Recent zbMATH articles in MSC 37J35https://zbmath.org/atom/cc/37J352023-05-08T18:47:08.967005ZWerkzeugStability conditions for refractive partially integrable piecewise smooth vector fieldshttps://zbmath.org/1507.370232023-05-08T18:47:08.967005Z"Buzzi, Claudio A."https://zbmath.org/authors/?q=ai:buzzi.claudio-aguinaldo"Rodero, Ana Livia"https://zbmath.org/authors/?q=ai:rodero.ana-livia"Teixeira, Marco A."https://zbmath.org/authors/?q=ai:teixeira.marco-antonioSummary: In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our main goal is to study stability problems inside the class of \(3\)-dimensional refractive piecewise smooth vector fields. Our concern is to study refractive vector fields that admit a first integral that leaves invariant any sphere centered at the origin. Global stability conditions on generic one-parameter families of refractive piecewise smooth vector fields on a two-dimensional sphere are presented and used to prove our main result, which establishes necessary conditions for the structural stability inside that class.Correction to: ``On curves with the Poritsky property''https://zbmath.org/1507.370342023-05-08T18:47:08.967005Z"Glutsyuk, Alexey"https://zbmath.org/authors/?q=ai:glutsyuk.alexey-aFrom the text: There were typographical errors in the original publication of the author's paper [ibid. 24, No. 2, Paper No. 35, 60 p. (2022; Zbl 1503.37044)]. The article has been corrected.On some invariants of Birkhoff billiards under conjugacyhttps://zbmath.org/1507.370352023-05-08T18:47:08.967005Z"Koudjinan, Comlan E."https://zbmath.org/authors/?q=ai:koudjinan.comlan-e"Kaloshin, Vadim"https://zbmath.org/authors/?q=ai:kaloshin.vadim-yuSummary: In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the ``normalized'' Mather's \(\beta \)-function are invariant under \(C^{\infty}\)-conjugacies. In contrast, we prove that any two elliptic billiard maps are \(C^0\)-conjugate near their respective boundaries, and \(C^{\infty}\)-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar.Non-abelian Toda lattice and analogs of Painlevé III equationhttps://zbmath.org/1507.370742023-05-08T18:47:08.967005Z"Adler, V. E."https://zbmath.org/authors/?q=ai:adler.vsevolod-eduardovich"Kolesnikov, M. P."https://zbmath.org/authors/?q=ai:kolesnikov.m-pSummary: In integrable models, stationary equations for higher symmetries serve as one of the main sources of reductions consistent with dynamics. We apply this method to the non-Abelian two-dimensional Toda lattice. It is shown that already the stationary equation of the simplest higher flow gives a non-trivial non-autonomous constraint that reduces the Toda lattice to a non-Abelian analog of pumped Maxwell-Bloch equations. The Toda lattice itself is interpreted as an auto-Bäcklund transformation acting on the solutions of this system. Further self-similar reduction leads to non-Abelian analogs of the Painlevé III equation.
{\copyright 2022 American Institute of Physics}On the open Toda chain with external forcinghttps://zbmath.org/1507.370752023-05-08T18:47:08.967005Z"Deift, Percy"https://zbmath.org/authors/?q=ai:deift.percy-a"Li, Luen-Chau"https://zbmath.org/authors/?q=ai:li.luen-chau"Spohn, Herbert"https://zbmath.org/authors/?q=ai:spohn.herbert"Tomei, Carlos"https://zbmath.org/authors/?q=ai:tomei.carlos"Trogdon, Thomas"https://zbmath.org/authors/?q=ai:trogdon.thomasAfter an interesting historical review of results about the Toda system, the original Toda Hamiltonian, i.e., the finite fixed-end case described by the Hamiltonian
\[
H_c(q,p)=\frac 12 \sum_{n=1}^Np_n^2+\sum_{n=1}^{N-1}e^{q_n-q_{n+1}}+c\sum_{n=1}^{N-1}(q_n-q_{n+1}),
\]
is studied. For \(c>0\), the last term of the Hamiltonian, an external forcing, stretches the lattice, while for \(c<0\) it compresses it. First, a numerical simulation is carried out, by using a Stormer-Verlet method, indicating that for \(c>0\) the system is integrable, while for \(c<0\) it can be either integrable or nearly integrable. Then, it is proved that solutions of the Hamilton equations are unique and exist globally for any value of \(c\). The analysis proceeds by considering only \(c>0\) and it is found that for \(t \to \infty\) the system splits into two subsets, one formed by points \(q_1\) and \(q_N\), the other by all the remaining points of the lattice, for which a precise asymptotic analysis is obtained. It is shown that for \(t \to \infty\) the particles behave as free particles and, eventually, the \(N\) commuting first integrals are determined and a Lax pair expression for the dynamics is given.
