arXiv research

The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.

The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.

Overview of integrable systems with symmetries, focusing on toric and semitoric systems.

Investigates integrable systems with linear periodic integral for e(3) Lie algebra.

New integrable systems derived from Nijenhuis geometry.

We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's splitting theorem for the integrable system is also studied giving some examples in whic…

Researchers prove integrability of magnetic systems on spheres up to dimension 6.

New method to derive integrable systems from existing Lax systems.

In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion of Darboux integrability for first order systems and second order scalar equatio…

New integrators for mechanical systems on Lie groups simplify based on group properties.

In integrable hydrodynamic systems, coordinates exist where generators and symmetries are simple.

The paper solves integrable systems of PDEs, including famous equations.

The paper proves stability of certain singularities in integrable systems.

New integrators preserve geometric structure in Hamiltonian systems.

Develops integrators for Hamiltonian systems in Jacobi manifolds.

We discuss a recurrent geometrical method, due to Élie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can va…

Geometric approach links hydrodynamic integrability to compatible nets.

In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical method, uncovers the fundamental geometric invariants of Darboux integrable syst…

We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow entropies", they describe the polynomial growth rate of the number of balls (fo…

This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the following aspects of integrable non-Hamiltonian systems: local and semi-local normal …

Extends integrability to cosymplectic manifolds.

Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according t…

In the present notes we explain the relationship between Calabi-Yau integrable systems and Hitchin systems based on work by Diaconescu-Donagi-Pantev and the author. Besides a review of these integrable systems, we highlight related topics, for example variations of Hodge structures, cameral curves and Slodowy slices, a…

There is a well-known example of integrable conservative system on $S^2$, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. Goryachev proposed a one-parameter family of examples of conservative systems on $S^2$ possessing an integral of fourth degree in moment…

Elliptic systems are characterized by Darboux integrability.

New integrable systems constructed for non-diagonal Killing tensors.

Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action …

Defines new bi-flat structures from integrable systems and flat coordinates.

Unified geometric framework for integrability of conservative and dissipative systems.

We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.

Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $ω$ and a projection onto $\mathbb{C}$. We provide a full characterization of the completely integrable systems that arise in this way.

We study holomorphic integrable systems on the hyperkähler manifold $G\times S_{\text{reg}}$, where $G$ is a complex semisimple Lie group and $S_{\text{reg}}$ is the Slodowy slice determined by a regular $\mathfrak{sl}_2(\mathbb{C})$-triple. Our main result is that this manifold carries a canonical \textit{abstract int…

Study integrable discretizations of cyclic systems with circular coordinate lines.

The paper analyzes errors in mechanical systems with external forces.

In this paper we deal with the classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on coefficients of the polynomial integral we consider the region (so called elliptic regions) where there are complex-conjugate eigenvalues. We show that for …

Study connects landslide flow to integrable systems for harmonic maps.

Quantizes Stäckel integrable systems into self-adjoint operators.

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference "Finite-dimensional Integrable Systems, FDIS 2017" held at CRM, Barcelona in July 2017.

Extends method for solving certain hydrodynamic systems.

We prove that the cosine law for spherical triangles and spherical tetrahedra defines integrable systems, both in the sense of multidimensional consistency and in the sense of dynamical systems.

Consider a Riemannian metric on two-torus. We prove that the question of existence of polynomial first integrals leads naturally to a remarkable system of quasi-linear equations which turns out to be a Rich system of conservation laws. This reduces the question of integrability to the question of existence of smooth (q…

Construct Lax pairs for BKM equations and related integrable hierarchies.

Polytopes connect Lie theory to physics, integrating integrable systems.

Survey of open problems linking integrable systems and Nijenhuis geometry.

We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and …

Paper constructs super integrable systems on color Lie algebra.

New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.

Develops integrators for contact Hamiltonian systems preserving geometric structure.