are not meant for experts in the field.
| 2007-05-23 |
solv-int/9901005 | Sergei Yu. Sakovich | Sergei Yu. Sakovich | Coupled KdV equations of Hirota-Satsuma type | null | J. Nonlinear Math. Phys. 6 (1999) 255-262 | 10.2991/jnmp.1999.6.3.2 | null | solv-int math-ph math.AP math.MP nlin.SI | null | It is shown that the system of two coupled Korteweg-de Vries equations passes
the Painlev\'e test for integrability in nine distinct cases of its
coefficients. The integrability of eight cases is verified by direct
construction of Lax pairs, whereas for one case it remains unknown.
| 2016-09-08 |
solv-int/9901006 | Alfred Ramani | B. Grammaticos (Paris VII) and A. Ramani (Ecole Polytechnique) | The hunting for the discrete Painlev\'e VI is over | 6 pages, Plain-TeX | null | null | null | solv-int nlin.SI | null | We present the discrete, q-, form of the Painlev\'e VI equation written as a
three-point mapping and analyse the structure of its singularities. This
discrete equation goes over to P_{VI} at the continuous limit and degenerates
towards the discrete q-P_{V} through coalescence. It possesses special
solutions in terms of the q-hypergeometric function. It can bilinearised and,
under the appropriate assumptions, ultradiscretised. A new discrete form for
P_{V} is also obtained which is of difference type, in contrast with the
`standard' form of the discrete P_{V}. Finally, we present the `asymmetric'
form of q-P_{VI}$ as a system of two first-order mappings involving seven
arbitrary parameters.
| 2007-05-23 |
solv-int/9901007 | David H. Sattinger | R. Beals, D.H. Sattinger, and J. Szmigielski | Acoustic Scattering and the Extended Korteweg deVries hierarchy | 18 pages | Advances in Mathematics, vol 140, (1998), 190-206 | null | null | solv-int nlin.SI | null | The acoustic scattering operator on the real line is mapped to a
Schr\"odinger operator under the Liouville transformation. The potentials in
the image are characterized precisely in terms of their scattering data, and
the inverse transformation is obtained as a simple, linear quadrature. An
existence theorem for the associated Harry Dym flows is proved, using the
scattering method. The scattering problem associated with the Camassa-Holm
flows on the real line is solved explicitly for a special case, which is used
to reduce a general class of such problems to scattering problems on finite
intervals.
| 2007-05-23 |
solv-int/9901008 | Fritz Gesztesy | Fritz Gesztesy and Helge Holden | Darboux-type transformations and hyperelliptic curves | LaTeX, 27 pages | null | null | null | solv-int nlin.SI | null | We systematically study Darboux-type transformations for the KdV and AKNS
hierarchies and provide a complete account of their effects on hyperelliptic
curves associated with algebro-geometric solutions of these hierarchies.
| 2007-05-23 |
solv-int/9901009 | null | A.N.W.Hone (Roma Tre) | Exact solutions of the associated Camassa-Holm equation | 11 pages | null | null | null | solv-int nlin.SI | null | Recently the associated Camassa-Holm (ACH) equation, related to the
Fuchssteiner-Fokas-Camassa-Holm equation by a hodograph transformation, was
introduced by Schiff, who derived B\"{a}cklund transformations by a loop group
technique and used these to obtain some simple soliton and rational solutions.
We show how the ACH equation is related to Schr\"{o}dinger operators and the
KdV hierarchy, and use this connection to obtain exact solutions (rational and
N-soliton solutions). More generally, we show that solutions of ACH on a
constant background can be obtained directly from the tau-functions of known
solutions of the KdV hierarchy on a zero background. We also present exact
solutions given by a particular case of the third Painlev\'{e} transcendent.
| 2007-05-23 |
solv-int/9901010 | Hendry Izaac Elim | Freddy P. Zen and Hendry I. Elim | Multi-soliton Solution of the Integrable Coupled Nonlinear Schrodinger
Equation of Manakov Type | 15 pages, LaTeX2e, PACS 42.65Sf | null | null | null | solv-int hep-th math-ph math.DS math.MP nlin.PS nlin.SI patt-sol | null | The general multi-soliton solution of the integrable coupled nonlinear
Schrodinger equation (NLS) of Manakov type is investigated by using
Zakharov-Shabat (ZS) scheme. We get the bright and dark multi-soliton solution
using inverse scattering method of ZS scheme. Elastic and inelastic collision
of N-solitons solution of the equation are also discussed.
| 2007-05-23 |
solv-int/9901011 | F. Delduc | Fran\c{c}ois Delduc, L. Gallot | A note on the third family of N=2 supersymmetric KdV hierarchies | null | J. Nonlinear Math. Phys. 6 (1999), no. 3, 332-343 | 10.2991/jnmp.1999.6.3.8 | JNMP 4/2002 (Article) | solv-int nlin.SI | null | We propose a hamiltonian formulation of the $N=2$ supersymmetric KP type
hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic
hamiltonian structure which allows for several reductions of the KP type
hierarchy. In particular, the third family of $N=2$ KdV hierarchies is
recovered. We also give an easy construction of Wronskian solutions of the KP
and KdV type equations.
| 2015-06-26 |
solv-int/9902001 | Takeo Kojima | N. Fukushima (Waseda Univ.), T. Kojima (Nihon Univ.) | Spontaneous polarization of the Kondo problem associated with the
higher-spin six-vertex model | 25 pages, LaTEX2e | J.Phys.A:Math.Gen.32,(1999) 6149-6168 | 10.1088/0305-4470/32/34/304 | null | solv-int hep-th nlin.SI | null | We study the multi-channel Kondo model associated with an integrable
higher-spin analogue of the anti-ferroelectric six-vertex model, which is
constructed by inserting spin 1/2 to spin 1 lines: $... C^3 \otimes C^3 \otimes
C^2 \otimes C^3 \otimes C^3 ... $. We formulate the problem in terms of
representation theory of quantum affine algebra $U_q(hat{sl}_2)$. We derive an
exact formula for the spontaneous staggered polarization for our model, which
corresponds to Baxter`s formula for the six-vertex model.
| 2009-10-31 |
solv-int/9902002 | Artur Sergyeyev | Artur G. Sergyeyev (= Arthur G. Sergheyev) (Institute of Mathematics
of NAS of Ukraine, Kyiv) | On time-dependent symmetries and formal symmetries of evolution
equations | 7 pages, Latex, no figures | Symmetry and perturbation theory (Rome, 1998), 303-308, World Sci.
