arXiv subjects

A. Ramani

Publications and source records attributed to A. Ramani.

At least 19 recordsLinked to original sources

arXiv2017

Miura transformations for discrete Painlevé equations coming from the affine E$_8$ Weyl group

We derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine Weyl group E$_8^{(1)}$. By deautonomisation we obtain two hitherto unknown systems, one of which turns out to be a linearisable one, and we show that both these systems arise from the deautonomisation of a non-QRT mapping. In order to unambiguously prove the integrability of these nonautonomous systems, we introduce a series of Miura transformations which allows us to prove that one of these systems is indeed a discrete Painlevé equation, related to the affine Weyl group E$_7^{(1)}$, and to cast it in canonical form. A similar sequence of Miura transformations allows us to effectively linearise the second system we obtain. An interesting off-shoot of our calculations is that the series of Miura transformations, when applied at the autonomous limit, allows one to transform a non-QRT invariant into a QRT one.

nlin.SI↗

arXiv2012

From Yang-Baxter maps to integrable quad maps and recurrences

Starting from known solutions of the functional Yang-Baxter equations, we exhibit Miura type of transformations leading to various known integrable quad equations. We then construct, from the same list of Yang-Baxter maps, a series of non-autonomous solvable recurrences of order two.

nlin.SI↗

arXiv2011

Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature.

cs.AI↗

arXiv2006

Nonintegrability of (2+1)-dimensional continuum isotropic Heisenberg spin system: Painlevé analysis

While many integrable spin systems are known to exist in (1+1) and (2+1) dimensions, the integrability property of the physically important (2+1) dimensional isotropic Heisenberg ferromagnetic spin system in the continuum limit has not been investigated in the literature. In this paper, we show through a careful singularity structure analysis of the underlying nonlinear evolution equation that the system admits logarithmic type singular manifolds and so is of non-Painlevé type and is expected to be nonintegrable.

nlin.SI↗

arXiv2004

Solvable Chaos

We present classes of discrete reversible systems which are at the same time chaotic and solvable.

math-ph↗

arXiv2003

A unified description of the asymmetric q-P_{v} and d-P_{iv} equations and their Schlesinger transformations

We present a geometric description, based on the affine Weyl group E_{6}^{(1)}, of two discrete analogues of the Painlevé VI equation, known as the asymmetric q-P_{V} and asymmetric d-P_{IV}. This approach allows us to describe in a unified way the evolution of the mapping along the independent variable and along the various parameters (the latter evolution being the one induced by the Schlesinger transformations). It turns out that both discrete Painlevé equations exhibit the property of self-duality: the same equation governs the evolution along any direction in the space of E_{6}^{(1)}.

nlin.SI↗

arXiv2002

The space of initial conditions for linearisable mappings

We apply the algebraic-geometric techniques developed for the study of mappings which have the singularity confinement property to mappings which are integrable through linearisation. The main difference with respect to the previous studies is that the linearisable mappings have generically unconfined singularities. Despite this fact we are able to provide a complete description of the dynamics of these mappings and derive rigorously their growth properties.

nlin.SI↗

arXiv2001

A Study of the Continuous and Discrete Gambier Systems

We present a systematic study of the Gambier system, which in the continuous case is given by two Riccati equations in cascade. We derive the condition for its integrability and show that the generic Gambier system contains one free function. We also derive the Schlesinger transformations for this system which allows in principle the systematic construction of the integrable cases. The above procedure is carried over to a discrete setting. We show thus how the discrete Gambier system can be expressed as a system of two homographic mappings in cascade. The integrable cases are obtained through the singularity confinement discrete integrability criterion. Finally the discrete Schlesinger transformations are also derived giving a handle to the construction of the integrable Gambier mapping.

nlin.SI↗

arXiv2001

Linearisable systems and the Gambier approach

A systematic study of the discrete second order projective system is presented, complemented by the integrability analysis of the associated multilinear mapping. Moreover, we show how we can obtain third order integrable equations as the coupling of a Riccati equation with second order Painlevé equations. This is done in both continuous and discrete cases.

