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Closed-form expressions for integrals of MKdV and sine-Gordon maps
Abstract
We present closed-form expressions for approximately N integrals of 2N -dimensional maps. The maps are obtained by travelling wave reductions of the modified Korteweg–de Vries equation and of the sine-Gordon equation, respectively. We provide the integrating factors corresponding to the integrals. Moreover we show how the integrals and the integrating factors relate to the staircase method.
... It is very important that the notion of Liouville-Arnold integrability also can be extended to symplectic maps with suitable modification of Liouville theorem [5], [1], [10]. Many examples of integrable autonomous ordinary difference equations were given, for example, in [3], [4], [8], [9] etc. Experience suggests that such equations are usually grouped in infinite classes of their own kind and their first integrals have a similar appearance. ...
... Ordinary difference equation (3) in this case looks as fifth-order one: ...
We consider two infinite classes of ordinary difference equations admitting
Lax pair representation. Discrete equations in these classes are parameterized
by two integers $k\geq 0$ and $s\geq k+1$. We describe the first integrals for
these two classes in terms of special discrete polynomials. We show an
equivalence of two difference equations belonged to different classes
corresponding to the same pair $(k, s)$. We show that solution spaces
$\mathcal{N}^k_s$ of different ordinary difference equations with fixed value
of $s+k$ are organized in chain of inclusions.
... Closed-form expressions for integrals of periodic reductions in the sine-Gordon equation were presented in [34] and their involutivity was proved in [33]. ...
We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document}: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_3$$\end{document} and A4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_4$$\end{document}, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.
... Closed-form expressions for integrals of periodic reductions of the sine-Gordon equation were presented in [23] and their involutivity was proved in [35]. ...
We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type $A_2$: this deforms to the Lyness family of integrable symplectic maps in the plane. For types $A_3$ and $A_4$ we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions of the discrete sine-Gordon equation.
... One of them, the method by describing its Lax-pair, can be found in [3,4,5]. There is a connection between the two classes, namely that many integrable maps can be obtained from integrable P∆E by imposing periodic boundary conditions [6]. By using the staircase method, P∆E mKdV can be reduced into O∆E mKdV [3].To study the dynamics and also the bifurcation, we need to have a parameter in the system. ...
The discrete modified Korteweg–de Vries (mKdV) is a class of discrete integrable systems that may be distinguished as integrable partial difference equations (PΔE) and integrable ordinary difference equations (OΔE). By considering traveling wave solutions, the OΔE mKdV can be obtained from PΔE mKdV. Meanwhile, a mapping can be constructed from an OΔE mKdV. In this paper, we will focus on producing a new map using a process (replacement), the interchange of a single parameter, and an integral and investigate its properties.
... In our article (Zakaria and Tuwankotta, 2016) a straight-forward generalization by adding parameters in the Lax pair of the ordinary discrete sine-Gordon partial difference equation has done. By using the standard staircase method, the resulted equation is then reduced to system of ordinary difference equations (see Van der Kamp et al., 2007 for the method). Note that this generalized sine-Gordon system is also analyzed in (Duistermaat, 2010). ...
We study the dynamics of a two dimensional map which is derived from another two dimensional map by re-parametrizing the parameter in the system. It is shown that some of the properties of the original map can be preserved by the choice of the re-parametrization. By means of performing stability analysis to the critical points, and also studying the level set of the integrals, we study the dynamics of the re-parametrized map. Furthermore, we present preliminary results on the existence of a set where iteration starts at a point in that set, in which it will go off to infinity after finite step.
... Integrable ordinary difference equations are equivalent to integrable mappings. By imposing a periodicity condition, not only integrable lattice equations can be reduced to ordinary difference equations (or mappings/maps) [3,6,2,1] but also there is a connection between the two classes so that many integrable maps can be obtained from integrable partial difference equations [5,4,6]. In this paper, we study the integrability of the discrete generalized ΔΔ modified Korteweg-de Vries (ΔΔ-mKdV) equation. ...
The integrability of traveling wave solution mappings can be obtained
as reductions of the discrete generalized ΔΔ modified Korteweg-de
Vries (ΔΔ-mKdV) equation. The properties of the integrable discrete
dynamical system can be examined through the level set of integral
function. In this paper, we show that the integral of a threedimensional
traveling wave solution mapping derived from
generalized ΔΔ-mKdV equation can be made in the normal form.
... One of the reasons for the interest to study discrete systems is that they are more fundamental than the continuous ones. Also it is of interest to understand whether or not the discrete systems derived from continuous nonlinear systems governed by ordinary or partial differential equation especially integrable ones preserve their integrability characteristics [1,2,5,11,[20][21][22][23][24][25][26]. In the last few decades, considerable progress has been accomplished and several integrable nonlinear ordinary and partial differential equations were discretized leading to differential-difference, ordinary difference equations or mappings preserving integrability characteristics of their counterpart [7, 12, 14, 27-29, 33, 34]. ...
A systematic investigation to derive nonlinear lattice equations governed by partial difference equations (PΔΔE) admitting specific Lax representation is presented. Further it is shown that for a specific value of the parameter the derived nonlinear PΔΔE's can be transformed into a linear PΔΔE's under a global transformation. Also it is demonstrated how to derive higher order ordinary difference equations (OΔE) or mappings in general and linearizable ones in particular from the obtained nonlinear PΔΔE's through periodic reduction. The question of measure preserving property of the obtained OΔE's and the construction of more than one integrals (or invariants) of them is examined wherever possible.