Publ., River Edge, NJ, 1999 | null | null | solv-int math-ph math.AP math.MP nlin.SI | null | We present the explicit formulae, describing the structure of symmetries and
formal symmetries of any scalar (1+1)-dimensional evolution equation. Using
these results, the formulae for the leading terms of commutators of two
symmetries and two formal symmetries are found. The generalization of these
results to the case of system of evolution equations is also discussed.
| 2017-09-29 |
solv-int/9902003 | Yuri B. Suris | Yuri B. Suris (TU Berlin) | Miura transformations for Toda--type integrable systems, with
applications to the problem of integrable discretizations | LaTeX, 58 pp | null | null | null | solv-int nlin.SI | null | We study lattice Miura transformations for the Toda and Volterra lattices,
relativistic Toda and Volterra lattices, and their modifications. In
particular, we give three successive modifications for the Toda lattice, two
for the Volterra lattice and for the relativistic Toda lattice, and one for the
relativistic Volterra lattice. We discuss Poisson properties of the Miura
transformations, their permutability properties, and their role as localizing
changes of variables in the theory of integrable discretizations.
| 2007-05-23 |
solv-int/9902004 | Oleg Kiselev | O.M.Kiselev, B.I.Suleimanov (Institute of Mathematics, Ufa Science
Centre, Russian Acad. of Sciences) | The solution of the Painleve equations as special functions of
catastrophes, defined by a rejection in these equations of terms with
derivative | Latex, 15 pages | null | null | null | solv-int nlin.SI | null | The relation between the Painleve equations and the algebraic equations with
the catastrophe theory point of view are considered. The asymptotic solutions
with respect to the small parameter of the Painleve equations different types
are discussed. The qualitative analysis of the relation between algebraic and
fast oscillating solutions is done for Painleve-2 as an example.
| 2009-09-25 |
solv-int/9902005 | Leon Jerome | J. Leon, M. Manna | Multiscale Analysis of Discrete Nonlinear Evolution Equations | published in J. Phys. A : Math. Gen. 32 (1999) 927-943 | null | 10.1088/0305-4470/32/15/012 | null | solv-int nlin.SI | null | The method of multiscale analysis is constructed for dicrete systems of
evolution equations for which the problem is that of the far behavior of an
input boundary datum. Discrete slow space variables are introduced in a general
setting and the related finite differences are constructed. The method is
applied to a series of representative examples: the Toda lattice, the nonlinear
Klein-Gordon chain, the Takeno system and a discrete version of the
Benjamin-Bona-Mahoney equation. Among the resulting limit models we find a
discrete nonlinear Schroedinger equation (with reversed space-time), a 3-wave
resonant interaction system and a discrete modified Volterra model.
| 2009-10-31 |
solv-int/9902006 | Monica Ugaglia | Monica Ugaglia | On the Hamiltonian and Lagrangian structures of time-dependent
reductions of evolutionary PDEs | 46 pages, Tex, to be published in "Differential Geometry and
Applications" | null | null | null | solv-int nlin.SI | null | In this paper we study the reductions of evolutionary PDEs on the manifold of
the stationary points of time-dependent symmetries. In particular we describe
how the finite dimensional Hamiltonian structure of the reduced system is
obtained from the Hamiltonian structure of the initial PDE and we construct the
time-dependent Hamiltonian function. We also present a very general Lagrangian
formulation of the procedure of reduction. As an application we consider the
case of the Painleve' equations PI, PII, PIII, PVI and also certain higher
order systems appeared in the theory of Frobenius manifolds.
| 2007-05-23 |
solv-int/9902007 | Oleg Kiselev | O.M.Kiselev (Institute of Mathematics, Ufa Science Centre, Russian
Acad. of Sciences) | Asymptotic approach for the rigid condition of appearance of the
oscillations in the solution of the Painleve-2 equation | Latex, 18 pages | null | null | null | solv-int nlin.SI | null | The asymptotic solution for the Painleve-2 equation with small parameter is
considered. The solution has algebraic behavior before point $t_*$ and fast
oscillating behavior after the point $t_*$. In the transition layer the
behavior of the asymptotic solution is more complicated. The leading term of
the asymptotics satisfies the Painleve-1 equation and some elliptic equation
with constant coefficients, where the solution of the Painleve-1 equation has
poles. The uniform smooth asymptotics are constructed in the interval,
containing the critical point $t_*$.
| 2009-09-25 |
solv-int/9902008 | Leonid Dickey | L.A.Dickey (Univ. of Oklahoma) | Modified KP and Discrete KP | LaTeX, 11 pages | null | null | null | solv-int nlin.SI | null | The discrete KP, or 1-Toda lattice hierarchy is the same as a properly
defined modified KP hierarchy.
| 2007-05-23 |
solv-int/9902009 | Michael Baake | M. Baake, U. Grimm and R. J. Baxter | A critical Ising model on the Labyrinth | 25 pages, 6 figures | Int. J. Mod. Phys. B 8 (1994) 3579-3600 | 10.1142/S0217979294001512 | null | solv-int nlin.SI | null | A zero-field Ising model with ferromagnetic coupling constants on the
so-called Labyrinth tiling is investigated. Alternatively, this can be regarded
as an Ising model on a square lattice with a quasi-periodic distribution of up
to eight different coupling constants. The duality transformation on this
tiling is considered and the self-dual couplings are determined. Furthermore,
we analyze the subclass of exactly solvable models in detail parametrizing the
coupling constants in terms of four rapidity parameters. For those, the
self-dual couplings correspond to the critical points which, as expected,
belong to the Onsager universality class.
