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At least 19 recordsLinked to original sources
arXiv2016

A construction of commuting systems of integrable symplectic birational maps. Lie-Poisson case

We give a construction of completely integrable ($2n$)-dimensional Hamiltonian systems with symplectic brackets of the Lie-Poisson type (linear in coordinates) and with quadratic Hamilton functions. Applying to any such system the so called Kahan-Hirota-Kimura discretization scheme, we arrive at a birational ($2n$)-dimensional map. We show that this map is symplectic with respect to a symplectic structure that is a perturbation of the original symplectic structure on $\mathbb R^{2n}$, and possesses $n$ independent integrals of motion, which are perturbations of the original Hamilton functions and are in involution with respect to the invariant symplectic structure. Thus, this map is completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original $n$-tuples of commuting vector fields, their Kahan-Hirota-Kimura discretizations also commute and share the invariant symplectic structure and the $n$ integrals of motion. This paper extends our previous ones, arXiv:1606.08238 [nlin.SI] and arXiv:1607.07085 [nlin.SI], where similar results were obtained for Hamiltonian systems with a constant (canonical) symplectic structure and cubic Hamilton functions.

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arXiv2007

N=2 supersymmetric unconstrained matrix GNLS hierarchies are consistent

We develop a pseudo-differential approach to the N=2 supersymmetric unconstrained matrix (k|n,m)-Generalized Nonlinear Schroedinger hierarchies and prove consistency of the corresponding Lax-pair representation (nlin.SI/0201026). Furthermore, we establish their equivalence to the integrable hierarchies derived in the super-algebraic approach of the homogeneously-graded loop superalgebra sl(2k+n|2k+m)\otimes C[{lambda},{lambda}^{-1}] (nlin.SI/0206037). We introduce an unconventional definition of N=2 supersymmetric strictly pseudo-differential operators so as to close their algebra among themselves.

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arXiv2009

Lagrangian multiform structure for the lattice KP system

We present a Lagrangian for the bilinear discrete KP (or Hirota-Miwa) equation. Furthermore, we show that this Lagrangian can be extended to a Lagrangian 3-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus we establish the multiform structure as proposed in arXiv:0903.4086v1 [nlin.SI] in a higher dimensional case.

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arXiv2012

Solutions to the modified Korteweg-de Vries equation

This is a continuation of Ref.[1](arXiv:nlin.SI/0603008). In the present paper we review solutions to the modified Korteweg-de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This is different from the case of the Korteweg-de Vries equation. We introduce an auxiliary matrix to deal with the complex operation and then we are able to give complete solution expressions for the matrix differential equation set. The obtained solutions to the modified Korteweg-de Vries equation can simply be categorized by two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg-de Vries equation in Wromskian form. This is derived with the help of the Galilean transformed modified Korteweg-de Vries equation. Finally, typical dynamics of the obtained solutions is analyzed and illustrated. We list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.

nlin.SI↗
arXiv2014

Integrability properties of some symmetry reductions

In our recent paper [H. Baran, I.S. Krasil'shchik, O.I. Morozov, P. Voj{č}{á}k, Symmetry reductions and exact solutions of Lax integrable $3$-dimensional systems, Journal of Nonlinear Mathematical Physics, Vol. 21, No. 4 (December 2014), 643--671; arXiv:1407.0246 [nlin.SI], DOI: 10.1080/14029251.2014.975532}], we gave a complete description of symmetry reduction of four Lax-integrable (i.e., possessing a zero-curvature representation with a non-removable parameter) $3$-dimensional equations. Here we study the behavior of the integrability features of the initial equations under the reduction procedure. We show that the ZCRs are transformed to nonlinear differential coverings of the resulting 2D-systems similar to the one found for the Gibbons-Tsarev equation in [A.V. Odesskii, V.V. Sokolov, Non-homogeneous systems of hydrodynamic type possessing Lax representations, arXiv:1206.5230, 2006]. Using these coverings we construct infinite series of (nonlocal) conservation laws and prove their nontriviality. We also show that the recursion operators are not preserved under reductions.

