Неабелевы обобщения уравнения Пенлеве IV тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Боброва Ирина Александровна

• Боброва Ирина Александровна
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Боброва Ирина Александровна

Contents

Introduction

Historical review

Statement of the problem

Main results

1. Matrix analogs

2. Fully non-Abelian analogs

Organization of the thesis

1 Painleve equations

1.1 Local analytical properties of ODEs

Example

1.2 Six classes

Example

1.3 Isomonodromic representations and Hamiltonian structure

Example

1.4 Auto-Backlund transformations

Example

1.5 Determinant structure of solutions

Example

1.6 Towards non-Abelian examples

2 Matrix setting

2.1 Matrix Painleve-Kovalevskaya test

2.2 Homogeneous integrable systems of P4 type

2.2.1 Matrix Painleve-Kovalevskaya test

2.3 Inhomogeneous integrable systems of P4 type

2.3.1 Case a = p =

2.3.2 Case a = 0, p =

2.3.3 Case a = 0, p =

2.4 Isomonodromic Lax pairs

2.4.1 Case P4

2.4.2 Case Pi

2.4.3 Case P^

2.5 Degeneracies

2.5.1 Case a = p =

2.5.2 Case a = 0, £ =

2.5.3 Case a = 0, p =

2.6 Remarks and further questions

2.6.1 Hamiltonian structure

2.6.2 ODEs on an associative algebra

2.6.3 Discussion

3 Fully non-Abelian setting

3.1 Quasideterminants and almost Hankel matrices

3.2 Non-commutative Toda equations

3.3 Backlund transformations and non-Abelian symmetric forms

3.3.1 Backlund transformations

3.3.2 Non-Abelian symmetric forms

3.4 Isomonodromic Lax pairs

3.5 Remarks and further questions

3.5.1 Hamiltonian structure

3.5.2 Discussion

4 On relations between two approaches

4.1 Matrix setting —> Fully non-Abelian setting

4.2 Matrix setting <— Fully non-Abelian setting

Conclusion

References

Введение диссертации (часть автореферата) на тему «Неабелевы обобщения уравнения Пенлеве IV»

Introduction

Our aim is to discover integrable matrix and non-abelian analogs of the differential Painlevé IV equation. Their solutions can be understood as new matrix and non-commutative transcendental functions. In addition, such equations can be regarded as quantum or non-abelian versions of the Painleve IV equation:

Historical review

The celebrated Painleve equations appeared as a result of a classification of complex second-order differential equations of the form

where P (z, y(z),y'(z)) is a meromorphic function in z and is a rational function in y (z), y'(z). In order to study these equations, the Painlevé property was introduced. Namely, their general solutions have no movable singular points except of poles . For the first time this property was used by S. Kovalevskaya in [ Kow89]. 50 classes of such equations were found by P. Painleve and his school [Pai00], [Pai02]. Later B. Gambier [Gam10] proved that only six of them define new special functions. The general solutions of the Painleve equations are currently known as the most general class of special functions called the Painlevé transcendents. These six classes of equations of the form ( ) are known as the Painlevé equations. They arise in a wide range of applications in mathematics and physics and have surprisingly rich mathematical structures.

In particular, they are related to a system of scalar differential equations [7uc0' ], [ Gar12] integrable in the sense of the Frobenius theorem. In the paper [ ?uc0' ], R. Fuchs studied the case of the sixth Painleve equation. The result of R. Fuchs was generalized by R. Garnier who considered irregular singularities [ Garl2 ] and, as a result, found such a representation for other Painleve equations. Let us note that L. Schlesinger and B. Malgrange also worked in this area (see, e.g. [Schl ], [ Ial7z ]). In the paper [JM81], it was established that the Painleve equations can be linearized. This fact is connected with monodromy preserving deformations related to vector bundles of rank 2. Thanks to the isomonodromic property, the space of solutions of the Painleve equations can be parameterized by the monodromy data . Namely, each of the

xIn the case of an rn-th order ODE with m > 2, this criteria is formulated in a different way. Namely, general solutions do not have critical movable points.

2Informally speaking.

P4

y''(z) = P (z,y(z),y'(z)) ,

y(z), z E C,

(1)

equations can be associated with the zero-locus of an affine cubic which is usually called the monodromy surface (e.g., [VDPS0! ]).

One of the reasons behind the ubiquitous appearance of the Painleve equations is that they are innately linked to the Toda hierarchy. In [DZ0z], B. Dubrovin and Y. Zhang proved that the r-function of a generic solution to the extended Toda hierarchy is annihilated by some combinations of the Virasoro operators. It is such Virasoro constraints that regulate the correlation functions of many systems in random matrix theory, in string theory and topological field theory. For instance, in [DZ05], expressions for the genus g > 1 total Gromov-Witten potential were obtained via the genus zero quantities derived from the Virasoro constraints.