| 2015-06-26 |
solv-int/9902010 | Hendry Izaac Elim | Freddy P. Zen and Hendry I. Elim | Lax Pair Formulation and Multi-soliton Solution of the Integrable Vector
Nonlinear Schrodinger Equation | 11 pages, LaTeX2.09 | null | null | null | solv-int adap-org hep-th math-ph math.DS math.MP nlin.AO nlin.PS nlin.SI patt-sol | null | The integrable vector nonlinear Schrodinger (vector NLS) equation is
investigated by using Zakharov-Shabat (ZS) scheme. We get a Lax pair
formulation of the vector NLS model. Multi-soliton solution of the equation is
also derived by using inverse scattering method of ZS scheme. We also find that
there is an elastic and inelastic collision of the bright and dark
multi-solitons of the system.
| 2016-09-08 |
solv-int/9902011 | Chie Bing Wang | Chie Bing Wang | Orthonormal Polynomials on the Unit Circle and Spatially Discrete
Painlev\'e II Equation | 16 pages | null | 10.1088/0305-4470/32/41/312 | null | solv-int nlin.SI | null | We consider the polynomials $\phi_n(z)= \kappa_n (z^n+ b_{n-1} z^{n-1}+
>...)$ orthonormal with respect to the weight $\exp(\sqrt{\lambda} (z+ 1/z))
dz/2 \pi i z$ on the unit circle in the complex plane. The leading coefficient
$\kappa_n$ is found to satisfy a difference-differential (spatially discrete)
equation which is further proved to approach a third order differential
equation by double scaling. The third order differential equation is equivalent
to the Painlev\'e II equation. The leading coefficient and second leading
coefficient of $\phi_n(z)$ can be expressed asymptotically in terms of the
Painlev\'e II function.
| 2009-10-31 |
solv-int/9902012 | John Harnad | J. Harnad and J. McKay (C.R.M., U. de Montreal and Concordia U.) | Modular Invariants and Generalized Halphen Systems | PlainTeX 15gs. Write - up of lecture by J. Harnad at International
meeting: SIDE III, Sabaudia, May 16-22, 1998. To appear in CRM Proceedings
and Lecture Notes series (1999/2000) | CRM Proceedings and Lecture Notes Series 25, Symmetries and
Integrability of Difference Equations, pp. 181- 195 (eds. Decio Levy and
Orlando Ragnisco, AMS, Providence R.I., 2000) | null | CRM 2597 (1999) | solv-int hep-th math-ph math.DS math.GR math.MP math.NT nlin.SI | null | Generalized Halphen systems are solved in terms of functions that uniformize
genus zero Riemann surfaces, with automorphism groups that are commensurable
with the modular group. Rational maps relating these functions imply subgroup
relations between their automorphism groups and symmetrization relations
between the associated differential systems.
| 2007-05-23 |
solv-int/9902013 | John Harnad | J. Harnad (C.R.M., U. de Montreal and Concordia U.) | Picard-Fuchs Equations, Hauptmoduls and Integrable Systems | PlainTeX 15gs. Write - up of lecture given at international meeting:
Integrability: Seiberg-Witten and Witham Equations, ICMS, Edinburgh,
September 14-19, 1998 | Integrability: The Seiberg-Witten and Witham Equations, Chapter 8,
pp. 137-152 ( Ed. H.W. Braden and I.M. Krichever, Gordon and Breach,
Amsterdam (2000)) | null | CRM 2596 (1999) | solv-int hep-th math-ph math.AG math.GR math.MP math.NT nlin.SI | null | The Schwarzian equations satisfied by certain Hauptmoduls (i.e., uniformizing
functions for Riemann surfaces of genus zero) are derived from the Picard-Fuchs
equations for families of elliptic curves and associated surfaces. The
inhomogeneous Picard-Fuchs equations associated to elliptic integrals with
varying endpoints are derived and used to determine solutions of equations that
are algebraically related to a class of Painlev\'e VI equations.
| 2007-05-23 |
solv-int/9902014 | Myrzakulov Ratbay | N.K.Bliev, G.N.Nugmanova, R.N.Syzdykova and R.Myrzakulov | Soliton equations in 2+1 dimensions: reductions, bilinearizations and
simplest solutions | 16 pages, Latex, no figures | null | null | CNLP-1997-05 | solv-int nlin.SI | null | Soliton equations in 2+1 and their 1+1 = 2+0 reductions are considered.
| 2007-05-23 |
solv-int/9902015 | S. E. Derkachov | S.E. Derkachov | Baxter's Q-operator for the homogeneous XXX spin chain | 23 pages, Latex | J.Phys.A32:5299-5316,1999 | 10.1088/0305-4470/32/28/309 | null | solv-int hep-th math-ph math.MP nlin.SI | null | Applying the Pasquier-Gaudin procedure we construct the Baxter's Q-operator
for the homogeneous XXX model as integral operator in standard representation
of SL(2). The connection between Q-operator and local Hamiltonians is
discussed. It is shown that operator of Lipatov's duality symmetry arises
naturally as leading term of the asymptotic expansion of Q-operator for large
values of spectral parameter.
| 2009-10-31 |
solv-int/9902016 | Vsevolod Adler | V.E. Adler, S.Ya. Startsev | Discrete analogues of the Liouville equation | LaTeX, 15 pages, submitted to Teor. i Mat. Fiz | Theoretical and Mathematical Physics 1999, Volume 121, Issue 2, pp
1484-1495 | 10.1007/BF02557219 | null | solv-int nlin.SI | null | The notion of Laplace invariants is transferred to the lattices and discrete
equations which are difference analogs of hyperbolic PDE's with two independent
variables. The sequence of Laplace invariants satisfy the discrete analog of
twodimensional Toda lattice. The terminating of this sequence by zeroes is
proved to be the necessary condition for existence of the integrals of the
equation under consideration. The formulae are presented for the higher
symmetries of the equations possessing integrals. The general theory is
illustrated by examples of difference analogs of Liouville equation.
