arXiv subjects

Adrian Stefan Carstea

Publications and source records attributed to Adrian Stefan Carstea.

8 recordsLinked to original sources

arXiv2025

Singularity confinement and proliferation of tau functions for a general differential-difference Sawada-Kotera equation

Blending Painlevé property with singularity confinement for a general arbitrary order Sawada-Kotera differential-difference equation, we find a proliferation of ``tau-functions'' (coming from strictly confined patterns). However only one of these function enters into the Hirota bilinear form (the others give multi-linear expressions) but has specific relations with all others. We also discuss the case of two modifications of Sawada-Kotera showing that periodic patterns appear in addition to strictly confined ones. Fully discretisations and express method for computing algebraic entropy are discussed.

nlin.SI↗

arXiv2025

On the geometry of a 4-dimensional extension of a $q$-Painlevé I equation with symmetry type $A_1^{(1)}$

We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a $q$-Painlevé I equation with symmetry of type $A_1^{(1)}$. By resolution of singularities it is lifted to a pseudo-automorphism of a rational variety obtained from $({\mathbb P}^1)^{\times 4}$ by blowing up along 28 subvarieties and we use this to establish its integrability in terms of conserved quantities and degree growth. We embed this rational variety into a family which admits an action of the extended affine Weyl group $\widetilde{W}(A_1^{(1)})\times \widetilde{W}(A_1^{(1)})$ by pseudo-isomorphisms. We use this to construct two 4-dimensional analogues of $q$-Painlevé equations, one of which is a deautonomisation of the original autonomous integrable map.

nlin.SI↗

arXiv2025

New envelope equations for shallow water waves and modulational instability

The dynamics of wave groups is studied for long waves, using the framework of the Benjamin-Bona-Mahony (BBM) equation and its generalizations. It is shown that the dynamics are richer than the corresponding results obtained just from the Korteweg-de Vries-type equation. First, a reduction to a nonlinear Schrödinger equation is obtained for weakly nonlinear wave packets, and it is demonstrated that either the focusing or the defocusing case can be obtained. This is in contrast to the corresponding reduction for the Korteweg-de Vries equation, where only the defocusing case is found. The focusing regime displays modulational instability responsible for the appearance of rogue waves. Next, the condition for modulational instability is obtained in the case of one and two monochromatic waves in interaction at slow space-time coordinates with equal scalings. Other new envelope equations are obtained starting from the general system describing shallow water waves found by Bona et al. [3]. A presumably integrable system is obtained form the integrable Kaup-Boussinesq one.

nlin.PS↗

arXiv2025

A fast algorithmic way to calculate the degree growth of birational mappings

We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings, based on Halburd's method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer arithmetic with additions and, in some cases, multiplications by small integers. It is therefore extremely fast. Several examples of integrable and non-integrable mappings are presented. In the latter case the dynamical degree we obtain from our method is always in agreement with that calculated by previously known methods.

nlin.SI↗

arXiv2019

An algebraically stable variety for a four-dimensional dynamical system reduced from the lattice super-KdV equation

In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by the traveling wave reduction from the lattice super-KdV equation in a case of finitely generated Grassmann algebra. The system is a coupling of a Quispel-Roberts-Thompson map and a linear map but does not satisfy the singularity confinement criterion. It was conjectured that the dynamical degree of this system grows quadratically. In this paper, constructing a rational variety where the system is lifted to an algebraically stable map and using the action of the map on the Picard lattice, we prove this conjecture. We also show that invariants can be found through the same technique.

math.DS↗

arXiv2018

Space of initial conditions and geometry of two 4-dimensional discrete Painlevé equations

A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they are lifted to pseudo-automorphisms of rational varieties obtained from $({\mathbb P}^1)^4$ by blowing-up along sixteen 2-dimensional subvarieties. The symmetry groups, the invariants and the degree growth rates are computed from the linearisation on the corresponding Néron-Severi bilattices. It turns out that the deautonomised version of one of the mappings is a Bäcklund transformation of a direct product of the fourth Painlevé equation which has $A_2^{(1)}+A_2^{(1)}$ type affine Weyl group symmetry, while that of the other mapping is of Noumi-Yamada's $A_5^{(1)}$ Painlevé equation.

math.DS↗

arXiv2017

Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. %By choosing a particular Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.

math.DS↗

arXiv2013

A note on minimization of rational surfaces obtained from birational dynamical systems

In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable mappings, including discrete versions of reduced Nahm equations.

math.DS↗