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Tsvetana Stoyanova

Publications and source records attributed to Tsvetana Stoyanova.

9 recordsLinked to original sources
arXiv2025

Non-integrability of the Sasano system of type $A^{(2)}_5$

The Sasano sytem of type $A^{(2)}_5$ is a four-dimensional non-linear system of ordinary differential equations, which has an affine Weyl group of symmetries of type $A^{(2)}_5$. It is also a tipe dependent Hamiltonian system, which can be considered as coupled Painlevé III systems. In this paper, utilizing the Morales-Ramis-Simó theory for integrability of Hamiltonian systems, we prove rigorously that for all values of the parameters, for which the Sasano system of type $A^{(2)}_5$ admits a particular rational solution, it is non-integrable by rational first integrals.

nlin.SI↗
arXiv2025

Non-Integrability of the Sasano System of Type $D_5^{(1)}$ and Stokes Phenomena

In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine Weyl group symmetry of type $D^{(1)}_l$, $l=4, 5, 6, \dots$. In this paper, we study the integrability of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl group $\widetilde{W}\bigl(D^{(1)}_5\bigr)$ and which we call the Sasano system of type $D^{(1)}_5$. We prove that one family of the Sasano system of type $D^{(1)}_5$ is not integrable by rational first integrals. We describe Stokes phenomena relative to a subsystem of the second normal variational equations. This approach allows us to compute in an explicit way the corresponding differential Galois group and therefore to determine whether the connected component of its unit element is not Abelian. Applying the Morales-Ramis-Simó theory, we establish a non-integrable result.

nlin.SI↗
arXiv2024

Non-integrability of the rational Sasano system of type $A^{(2)}_4$

In this paper we study the integrability of the Sasano system of type $A^{(2)}_4$ from the point of view of the Hamiltonian dynamics. We prove rigorously that for these values of the parameters for which the Sasano system of type $A^{(2)}_4$ has a particular rational solution it is not integrable by rational first integrals. By an explicit computation we show that the connected component $G^0$ of the unit element of the differential Galois group of the normal variational equations along a simple particular rational solution is a direct product of two groups $SL_2(\mathbb{C})$. This the Morales-Ramis theory leads to a non-integrable result. Moreover using Bäcklund transformations we extend the obtained particular non-integrable result to the entire orbit of the parameters.

math-ph↗
arXiv2023

The reducible double confluent Heun equation and a general symmetric unfolding of the origin

The reducible double confluent Heun equation (DCHE) is the only DCHE whose general symmetric unfolding leads to a Fuchsian equation. Contrary to general Heun equation the unfolded Fuchsian equation has 5 singular points : $x_L=-\sqrt{\varepsilon}, x_R=\sqrt{\varepsilon}, x_{LL}=-1/\sqrt{\varepsilon}, x_{RR}=1/\sqrt{\varepsilon}$ and $x_{\infty}=\infty$. We prove that the monodromy matrix around the regular resonant singularity at the origin is realizable as a limit of the product of the monodromy matrices around resonant singularities $x_L$ and $x_R$ when $\sqrt{\varepsilon} \to 0$ while the Stokes matrix at the irregular singularity at the origin is a limit of the part of the monodromy matrix around the resonant singularity $x_L$. We also show that the reducible DCHE possesses a holomorphic solution in the whole $\mathbb{C}^*$ if and only if the parameters of the equation are connected by a Bessel function of first kind and order depending on the non-zero chracteristic exponent at the origin.

math.CA↗
arXiv2023

Non-integrability of the Painlevé IV equation in the Liouville-Arnold sense and Stokes phenomena

In this paper we study the integrability of the Hamiltonian system associated to the fourth Painlevé equation. We prove that one two parametric family of this Hamiltonian system is not integrable in the sense of the Liouville-Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations we deduce that the connected component of the unit element of the corresponding Galois grou is not Abelian. Thus the Morales-Ramis-Simó theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for allvalues of the parameters for which the $P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in $t$.

nlin.SI↗
arXiv2020

Stokes matrices of a reducible equation with two irregular singularities of Poincaré rank 1 via monodromy matrices of a reducible Heun type equation

We consider a second order reducible equation having non-resonant irregular singularities at $x=0$ and $x=\infty$. Both of them are of Poincaré rank 1. We introduce a small complex parameter $\varepsilon$ that splits together $x=0$ and $x=\infty$ into four different Fuchsian singularities $x_L=-\sqrt{\varepsilon}, x_R=\sqrt{\varepsilon}$ and $x_{LL}=-1/\sqrt{\varepsilon}, x_{RR}=1/\sqrt{\varepsilon}$, respectively. The perturbed equation is a second order reducible Fuchsian equation with 4 different singularities, i.e. a Heun type equation. Then we prove that when the perturbed equation has exactly two resonant singularities of different type, all the Stokes matrices of the initial equation are realized as a limit of the nilpotent parts of the monodromy matrices of the perturbed equation when $\varepsilon \rightarrow 0$ in the real positive direction. To establish this result we combine a direct computation with a theoretical approach.

math.CA↗
arXiv2019

Stokes matrices for a class of reducible equations

This paper is a continuation of our previous work \cite{St} where we have studied the Stokes phenomenon for a particular family of equation \eqref{initial} with \eqref{form-0}-\eqref{npe} from a perturbative point of view. Here we focus on the explicit computation of the Stokes matrices at the non-resonant irregular singularity for a more general situation. In particular, utilizing Borel-Laplace summation, the iterated integrals approach and some properties of the hypergeometric series we compute by hand the Stokes matrices of three families of equation \eqref{initial}-\eqref{form-0}-\eqref{npe} under assumptions that $β_j$'s are distinct and $|β_3-β_1| < |β_3-β_2|$. Moreover, these results remain valid for these distinct $β_j$'s for which $|β_3-β_1|=|β_3-β_2|$ but $β_3-β_1 \neq \pm (β_3-β_2)$ on condition that $\mathcal{Re} (α_2-α_1) > -1$. In addition, iterated integrals approach allows us to give, under some restrictions, an explicit representation of the 1-sum of the product of two certain divergent 1-summable power series, that have different singular directions.

math.CA↗
arXiv2018

Zero level perturbation of a certain third-order linear solvable ODE with an irregular singularity at the origin of Poincaré rank 1

We study an irregular singularity of Poincaré rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter $\varepsilon\in (\mathbb{R}_+, 0)\,(\varepsilon < 1)$ which causes the splitting of the irregular singularity into two finite Fuchsian singularities. We show that when the solutions of the perturbed equation contain logarithmic terms, the Stokes matrices of the initial equation are limits of the part of the monodromy matrices around the finite resonant Fuchsian singularities of the perturbed equation.

math.CA↗