| 2014-08-27 |
solv-int/9902017 | Nicolas Regnault | Denis Bernard, Nicolas Regnault (Saclay-CNRS) | Vertex Operator Solutions of 2d Dimensionally Reduced Gravity | 20 pages; added comparison with Belinskii-Zakharov method and remarks | Commun.Math.Phys.210:177-201,2000 | 10.1007/s002200050776 | null | solv-int gr-qc hep-th nlin.SI | null | We apply algebraic and vertex operator techniques to solve two dimensional
reduced vacuum Einstein's equations. This leads to explicit expressions for the
coefficients of metrics solutions of the vacuum equations as ratios of
determinants. No quadratures are left. These formulas rely on the
identification of dual pairs of vertex operators corresponding to dual metrics
related by the Kramer-Neugebauer symmetry.
| 2008-11-26 |
solv-int/9903001 | V. V. Mangazeev | H.E. Boos and V.V. Mangazeev | Functional relations and nested Bethe ansatz for sl(3) chiral Potts
model at q^2=-1 | 20 pages, 6 figures, to appear in J. Phys. A: Math. and Gen | null | 10.1088/0305-4470/32/16/012 | Math. Res. Report No. MRR 051-98, ANU | solv-int nlin.SI | null | We obtain the functional relations for the eigenvalues of the transfer matrix
of the sl(3) chiral Potts model for q^2=-1. For the homogeneous model in both
directions a solution of these functional relations can be written in terms of
roots of Bethe ansatz-like equations. In addition, a direct nested Bethe ansatz
has also been developed for this case.
| 2009-10-31 |
solv-int/9903002 | Nugzar Makhaldiani | Dumitru Baleanu and Nugzar Makhaldiani | Nambu--Poisson reformulation of the finite dimensional dynamical systems | 6 pages, latex, no figures | null | null | Communications of the JINR, Dubna, E2-98-348, 1998 | solv-int nlin.SI | null | In this paper we introduce a system of nonlinear ordinary differential
equations which in a particular case reduces to Volterra's system. We found in
two simplest cases the complete sets of the integrals of motion using
Nambu--Poisson reformulation of the Hamiltonian dynamics. In these cases we
have solved the systems by quadratures.
| 2007-05-23 |
solv-int/9903003 | Ziad Maassarani | Z. Maassarani (University of Virginia) | New Integrable Models from Fusion | 11 pages, Latex. v2: statement concerning symmetries qualified, 3
minor misprints corrected. J. Phys. A (1999) in press | J.Phys.A32:5123-5132,1999 | 10.1088/0305-4470/32/27/310 | null | solv-int cond-mat math-ph math.MP nlin.SI | null | Integrable multistate or multiflavor/color models were recently introduced.
They are generalizations of models corresponding to the defining
representations of the U_q(sl(m)) quantum algebras. Here I show that a similar
generalization is possible for all higher dimensional representations. The
R-matrices and the Hamiltonians of these models are constructed by fusion. The
sl(2) case is treated in some detail and the spin-0 and spin-1 matrices are
obtained in explicit forms. This provides in particular a generalization of the
Fateev-Zamolodchikov Hamiltonian.
| 2008-11-26 |
solv-int/9903004 | V. E. Vekslerchik | V.E. Vekslerchik | Finite genus solutions for the Ablowitz-Ladik hierarchy | 14 pages, LaTeX 2e | Journal of Physics A, 32 (1999) 4983 | 10.1088/0305-4470/32/26/316 | null | solv-int nlin.SI | null | The question of constructing the finite genus quasiperiodic solutions for the
Ablowitz-Ladik hierarchy (ALH) is considered by establishing relations between
the ALH and the Fay's identity for the theta-functions. It is shown that using
a limiting procedure one can derive from the latter an infinite number of
differential identities which can be arranged as an infinite set of
differential-difference equations coinciding with the equations of the ALH, and
that the original Fay's identity can be rewritten in a form similar to the
functional equations representing the ALH which have been derived in the
previous works of the author. This provides an algorithm for obtaining some
class of quasiperiodic solutions for the ALH, which can be viewed as an
alternative to the inverse scattering transform or the algebro-geometrical
approach.
| 2012-11-09 |
solv-int/9903005 | Pierre van Moerbeke | M. Adler, E. Horozov, P. van Moerbeke | The Pfaff lattice and skew-orthogonal polynomials | 21 pages | Intern. Math. Research Notices, 1999 | null | null | solv-int nlin.SI | null | Consider a semi-infinite skew-symmetric moment matrix, $m_{\iy}$ evolving
according to the vector fields $\pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} ,$ where
$\Lb$ is the shift matrix. Then the skew-Borel decomposition $ m_{\iy}:= Q^{-1}
J Q^{\top -1} $ leads to the so-called Pfaff Lattice, which is integrable, by
virtue of the AKS theorem, for a splitting involving the affine symplectic
algebra. The tau-functions for the system are shown to be pfaffians and the
wave vectors skew-orthogonal polynomials; we give their explicit form in terms
of moments. This system plays an important role in symmetric and symplectic
matrix models and in the theory of random matrices (beta=1 or 4).
| 2007-05-23 |
solv-int/9903006 | Marcio J. Martins | M.J. Martins | Integrable mixed vertex models from braid-monoid algebra | To appear in the Festschriff in honor of Prof. Mc Guire published by
World Scientific, plain latex, 10 pages | null | null | UFSCARF-TH-98-12 | solv-int nlin.SI | null | We use the braid-monoid algebra to construct integrable mixed vertex models.