Reviewer: Giovanni Rastelli (Vercelli)More on superintegrable models on spaces of constant curvaturehttps://zbmath.org/1507.370762023-05-08T18:47:08.967005Z"Gonera, Cezary"https://zbmath.org/authors/?q=ai:gonera.cezary"Gonera, Joanna"https://zbmath.org/authors/?q=ai:gonera.joanna"Lucas, Javier de"https://zbmath.org/authors/?q=ai:de-lucas.javier"Szczesek, Wioletta"https://zbmath.org/authors/?q=ai:szczesek.wioletta"Zawora, Bartosz M."https://zbmath.org/authors/?q=ai:zawora.bartosz-mSummary: A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalized Kepler potential. The angular components, on the contrary, are given implicitly by a generally transcendental equation. In the present note, devoted to the previously less studied models with the radial potential of the generalized Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator.Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree \(-4\)https://zbmath.org/1507.370772023-05-08T18:47:08.967005Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Tian, Yuzhou"https://zbmath.org/authors/?q=ai:tian.yuzhouSummary: We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian \(H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) \), being \(P(q_1, q_2)\) a homogeneous polynomial of degree 4 of one of the following forms \(\pm q_1^4\), \(4q_1^3q_2\), \(\pm 6q_1^2q_2^2\), \(\pm \left(q_1^2+q_2^2\right)^2\), \(\pm q_2^2\left(6q_1^2-q_2^2\right)\), \(\pm q_2^2\left(6q_1^2+q_2^2\right)\), \(q_1^4+6\mu q_1^2q_2^2-q_2^4\), \(-q_1^4+6\mu q_1^2q_2^2+q_2^4\) with \(\mu>-1/3\) and \(\mu\neq 1/3 \), and \(q_1^4+6\mu q_1^2q_2^2+q_2^4\) with \(\mu \neq \pm 1/3 \). We note that any homogeneous polynomial of degree 4 after a linear change of variables and a rescaling can be written as one of the previous polynomials. We remark that for the polynomial \(q_1^4+6\mu q_1^2q_2^2+q_2^4\) when \(\mu\in\left\{-5/3, -2/3\right\}\) we only can prove that it has no a polynomial first integral.Pairs of commuting quadratic elements in the universal enveloping algebra of Euclidean algebra and integrals of motionhttps://zbmath.org/1507.370782023-05-08T18:47:08.967005Z"Marchesiello, A."https://zbmath.org/authors/?q=ai:marchesiello.antonella"Šnobl, L."https://zbmath.org/authors/?q=ai:snobl.liborSummary: Motivated by the consideration of integrable systems in three spatial dimensions in Euclidean space with integrals quadratic in the momenta we classify three-dimensional Abelian subalgebras of quadratic elements in the universal enveloping algebra of the Euclidean algebra under the assumption that the Casimir invariant \(\vec{p}\cdot\vec{l}\) vanishes in the relevant representation. We show by means of an explicit example that in the presence of magnetic field, i.e. terms linear in the momenta in the Hamiltonian, this classification allows for pairs of commuting integrals whose leading order terms cannot be written in the famous classical form of \textit{A. A. Makarov} et al. [``A systematic search for nonrelativistic systems with dynamical symmetries'', Nuovo Cimento, X. Ser. A 52, 1061--1084 (1967, \url{doi:10.1007/BF02755212})]. We consider limits simplifying the structure of the magnetic field in this example and corresponding reductions of integrals, demonstrating that singularities in the integrals may arise, forcing structural changes of the leading order terms.A representation of the Dunkl oscillator model on curved spaces: factorization approachhttps://zbmath.org/1507.370792023-05-08T18:47:08.967005Z"Najafizade, Amene"https://zbmath.org/authors/?q=ai:najafizade.amene"Panahi, Hossein"https://zbmath.org/authors/?q=ai:panahi.hossein"Chung, Won Sang"https://zbmath.org/authors/?q=ai:chung.won-sang"Hassanabadi, Hassan"https://zbmath.org/authors/?q=ai:hassanabadi.hassanSummary: In this paper, we study the Dunkl oscillator model in a generalization of superintegrable Euclidean Hamiltonian systems to the two-dimensional curved ones with a \(m: n\) frequency ratio. This defined model of the two-dimensional curved systems depends on a curvature/deformation parameter of the underlying space involving reflection operators. The curved Hamiltonian \(\mathcal{H}_\kappa\) admits the separation of variables in both geodesic parallel and polar coordinates, which generalizes the Cartesian coordinates of the plane. Similar to the behavior of the Euclidean case, which is the \(\kappa\rightarrow0\) limit case of the curved space, the superintegrability of a curved Dunkl oscillator is naturally understood from the factorization approach viewpoint in that setting. Therefore, their associated sets of polynomial constants of motion (symmetries) as well as algebraic relations are obtained for each of them separately. The energy spectrum of the Hamiltonian \(\mathcal{H}_\kappa\) and the separated eigenfunctions are algebraically given in terms of hypergeometric functions and in the special limit case of null curvature occur in the Laguerre and Jacobi polynomials. Finally, the overlap coefficients between the two bases of the geodesic parallel and polar coordinates are given by hypergeometric polynomials.