... They also are a special case of a much bigger class of polynomials which has been introduced in [24]. A different class of (non-polynomial) multi-sums of products, θ, was introduced in [21]. For these multi-sums of products similar relations to (1.1) were derived in [20,Lemma 1]. ...
We construct and study certain Liouville integrable, superintegrable, and
non-commutative integrable systems, which are associated with multi-sums of
products.
... The complete integrability of some particular KdV-type maps was proved in [5], and progress has been made recently with other families of maps. For maps obtained as reductions of the equations in the Adler-Bobenko-Suris (ABS) classification [1], and for reductions of the sine-Gordon and modified Korteweg-de Vries (mKdV) equations, first integrals were given in closed form by using the staircase method and the noncommutative Vieta expansion [32, 17]. In particular, the complete integrability of mappings obtained as reductions of the discrete sine-Gordon, mKdV and potential KdV (pKdV) equations was studied in detail in [33, 34]. ...
We study the integrability of mappings obtained as reductions of the
discrete Korteweg-de Vries (KdV) equation and of two copies of the
discrete potential Korteweg-de Vries equation (pKdV). We show that the
mappings corresponding to the discrete KdV equation, which can be
derived from the latter, are completely integrable in the
Liouville-Arnold sense. The mappings associated with two copies of the
pKdV equation are also shown to be integrable.
We show how some integrable third-order difference equations recently
given in the literature are related to one another by the process of
interchanging parameters and integrals. Using the same process, we then
create a 21-parameter family of integrable third-order difference
equations that contains the previous examples as special cases. Our
methodology illustrates that the combination of finding 2-integrals
(i.e. integrals of the second iterate of the map), exploiting linear
parameter dependence and using the interchange process provides a
powerful way to relate and create higher-dimensional discrete integrable
systems.
In this paper, we first give a terse survey of symplectic maps, their canonical formulation and integrability. Then, we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices. A number of illustrative examples are provided.
A new class of discrete dynamical systems is introduced via a duality relation for discrete dynamical systems with a number of explicitly known integrals. The dual equation can be defined via the difference of an arbitrary linear combination of integrals and its upshifted version. We give an example of an integrable mapping with two parameters and four integrals leading to a (four-dimensional) dual mapping with four parameters and two integrals. We also consider a more general class of higher-dimensional mappings arising via a travelling-wave reduction from the (integrable) MKdV partial-difference equation. By differencing the trace of the monodromy matrix we obtain a class of novel dual mappings which is shown to be integrable as level-set-dependent versions of the original ones.
We classify integrable equations which have the form , where , and K a quadratic polynomial in derivatives of v. This is done using the symbolic calculus, biunit coordinates and the Lech–Mahler theorem. Furthermore we present a method, based on resultants, to determine whether an equation is in a hierarchy of lower order.
The Carg??se summer school celebrated the 100th anniversary of the Painlev?? property, the property that was introduced by Painlev?? and subsequently by Gambier and their school to classify ordinary differential equations (ODEs) according to the singularity behavior of their solutions, [1???4]. In the other contributions to this volume the various implications of the Painlev?? property as well as of the particular differential equations in which this property is manifested are explained in detail.
CONTENTS Introduction Chapter I. Integrable Lagrange systems with discrete time § 1. Hamiltonian theory of Lagrange systems with discrete time § 2. Discrete analogues of classical integrable systems and factorization of matrix polynomials § 3. Spectral theory of difference operators and discrete systems Chapter II. Integrability in dynamics of general algebraic maps § 1. Commuting polynomial maps and simple Lie groups § 2. Integrable polynomial automorphisms of the plane § 3. Algebraic relations and the Yang-Baxter equation. The growth of the number of images of iterations of multivalued maps References
A classification of discrete integrable systems on quad–graphs, i.e. on surface cell decompositions with quadrilateral faces,
is given. The notion of integrability laid in the basis of the classification is the three–dimensional consistency. This property
yields, among other features, the existence of the discrete zero curvature representation with a spectral parameter. For all
integrable systems of the obtained exhaustive list, the so called three–leg forms are found. This establishes Lagrangian and
symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations
of these ideas to the three–dimensional integrable systems and to the quantum context are also discussed.
These lectures are devoted to discrete integrable Lagrangian models. A large collection of integrable models is presented in the Lagrangian fashion, along with their integrable discretizations: the Neumann system, the Garnier system, three systems from the rigid-body dynamics (multidimensional versions of the Euler top, the Lagrange top, and the top in a quadratic potential), the Clebsch case of the Kirchhoff equations for a rigid body in an ideal fluid, and certain lattice systems of the Toda type. The presentation of examples is preceded by the relevant theoretical background material on Hamiltonian mechanics on Poisson and symplectic manifolds, complete integrability and Lax representations, Lagrangian mechanics with continuous and discrete time on general manifolds and, in particular, on Lie groups.
Periodic initial value problems of time and space discretizations of integrable partial differential equations give rise to multi-dimensional integrable mappings. Using the associated linear spectral problems (Lax pairs), a systematic derivation is given of the corresponding sets of polynomial invariants. The level sets are algebraic varieties on which the trajectories of the corresponding dynamical systems lie.
We construct higher dimensional analogues of two-dimensional integrable mappings preserving one biquadratic integral. We also provide their non-autonomous forms.