The transfer matrix of a mixed SU(N) model is diagonalized by nested Bethe
ansatz approach.
| 2007-05-23 |
solv-int/9903007 | Oliver B. Fringer | O. B. Fringer (1,2) and D. D. Holm (2) ((1) Stanford University, (2)
Los Alamos National Laboratory) | Integrable vs Nonintegrable Geodesic Soliton Behavior | 49 pages, 29 figures, animations at:
http://rossby.stanford.edu/~fringer/pulsons | null | null | null | solv-int nlin.SI | null | We study confined solutions of certain evolutionary partial differential
equations (pde) in 1+1 space-time. The pde we study are Lie-Poisson Hamiltonian
systems for quadratic Hamiltonians defined on the dual of the Lie algebra of
vector fields on the real line. These systems are also Euler-Poincare equations
for geodesic motion on the diffeomorphism group in the sense of the Arnold
program for ideal fluids, but where the kinetic energy metric is different from
the L2 norm of the velocity. These pde possess a finite-dimensional invariant
manifold of particle-like (measure-valued) solutions we call ``pulsons.'' We
solve the particle dynamics of the two-pulson interaction analytically as a
canonical Hamiltonian system for geodesic motion with two degrees of freedom
and a conserved momentum. The result of this two-pulson interaction for
rear-end collisions is elastic scattering with a phase shift, as occurs with
solitons. In contrast, head-on antisymmetric collisons of pulsons tend to form
singularities.
| 2007-05-23 |
solv-int/9903008 | Nugmanova G. N. | R.Myrzakulov | Singularity Structure Analysis, Integrability, Solitons and Dromions in
(2+1)-Dimensional Zakharov Equations | 15 pages, Latex, no figures | null | null | CNLP-1997-03 | solv-int nlin.SI | null | The (2+1)-dimensional integrable Zakharov equations and their reductions are
considered
| 2007-05-23 |
solv-int/9903009 | Pierre van Moerbeke | M. Adler & P. van Moerbeke | Symmetric random matrices and the Pfaff lattice | 44 pages | null | null | null | solv-int nlin.SI | null | Consider a symmetric (finite) matrix ensemble, with a certain probability
distribution. What is the probability that the spectrum belongs to a certain
interval or union of intervals on the real line? In this paper, we show that,
upon introducing an appropriate time parameter, this probability is intimately
related to Pfaffians, which as a vector satisfy the so-called Pfaff lattice.
The latter is a particular reduction of the 2d-Toda lattice. In particular,
they satisfy a KP-like equation, but with a right hand side, depending on
nearest neighbors. They also satisfy Virasoro constraints, which combined with
the KP-like equation lead to inductive equations for the probabilities.
| 2007-05-23 |
solv-int/9903010 | V. V. Mangazeev | H.E. Boos and V.V. Mangazeev | Bethe ansatz for the three-layer Zamolodchikov model | 22 pages, LaTeX, 5 figures | null | 10.1088/0305-4470/32/28/308 | null | solv-int nlin.SI | null | This paper is a continuation of our previous work (solv-int/9903001). We
obtain two more functional relations for the eigenvalues of the transfer
matrices for the $sl(3)$ chiral Potts model at $q^2=-1$. This model, up to a
modification of boundary conditions, is equivalent to the three-layer
three-dimensional Zamolodchikov model. From these relations we derive the Bethe
ansatz equations.
| 2009-10-31 |
solv-int/9903011 | David H. Sattinger | R. Beals, D.H. Sattinger, and J. Szmigielski | Multipeakons and a theorem of Stieltjes | 6 pages | Inverse Problems, volume 15 (1999), Letters, L1-L4 | 10.1088/0266-5611/15/1/001 | null | solv-int nlin.SI | null | A closed form of the multi-peakon solutions of the Camassa-Holm equation is
found using a theorem of Stieltjes on continued fractions. An explicit formula
is obtained for the scattering shifts.
| 2009-10-31 |
solv-int/9903012 | Daisuke Takahashi | Daisuke Takahashi and Kenji Kajiwara | On integrability test for ultradiscrete equations | 10 pages including 5 figures | null | null | null | solv-int nlin.SI | null | We consider an integrability test for ultradiscrete equations based on the
singularity confinement analysis for discrete equations. We show how
singularity pattern of the test is transformed into that of ultradiscrete
equation. The ultradiscrete solution pattern can be interpreted as a perturbed
solution. We can also check an integrability of a given equation by a
perturbation growth of a solution, namely Lyapunov exponent. Therefore,
singularity confinement test and Lyapunov exponent are related each other in
ultradiscrete equations and we propose an integrability test from this point of
view.
| 2007-05-23 |
solv-int/9903013 | Takayuki Tsuchida | T. Tsuchida, H. Ujino, M. Wadati (University of Tokyo) | Integrable semi-discretization of the coupled nonlinear Schr\"{o}dinger
equations | 27 pages, LaTeX2e (IOP style) | J. Phys. A: Math. Gen. 32 (1999) 2239-2262 | 10.1088/0305-4470/32/11/016 | null | solv-int nlin.SI | null | A system of semi-discrete coupled nonlinear Schr\"{o}dinger equations is
studied. To show the complete integrability of the model with multiple
components, we extend the discrete version of the inverse scattering method for
the single-component discrete nonlinear Schr\"{o}dinger equation proposed by
Ablowitz and Ladik. By means of the extension, the initial-value problem of the
model is solved. Further, the integrals of motion and the soliton solutions are
constructed within the framework of the extension of the inverse scattering
method.
| 2007-05-23 |
solv-int/9903014 | Masuda | K. Kajiwara, T. Masuda (U. of Doshisha) | A generalization of determinant formulas for the solutions of Painlev\'e
II and XXXIV equations | 20 pages, LaTeX 2.09(IOP style), submitted to J. Phys. A | null | 10.1088/0305-4470/32/20/309 | null | solv-int nlin.SI | null | A generalization of determinant formulas for the classical solutions of
Painlev\'e XXXIV and Painlev\'e II equations are constructed using the
technique of Darboux transformation and Hirota's bilinear formalism. It is
shown that the solutions admit determinant formulas even for the transcendental
case.
| 2009-10-31 |
solv-int/9903015 | Masuda | K. Kajiwara, T. Masuda (U. of Doshisha) | On the Umemura Polynomials for the Painlev\'e III equation | 10 pages, LaTeX 2.09(Elsevier style), submitted to Phys. Lett. A | null | 10.1016/S0375-9601(99)00577-0 | null | solv-int nlin.SI | null | A determinant expression for the rational solutions of the Painlev\'e III
(P$_{\rm III}$) equation whose entries are the Laguerre polynomials is given.
