Wydział Fizyki UW > Badania > Seminaria i konwersatoria > Seminarium "Nieliniowość i Geometria"

Seminarium "Nieliniowość i Geometria"

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Zapraszamy do sali B0.17, ul. Pasteura 5 o godzinie 11:15

Piotr Goldstein (NCBJ)

A system of second-order ordinary differential equations with exponential nonlinearity

A system of real ordinary differential equations (ODE), $\ddot{y}_k=\sum_{l=1}^n M_{kl}\exp y_l,~k=1,...,n$, where $M$ is a real $n\times n$ constant symmetric matrix, is discussed in detail for general $n$. After a symmetry analysis and a discussion of the Lagrangian-Hamiltonian structure, a one-parameter family of exact solutions is found and proved to be unstable to small perturbations. Next, the system is tested for the Painlev\'e property. The result shows that the only completely integrable cases of this system are equivalent to the Toda lattice by similarity transformations of matrix $M$. A few physical applications are discussed for the general and some special $n$'s.

Zapraszamy do sali B0.17, ul. Pasteura 5 o godzinie 11:15

Debora Choińska (Wydział Fizyki, Uniwersytet Warszawski)

Solutions of the hyperbolic Ernst equation

Ernst-type equations are elegant reformulations of Einstein's vacuum equations of general relativitywhen the existence of two commuting Killing vector fields is assumed.Axisymmetric, stationary spacetimes such as rotating black holes and planar gravitational waves areexamples of solutions of the Ernst-type equations.An important mathematical feature of the Ernst-type equations is that they are integrable nonlinear differential equations,allowing the application of methods developed for solving integrable systems.In particular, the inverse scattering method leading to the dressing method enabled to generate n-soliton solutions on the Kasner background [1,2].We will use the method that dates back to Bianchi's work i.e. and is based on a Bäcklund transformation and a nonlinear superposition principle [3,4]. We will compare the two methods showing differences between two soliton solutions. We will also present some solutions without singularities.[1] V. A. Belinsky and V. E. Zakharov, Sov. Phys. JETP, 48:985–994, 1978.[2] Belinski V, Verdaguer E. Gravitational Solitons. Cambridge University Press; 2001.[3] L. Bianchi. Memorie della Societ`a Italiana delle Scienze, detta dei XL,13:261–289, 1905.[4] M. Nieszporski. The multicomponent Ernst equation and the Moutard transformation. Physics Letters A, 272(1):74–79, 2000.

Zapraszamy do sali B0.17, ul. Pasteura 5 o godzinie 11:45

Jan L. Cieśliński (Wydział Fizyki, Uniwersytet w Białymstoku)

Dokładna dyskretyzacja klasycznego problemu Keplera

Przedstawiony zostanie schemat numeryczny, który zachowuje wszystkie całki ruchu,a zatem i trajektorie. Co więcej, udało się nawet dokładnie odtworzyć położenie punktów na orbicie,gdyż prawdziwym parametrem jest tzw. anomalia prawdziwa (kąt między kolejnymi wektorami położenia jest,z założenia, stały), zatem tzw. równanie Keplera wykorzystywane jest, w przeciwieństwie do innych metod,w sposób jawny. Preprint: https://arxiv.org/abs/2512.12099

Zapraszamy do sali 2.21, ul. Pasteura 5 o godzinie 11:15

Adam Doliwa (Uniwersytet Warmińsko-Mazurski w Olsztynie)

Non-commutative multiple bi-orthogonal polynomials: formal approach and integrability

We define the non-commutative multiple bi-orthogonal polynomial systems, which simultaneously generalize the concepts of multiple orthogonality, matrix orthogonal polynomials and of the bi-orthogonality. We present quasideterminantal expressions for such polynomial systems in terms of formal bi-moments. The normalization functions for such monic polynomials satisfy the non-commutative Hirota equations, while the polynomials provide solution of the corresponding linear system. This shows, in particular, that our polynomial systems form a part of the theory of integrable systems. We study also a specialization of the problem to non-commutative multiple orthogonal polynomials, what results in the corresponding Hankel-type quasideterminantal expressions in terms of the moments. Moreover, such a reduction allows to introduce in a standard way the discrete-time variable and gives rise to an integrable system which is a non-commutative version of the multidimensional discrete-time Toda equations.https://arxiv.org/abs/2510.02207

Zapraszamy do sali 2.21, ul. Pasteura 5 o godzinie 12:15

Ralph Willox (University of Tokyo)

How to detect `integrability' in bi-rational maps?

In recent years, stating that a rational map is `integrable', has become synonymous with saying that its iterates exhibit `polynomial degree growth' in terms of the initial conditions. For higher-dimensional maps, however, it is in general very difficult to calculate the degrees of their iterates in any systematic way and there is a pressing need to develope reliable, rigorous, algorithms to this end.In this seminar, I present two such calculation methods that can be applied to rational maps of any order. Both methods are based on the singularity structure of the map, one is a method that rigorously calculates the algebraic entropy of the map while the other provides an ultra-efficient algorithm for generating the degree sequence for the iterates of the map.
This talk is (partly) based on on [A. Stokes, T. Mase, R. Willox and B. Grammaticos, "Deautonomisation by singularity confinement and degree growth", The Journal of Geometric Analysis 35, 65 (2025)] and [B. Grammaticos, A. Ramani, A.S. Carstea and R. Willox, "A Fast Algorithmic Way to Calculate the Degree Growth of Birational Mappings", Mathematics 13, no. 5: 737 (2025)]

Zapraszamy do sali 2.21, ul. Pasteura 5 o godzinie 12:15

Marta Dell ‘Atti (MIM UW)

Pfaff lattice: reductions and continuum limit

Motivated by the theory of orthogonal random matrix ensembles, we investigate the integrable semi-discrete hierarchy of the Pfaff lattice. With the use of a mapping connecting this system with the Toda lattice (related to the unitary ensemble), we derive the initial datum for the first flow of Pfaff. This initial datum provides the most natural approach for constructing the continuum limit, while also suggesting a natural reduction of the system. Interestingly, both the semi-discrete system and the hydrodynamic chain emerging in the continuum limit reduce to the same novel semi-discrete system. The talk is based on the work in collaboration with C. Benassi and A. Moro (https://doi.org/10.48550/arXiv.2504.11296).