nlin.SI↗

arXiv2001

Blending two discrete integrability criteria: singularity confinement and algebraic entropy

We confront two integrability criteria for rational mappings. The first is the singularity confinement based on the requirement that every singularity, spontaneously appearing during the iteration of a mapping, disappear after some steps. The second recently proposed is the algebraic entropy criterion associated to the growth of the degree of the iterates. The algebraic entropy results confirm the previous findings of singularity confinement on discrete Painlevé equations. The degree-growth methods are also applied to linearisable systems. The result is that systems integrable through linearisation have a slower growth than systems integrable through isospectral methods. This may provide a valuable detector of not just integrability but also of the precise integration method. We propose an extension of the Gambier mapping in $N$ dimensions. Finally a dual strategy for the investigation of the integrability of discrete systems is proposed based on both singularity confinement and the low growth requirement.

nlin.SI↗

arXiv2001

Discrete systems related to some equations of the Painlevé-Gambier classification

We derive integrable discrete systems which are contiguity relations of two equations in the Painlevé-Gambier classification depending on some parameter. These studies extend earlier work where the contiguity relations for the six transcendental Painlevé equations were obtained. In the case of the Gambier equation we give the contiguity relations for both the continuous and the discrete system.

nlin.SI↗

arXiv2001

Linearisable Mappings and the Low-Growth Criterion

We examine a family of discrete second-order systems which are integrable through reduction to a linear system. These systems were previously identified using the singularity confinement criterion. Here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates. We show that the linearisable mappings are characterised by a very special degree growth. The ones linearisable by reduction to projective systems exhibit zero growth, i.e. they behave like linear systems, while the remaining ones (derivatives of Riccati, Gambier mapping) lead to linear growth. This feature may well serve as a detector of integrability through linearisation.

nlin.SI↗

arXiv1999

Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations

We examine the validity of the results obtained with the singularity confinement integrability criterion in the case of discrete Painlevé equations. The method used is based on the requirement of non-exponential growth of the homogeneous degree of the iterate of the mapping. We show that when we start from an integrable autonomous mapping and deautonomise it using singularity confinement the degrees of growth of the nonautonomous mapping and of the autonomous one are identical. Thus this low-growth based approach is compatible with the integrability of the results obtained through singularity confinement. The origin of the singularity confinement property and its necessary character for integrability are also analysed.

solv-int↗

arXiv1999

The hunting for the discrete Painlevé VI is over

We present the discrete, q-, form of the Painlevé 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.

solv-int↗

arXiv1998

Discrete and Continuous Linearizable Equations

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 Painlevé 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.

solv-int↗

arXiv1998

On Discrete Painleve Equations Associated with the Lattice KdV Systems and the Painleve VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented which can be considered to be a discrete analogue of the Painleve V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial difference equations of KdV type. The new equation which is referred to as GDP (generalised discrete Painleve equation) contains various ``discrete Painleve equations'' as subcases for special values/limits of the parameters, some of which were already given before in the literature. The general solution of the GDP can be expressed in terms of Painleve VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Baecklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.

solv-int↗

arXiv1998

A Bilinear Approach to Discrete Miura Transformations

We present a systematic approach to the construction of Miura transformations for discrete Painlevé equations. Our method is based on the bilinear formalism and we start with the expression of the nonlinear discrete equation in terms of $τ$-functions. Elimination of $τ$-functions from the resulting system leads to another nonlinear equation, which is a ``modified'' version of the original equation. The procedure therefore yields Miura transformations. In this letter, we illustrate this approach by reproducing previously known Miura transformations and constructing new ones.

solv-int↗

arXiv1998

Schlesinger Transformations for Linearisable Equations

We introduce the Schlesinger transformations of the Gambier equation. The latter can be written, in both the continuous and discrete cases, as a system of two coupled Riccati equations in cascade involving an integer parameter n. In the continuous case the parameter appears explicitly in the equation while in the discrete case it corresponds to the number of steps for singularity confinement. Two Schlesinger transformations are obtained relating the solutions for some value $n$ to that corresponding to either n+1 or n+2.

solv-int↗