Degeneration of this determinant expression to that for the rational solutions
of P$_{\rm II}$ is discussed by applying the coalescence procedure.
| 2009-10-31 |
solv-int/9903016 | Evgueni Sklyanin | E. K. Sklyanin | Canonicity of Baecklund transformation: r-matrix approach. I | 7 pages, LATEX-2e, macros included | Translations of the American Mathematical Society-Series 2, 201
(2000) 277-282 | null | null | solv-int nlin.SI | null | For the Hamiltonian integrable systems governed by SL(2)-invariant r-matrix
(such as Heisenberg magnet, Toda lattice, nonlinear Schroedinger equation) a
general procedure for constructing Baecklund transformation is proposed. The
corresponding BT is shown to preserve the Poisson bracket. The proof is given
by a direct calculation using the r-matrix expression for the Poisson bracket.
| 2015-11-12 |
solv-int/9903017 | Evgueni Sklyanin | E. K. Sklyanin | Canonicity of Baecklund transformation: r-matrix approach. II | 7 pages, LATEX 2e, macros included | Proceedings of the Steklov Institute of Mathematics 226 (1999)
121-126 | null | null | solv-int nlin.SI | null | This is the second part of the paper devoted to the general proof of
canonicity of Baecklund transformation (BT) for a Hamiltonian integrable system
governed by SL(2)-invariant r-matrix. Introducing an extended phase space from
which the original one is obtained by imposing a 1st kind constraint, we are
able to prove the canonicity of BT in a new way. The new proof allows to
explain naturally the fact why the gauge transformation matrix M associated to
the BT has the same structure as the Lax operator L. The technique is
illustrated on the example of the DST chain.
| 2015-11-11 |
solv-int/9904001 | Bernard Deconinck | Bernard Deconinck and Harvey Segur | Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation | 22 pages, 12 figures. Submitted for publication | null | null | MSRI 1999-020 | solv-int nlin.SI | null | The real, nonsingular elliptic solutions of the Korteweg-deVries equation are
studied through the time dynamics of their poles in the complex plane. The
dynamics of these poles is governed by a dynamical system with a constraint.
This constraint is shown to be solvable for any finite number of poles located
in the fundamental domain of the elliptic function, often in many different
ways. Special consideration is given to those elliptic solutions that have a
real nonsingular soliton limit.
| 2007-05-23 |
solv-int/9904002 | Pilar G. Estevez | P. G. Estevez (Universidad de Salamanca, Spain), P.A. Clarkson
(University of Kent at Canterbury, UK) | Discrete equations and the singular manifold method | to appear in SIDE III proceedings | null | null | null | solv-int nlin.SI | null | The Painleve expansion for the second Painleve equation (PII) and fourth
Painleve equation (PIV) have two branches. The singular manifold method
therefore requires two singular manifolds. The double singular manifold method
is used to derive Miura transformations from PII and PIV to modified Painleve
type equations for which auto-Backlund transformations are obtained. These
auto-Backlund transformations can be used to obtain discrete equations.
| 2007-05-23 |
solv-int/9904003 | Vadim Kuznetsov | A.N.W. Hone, V.B. Kuznetsov, O. Ragnisco | Backlund transformations for many-body systems related to KdV | LaTeX2e, 8 pages | J.Phys. A32 (1999) L299-L306 | 10.1088/0305-4470/32/27/102 | null | solv-int hep-th math-ph math.MP nlin.SI | null | We present Backlund transformations (BTs) with parameter for certain
classical integrable n-body systems, namely the many-body generalised
Henon-Heiles, Garnier and Neumann systems. Our construction makes use of the
fact that all these systems may be obtained as particular reductions
(stationary or restricted flows) of the KdV hierarchy; alternatively they may
be considered as examples of the reduced sl(2) Gaudin magnet. The BTs provide
exact time-discretizations of the original (continuous) systems, preserving the
Lax matrix and hence all integrals of motion, and satisfy the spectrality
property with respect to the Backlund parameter.
| 2009-10-31 |
solv-int/9904004 | Andrei Maltsev | B.A.Dubrovin, A.Ya.Maltsev | Recurrent procedure for the determination of the Free Energy
$\epsilon^{2}$-expansion in the Topological String Theory | 18 pages, Latex | null | null | null | solv-int hep-th nlin.SI | null | We present here the iteration procedure for the determination of free energy
$\epsilon^{2}$-expansion using the theory of KdV - type equations. In our
approach we use the conservation laws for KdV - type equations depending
explicitly on times $t_{1}, t_{2}, ...$ to find the $\epsilon^{2}$-expansion of
$u(x,t_{1},t_{2},...)$ after the infinite number of shifts of $u(x,0,0,...)
\equiv x$ along $t_{1}, t_{2}, ...$ in recurrent form. The formulas for the
free energy expansion are just obtained then as a result of quite simple
integration procedure applied to $u_{n}(x)$.
| 2007-05-23 |
solv-int/9904005 | Igor G. Korepanov | I.G. Korepanov | The tetrahedral analog of Veneziano amplitude | LaTeX, 7 pages. v2: some minor editing | null | null | null | solv-int nlin.SI | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ | In solv-int/9812016 it was shown that the Veneziano amplitude in string
theory comes naturally from one of the simplest solutions of the functional
pentagon equation (FPE). More generally, FPE is intimately connected with the
duality condition for scattering processes. Here I find the amplitude that
comes the same way from a solution of the functional tetrahedron equation, with
the duality replaced by the local Yang - Baxter equation.