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arXiv2001

The affine $A_{n-1}^{(1)}$ Toda fields with boundary reflection

We study the affine $A_{n-1}^{(1)}$ Toda fields with boundary reflection. Our approach is based on the free field approach. We construct free field realizations of the boundary state and its dual. For an application of these realizations, we present integral representations for the form factors of the local operators. In a limiting case $ρ\to \infty$, our integral representations reproduce those of form factors for the SU(n) invariant massive Thirring model with boundary reflection [nlin.SI/0010020,Int.J.Mod.Phys.A16,no.15 (2001)2665-2689].

nlin.SI↗
arXiv2002

Baxter Q-operators for integrable DST chain

Following the procedure, described in the paper nlin.SI/0003002, for the integrable DST chain we construct Baxter Q-operators as the traces of monodromy of some M-operators, that act in quantum and auxiliary spaces. Within this procedure we obtain two basic M-operators and derive some functional relations between them such as intertwining relations and wronskian-type relations between two basic Q-operators.

nlin.SI↗
arXiv2006

Radial Loewner equation and dispersionless cmKP hierarchy

It has been shown that the dispersionless KP hierarchy (or the Benney hierarchy) is reduced to the chordal Löwner equation. We show that the radial Löwner equation also gives reduction of a dispersionless type integrable system. The resulting system acquires another degree of freedom and becomes the dcmKP hierarchy, which is a ``half'' of the dispersionless Toda hierarchy. The results of this article was announced in nlin.SI/0512008.

nlin.SI↗
arXiv2007

Generalized N=4 supersymmetric Toda lattice hierarchy and N=4 superintegrable mapping

It is shown that the one-dimensional generalized N=4 supersymmetric Toda lattice (TL) hierarchy (nlin.Si/0311030) contains the N=4 super-KdV hierarchy with the first flow time in the role of space coordinate. Two different N=2 superfield forms of the generalized N=4 supersymmetric TL equation, which are useful when solving the N=4 super-KdV and (1,1)-GNLS hierarchies, are discussed.

nlin.SI↗
arXiv2007

Differential Geometry of Hydrodynamic Vlasov Equations

We consider hydrodynamic chains in $(1+1)$ dimensions which are Hamiltonian with respect to the Kupershmidt-Manin Poisson bracket. These systems can be derived from single $(2+1)$ equations, here called hydrodynamic Vlasov equations, under the map $A^n =\int_{-\infty}^\infty p^n f dp.$ For these equations an analogue of the Dubrovin-Novikov Hamiltonian structure is constructed. The Vlasov formalism allows us to describe objects like the Haantjes tensor for such a chain in a much more compact and computable way. We prove that the necessary conditions found by Ferapontov and Marshall in (arXiv:nlin.SI/0505013) for the integrability of these hydrodynamic chains are also sufficient.

nlin.SI↗
arXiv2004

A strange recursion operator demystified

We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanli) et al. (arXiv: nlin.SI/0203036) is bihamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators. Using this, we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution, and the evolution systems corresponding to these symmetries are bihamiltonian as well. We also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form, with the integration operator $D^{-1}$ appearing in each term at most once.

nlin.SI↗
arXiv2009

Soliton Solutions for ABS Lattice Equations II: Casoratians and Bilinearization

In Part I [arXiv:0902.4873 [nlin.SI]] soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between the Q3 equation and the NQC equations, and then by reductions. In that work central role was played by a Cauchy matrix. In this work we use a different approach, we derive the $N$-soliton solutions following Hirota's direct and constructive method. This leads to Casoratians and bilinear difference equations. We give here details for the H-series of equations and for Q1; the results for Q3 have been given earlier.

nlin.SI↗
arXiv2009

Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy

The lattice Gel'fand-Dikii hierarchy was introduced by Nijhoff, Papageorgiou, Capel and Quispel in 1992 as the family of partial difference equations generalizing to higher rank the lattice Korteweg-de Vries systems, and includes in particular the lattice Boussinesq system. We present a Lagrangian for the generic member of the lattice Gel'fand-Dikii hierarchy, and show that it can be considered as a Lagrangian 2-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus the multiform structure proposed in arXiv:0903.4086v2 [nlin.SI] is extended to a multi-component system.

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arXiv2017

Bäcklund Transformation and Quasi-Integrable Deformation of Mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models

In this paper we study a non-linear partial differential equation (PDE), proposed by N. Kudryashov [arXiv:1611.06813v1[nlin.SI]], using continuum limit approximation of mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models. This generalized semi-discrete equation can be considered as a model for the description of non-linear dislocation waves in crystal lattice and the corresponding continuous system can be called mixed generalized potential KdV and sine-Gordon equation. We obtain the Bäcklund transformation of this equation in Riccati form in inverse method. We further study the quasi-integrable deformation of this model.

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