Note that the isomonodromic r-function is closely connected with the so-called sigma form for the Painleve equations. This issue was investigated by K. Okamoto [Oka8 ], who developed the Hamiltonian theory of the Painleve differential equations [0ka80] and showed that all Backlund transformations can be obtained as natural affine Weyl groups actions on the sigma form [ )ka87a], [0ka87 ], [0ka8 ], [0ka87< ].

Regarding applications of the Painleve equations in integrable systems, it turns out that these equations can be obtained as reduced ODEs of some integrable PDEs. The Ablowitz-Ramani-Segur conjecture [ARS8( ] states that a nonlinear PDE is solvable by the inverse scattering method [ZS74] only if every nonlinear ODE obtained by an exact reduction has the Painleve property. For example, the first and second Painleve equations are the reductions of the KdV equation, the ODE reduction of the sine-Gordon equation is the third Painleve equation, and the sixth Painleve equation is the reduction of the nonlinear Schrodinger equation. In the paper [JKT07], it was shown that the Painleve III-VI equations are the reductions of the three-wave resonant system.

In recent years, quantum, or more generally, non-abelian extensions of various integrable systems have acquired considerable attention. It was motivated by problematics and needs of modern quantum physics as well as by a natural attempts of mathematicians to extend and to generalize the "classical" integrable structures and systems. In particular, the Painleve transcendents provide a good example of this phenomena.

Quantum versions of the Painleve equations were obtained in [NGR+0 ], where the authors quantized the Poisson brackets related to the so-called symmetric form of the Painleve equations. There is also an interesting non-commutative family of non-autonomous many-particle integrable systems that were introduced in [BCR18]. The authors defined an isomonodromic representation of Takasaki Hamiltonian systems of Painleve-Calogero families [Tak01]. A fully non-abelian version for the second Painleve equation was presented in [RR10]. The authors

3In the authors terminology.

investigated a Hankel quasideterminant structure of the P2 solutions caused by their relation to the non-commutative Toda equations [GR92a]. While it is already known [Kaw15] that each of the Painleve equations has only one matrix Hamiltonian analog, in the recent paper [AS21b], it was shown that the Painleve II equation has at least three non-equivalent matrix generalizations. In order to derive them, the authors have used the matrix Painleve-Kovalevskaya test introduced in [ S98 ]. Since the Painleve equations are close to the orthogonal polynomials, several authors have derived their matrix analogs, by using matrix generalizations of the orthogonal polynomials (e.g., [ CM1 ], [ CM+18]).

This historical review do not pretend to provide full overview of the Painleve theory, so we refer the reader to excellent books [CM0 ], [7IN+0 ], as well as reviews cited in them.

Заключение диссертации по теме «Другие cпециальности», Боброва Ирина Александровна

Conclusion

In the present thesis, we have discussed different non-abelian analogs for the Painleve IV equation that are integrable in the sense of Definition 0.1. We considered two settings: matrix and fully non-commutative. We submit the results given in Subsections 2.2 - 2.5, 2.6.1 (matrix case) and 3.3 - 3.- , 3.5.1 (fully non-abelian case) for the defense of this PhD thesis.

In the matrix setting, it turns out that the Painleve IV equation has three different analogs labeled as P4, P;J, and P^. They were obtained by using the matrix generalization of the Painleve-Kovalevskaya test, suggested in [ ]. The system can be written only as a system of first-order ODEs, while the remaining systems, P^ and P , reduces to a second-order ODE. All this systems have arbitrary matrix constants. In addition, each of these systems admits the isomonodromic representation. Some limiting transitions connect the P^ - P^ systems with known matrix generalizations of the Painleve II equation with arbitrary constant matrix coefficients [ lS211 ].

In the fully non-abelian setting, there exists only one analog for the Painleve IV equation. This analog generalizes the quantum Painleve IV equation and the matrix P^ system. It can be written as a system only and was derived by using the solutions of the infinite non-commutative one dimensional Toda lattice. Namely, the solutions of the fully non-commutative Painleve IV system can be expressed via the solutions of the infinite non-commutative Toda system. The latter possesses solutions in terms of the Hankel quasideterminants. This implies that the fully non-commutative P4 analog also admits solutions in the Hankel quasideterminants. Besides this property, the fully non-abelian analog has the zero-curvature representation.

The problems for the further study and generalization are discussed in Subsections 2.6.3 and 3.5.2.

Список литературы диссертационного исследования кандидат наук Боброва Ирина Александровна, 2023 год

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