| 2009-11-30 |
solv-int/9904006 | R. Radhakrishnan | R. Radhakrishnan and M. Lakshmanan | Suppression and Enhancement of Soliton Switching During Interaction in
Periodically Twisted Birefringent Fiber | 10 pages, 4 figures, Latex, submitted to Phys. Rev. E | null | 10.1103/PhysRevE.60.2317 | null | solv-int nlin.SI | null | Soliton interaction in periodically twisted birefringent optical fibers has
been analysed analytically with refernce to soliton switching. For this purpose
we construct the exact general two-soliton solution of the associated coupled
system and investigate its asymptotic behaviour. Using the results of our
analytical approach we point out that the interaction can be used as a switch
to suppress or to enhance soliton switching dynamics, if one injects
multi-soliton as an input pulse in the periodically twisted birefringent fiber.
| 2009-10-31 |
solv-int/9904007 | null | Q-Han Park and H.J. Shin(Kyunghee U.) | Complex sine-Gordon Equation in Coherent Optical Pulse Propagation | null | Journal of Korean Physical Society, Vol. 30; 336-340, 1997 | null | null | solv-int nlin.SI | null | It is shown that the McCall-Hahn theory of self-induced transparency in
coherent optical pulse propagation can be identified with the complex
sine-Gordon theory in the sharp line limit. We reformulate the theory in terms
of the deformed gauged Wess-Zumino-Witten sigma model and address various new
aspects of self-induced transparency.
| 2007-05-23 |
solv-int/9904008 | null | Jongbae Kim (ETRI), Q-Han Park and H.J. Shin (Kyunghee U.) | Conservation Laws in Higher-Order Nonlinear Optical Effects | null | Phys. Rev. E {\bf 58} 6746, 1998 | 10.1103/PhysRevE.58.6746 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9904009 | null | Q-Han Park and H.J. Shin (Kyunghee U.) | Painlev\'{e} analysis of the coupled nonlinear Schr\"{o}dinger equation
for polarized optical waves in an isotropic medium | null | Phys. Rev. E {\bf 59} 2373, 1999 | 10.1103/PhysRevE.59.2373 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9904010 | Baryakhtar | I.V.Baryakhtar, V.G.Baryakhtar, E.N.Economou | Kinetic and Transport Equations for Localized Excitations in Sine-Gordon
Model | 23 pages, latex, no figures | null | 10.1103/PhysRevE.60.6645 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9904011 | Nugmanova G. N. | R. Myrzakulov | On the M-XX equation | 9 pages, Latex, no figures | null | null | null | solv-int nlin.SI | null | The (2+1)-dimensional integrable M-XX equation is considered.
| 2007-05-23 |
solv-int/9904012 | Paul Fendley | P. Fendley and H. Saleur | Differential equations and duality in massless integrable field theories
at zero temperature | 18 pages, harvmac | Nucl.Phys.B574:571-586,2000 | 10.1016/S0550-3213(99)00832-9 | null | solv-int hep-th nlin.SI | null | 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.
| 2008-11-26 |
solv-int/9904013 | Sergei M. Sergeev | S. Sergeev | Solitons in a 3d integrable model | LaTeX, 6 pages | null | 10.1016/S0375-9601(99)00849-X | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9904014 | Robert Milson | Niky Kamran, Robert Milson, Peter Olver | Invariant Modules and the Reduction of Nonlinear Partial Differential
Equations to Dynamical Systems | 28 pages. To appear in Advances in Mathematics | null | null | null | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9904015 | Nikolay Asenov Kostov | N.A.Kostov, Z.T. Kostova | Nonlinear waves, differential resultant, computer algebra and completely
integrable dynamical systems | 33 pages, no figures | null | null | null | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9904016 | Nikolay Asenov Kostov | N.A. Kostov | Korteweg-de Vries hierarchy and related completely integrable systems:
I. Algebro-geometrical approach | 31 pages, no figures | null | null | Preprint TH-98/4, 1998, INRNE, BAS, Sofia | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9904017 | Nikolay Asenov Kostov | P.L. Christiansen, J.C. Eilbeck, V.Z. Enolskii, and N.A. Kostov | Quasi-Periodic and Periodic Solutions for Systems of Coupled Nonlinear
SCHR\"Odinger Equations | 21 pages, no figures | null | 10.1098/rspa.2000.0612 | TH/2 INRNE, BAS, Sofia | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9904018 | Sudipta Nandy | Sasanka Ghosh, Anjan Kundu, Sudipta Nandy | Soliton solutions, Liouville integrability and gauge equivalence of Sasa
Satsuma equation | 14 pages, to be published in J. Math. Phys. April-May, 1999 | null | 10.1063/1.532845 | null | solv-int nlin.SI | null | 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
| 2015-06-26 |
solv-int/9904019 | Sudipta Nandy | Sasanka Ghosh, Sudipta Nandy | Optical solitons in higher order nonlinear Schrodinger equation | 10 pages | null | null | null | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9904020 | Dr P. K. Panigrahi | C. Nagaraja Kumar and Prasanta K. Panigrahi (School of Physics,
University of Hyderabad, Hyderabad, India) | Compacton-like Solutions for Modified KdV and other Nonlinear Equations | 4 pages, RevTex | null | null | null | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9904021 | Sudipta Nandy | Sasanka Ghosh and Sudipta Nandy | Inverse scattering method and vector higher order nonlinear Schrodinger
equation | 28 pages | null | 10.1016/S0550-3213(99)00484-8 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9904022 | Unal Goktas | Willy Hereman (Colorado School of Mines), Unal Goktas (Wolfram
Research, Inc.) | Integrability Tests for Nonlinear Evolution Equations | uses 31x47jw.sty, chapter in: Computer Algebra Systems: A Practical
Guide (Ed. Michael Wester), Wiley and Sons, New York, in press | null | null | null | solv-int nlin.SI | null | Discusses several integrability tests for nonlinear evolution equations.
| 2007-05-23 |
solv-int/9904023 | Andrew Pickering | Pilar R. Gordoa, Nalini Joshi and Andrew Pickering | Mappings preserving locations of movable poles: a new extension of the
truncation method to ordinary differential equations | To appear in Nonlinearity (22 pages) | null | 10.1088/0951-7715/12/4/313 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9904024 | Svetlana Pacheva-Nissimov | Henrik Aratyn, Emil Nissimov and Svetlana Pacheva | Multi-Component Matrix KP Hierarchies as Symmetry-Enhanced Scalar KP
Hierarchies and Their Darboux-B"acklund Solutions | 10 pages, LaTeX209 | null | null | BGU-99/20/Apr-PH | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9905001 | Polterovich Iosif | Iosif Polterovich | From Agmon-Kannai expansion to Korteweg-de Vries hierarchy | 7 pages, AMS-LaTeX | null | null | null | solv-int math-ph math.MP nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9905002 | Katrina Elfrieda Hibberd | Katrina Hibberd, Itzhak Roditi, Jon Links and Angela Foerster | Bethe ansatz solution of the closed anisotropic supersymmetric U model
with quantum supersymmetry | 7 pages (revtex), minor modifications. To appear in Mod. Phys. Lett.
A | null | 10.1142/S021773230000013X | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9905003 | Alexander Turbiner | Alexander Turbiner, Pavel Winternitz | Solutions of Non-linear Differential and Difference Equations with
Superposition Formulas | 17 pages, REVTeX, submitted to Journ.Math.Phys | null | null | ICN-UNAM 99-03 (Mexico), CRM-2606 (Montreal) | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9905004 | Takayuki Tsuchida | T. Tsuchida, M. Wadati | New integrable systems of derivative nonlinear Schr\"{o}dinger equations
with multiple components | 15 pages, LaTeX209, to appear in Phys. Lett. A | Phys. Lett. A 257 (1999) 53-64 | 10.1016/S0375-9601(99)00272-8 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9905005 | Luis Martinez Alonso | Boris Konopelchenko and Luis Martinez Alonso | The KP Hierarchy in Miwa Coordinates | 14 pages Latex2e | null | 10.1016/S0375-9601(99)00373-4 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9905006 | R. Radhakrishnan | R. Radhakrishnan, A. Kundu and M. Lakshmanan | Coupled nonlinear Schrodinger equations with cubic-quintic nonlinearity:
Integrability and soliton interaction in non-Kerr media | 13 pages, 5 figures, LaTex, Submitted to Phys. Rev. E | null | 10.1103/PhysRevE.60.3314 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9905007 | Hubert Saleur | H. Saleur | The continuum limit of sl(N/K) integrable super spin chains | null | Nucl.Phys. B578 (2000) 552-576 | 10.1016/S0550-3213(00)00002-X | USC-99-002 | solv-int cond-mat hep-th nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9905008 | Ruslan Sharipov | R. F. Bikbaev, R. A. Sharipov | Magnetization waves in Landau-Lifshitz Model | AmSTeX, Ver. 2.1h, 5 pages, amsppt style, 1 figure | Phys. Lett. 134A (1988), no. 2, 105-107. | 10.1016/0375-9601(88)90943-7 | null | solv-int nlin.SI | null | 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.
| 2015-06-26 |
solv-int/9905009 | Takeo Kojima | H. Furutsu and T. Kojima (Nihon Univ.) | $U_q(\hat{sl}_n)$-analog of the XXZ chain with a boundary | 24 pages, LaTEX2e | J.Math.Phys. 41 (2000) 4413-4436 | 10.1063/1.533351 | null | solv-int hep-th nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9905010 | Nalini Joshi | Nalini Joshi, Johannes A. Petersen, and Luke M. Schubert | Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili
equations | 17 pages in LaTeX2e, to appear in Stud. Appl. Math | null | null | null | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9905011 | Askold Perelomov | L. Gavrilov (U. Paul Sabatier, Toulouse), A. Perelomov (MPIM Bonn) | On the explicit solutions of the elliptic Calogero system | 18 pages, Latex | null | 10.1063/1.533096 | null | solv-int nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9905012 | Doc. Dr. Ayse Humeyra Bilge | Ayse Humeyra Bilge | A System with a Recursion Operator but One Higher Local Symmetry of the
Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ | 7 pages, no figures | Lie Groups and Their Applications, Vol.1, No 2, pp.132-139, (1994) | null | null | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9905013 | H. J. S. Dorren | H.J.S. Dorren and J.J.B. van den Heuvel | On pulse broadening for optical solitons | 11 pages, the manuscript has undergone major revisions | null | null | null | solv-int nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9906001 | David H. Sattinger | Richard Beals, D.H. Sattinger, and J. Szmigielski | Multipeakons and the Classical Moment Problem | 32 pages, 3 figures; to appear in Advances in Mathematics | null | null | null | solv-int nlin.SI | null | 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
| 2007-05-23 |
solv-int/9906002 | David H. Sattinger | Yi Li and D.H. Sattinger | Soliton Collisions in the Ion Acoustic Plasma Equations | 13 pages, 3 figures | Journal of Mathematical Fluid Mechanics, volume 1, (1999), pp.
117-130 | 10.1007/s000210050006 | null | solv-int nlin.SI | null | 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
| 2009-10-31 |
solv-int/9906003 | Antonio Lima Santos | A. Lima-Santos | Reflection K-Matrices for 19-Vertex Models | 35 pages, LaTex | Nucl. Phys. B 558 [PM] 637-667 | 10.1016/S0550-3213(99)00456-3 | null | solv-int cond-mat.stat-mech hep-th nlin.SI | null | 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.
| 2009-10-31 |
solv-int/9906004 | John Harnad | J. Harnad (C.R.M., U. de Montreal and Concordia U.) | On the bilinear equations for Fredholm determinants appearing in random
matrices | arxiv version is already official | J. Nonlinear Math. Phys., volume 9, no. 4 (2002) 530-550 | null | null | solv-int hep-th math-ph math.MP nlin.SI | null | 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.
| 2007-05-23 |
solv-int/9906005 | Runliang Lin | Yunbo Zeng (1), Wen-Xiu Ma (2) ((1)Tsinghua University, Beijing,
China, (2) City University of Hong Kong, China) | Families of quai-bi-Hamiltonian systems and separability | 27 pages, Amstex, no figues, to be published in Journal of
Mathematical Physics | null | 10.1063/1.532979 | null | solv-int nlin.SI | null | 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.
| 2015-06-26 |