{\copyright 2022 American Institute of Physics}An introduction to classical monodromy: applications to molecules in external fieldshttps://zbmath.org/1507.370802023-05-08T18:47:08.967005Z"Omiste, Juan J."https://zbmath.org/authors/?q=ai:omiste.juan-j"González-Férez, Rosario"https://zbmath.org/authors/?q=ai:gonzalez-ferez.rosario"Ortega, Rafael"https://zbmath.org/authors/?q=ai:ortega.rafaelSummary: An integrable Hamiltonian system presents monodromy if the action-angle variables cannot be defined globally. As a prototype of classical monodromy with azimuthal symmetry, we consider a linear molecule interacting with external fields and explore the topology structure of its phase space. Based on the behavior of closed orbits around singular points or regions of the energy-momentum plane, a semi-theoretical method is derived to detect classical monodromy. The validity of the monodromy test is numerically illustrated for several systems with azimuthal symmetry.
{\copyright 2022 American Institute of Physics}On the linear stability of a vortex pair equilibrium on a Riemann surface of genus zerohttps://zbmath.org/1507.370812023-05-08T18:47:08.967005Z"Rodrigues, Adriano Regis"https://zbmath.org/authors/?q=ai:rodrigues.adriano-regis"Castilho, César"https://zbmath.org/authors/?q=ai:castilho.cesar"Koiller, Jair"https://zbmath.org/authors/?q=ai:koiller.jairSummary: We present a simple procedure to perform the linear stability analysis of a vortex pair equilibrium on a genus zero surface with an arbitrary metric. It consists of transferring the calculations to the round sphere in \(\mathbb{R}^3\), with a conformal factor, and exploring the Möbius invariance of the conformal structure, so that the equilibria, \textit{seen on the representing sphere}, appear in the north/south poles. Three example problems are analyzed: \(i)\) For a surface of revolution of genus zero, a vortex pair located at the poles is nonlinearly stable due to integrability. We compute the two frequencies of the linearization. One is for the reduced system, the other is related to the reconstruction. Exceptionally, one of them can vanish. The calculation requires only the local profile at the poles and one piece of global information (given by a quadrature). \(ii)\) A vortex pair on a double faced elliptical region, limiting case of the triaxial ellipsoid when the smaller axis goes to zero. We compute the frequencies of the pair placed at the centers of the faces. \(iii)\) The stability, to a restricted set of perturbations, of a vortex equilateral triangle located in the equatorial plane of a spheroid, with polar vortices added so that the total vorticity vanishes.On integrable systems outside Nijenhuis and Haantjes geometryhttps://zbmath.org/1507.370822023-05-08T18:47:08.967005Z"Tsiganov, A. V."https://zbmath.org/authors/?q=ai:tsiganov.andrey-vladimirovichThe article deals with the construction of completely integrable \(n\)-dimensional mechanical systems on Riemannian or pseudo-Riemannian manifolds \((M,g)\) with quadratic (in momenta) Hamiltonians:
\[
H=\sum_{i,j=1}^{n}g^{ij}(q)p_{i}p_{j}+V(q).
\]
Such systems are separable via \textit{S. Benenti}'s scheme [SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 013, 21 p. (2016; Zbl 1386.70036)] by means of a valence-two characteristic Killing tensor satisfying some technical conditions.
As the quadratic conservation laws in many cases can be determined using only two quadratic integral of motions, the author claims that \(n-1\) quadratic and one quartic constants of motion can be also determined using only two quadratic integrals of motion:
\[
H_{1}=\sum_{i,j=1}^{n}g^{ij}(q)p_{i}p_{j}+V(q),\qquad H_{2}= \sum_{i,j=1}^{n}K^{ij}(q)p_{i}p_{j}+U(q),
\]
where \(K\) is the completely non-invariant tensor outside Nijenhuis and Haantjes geometry. Some special results on the three-dimensional Euclidean space, where Killing tensors are characterized by some special non-invariance conditions on the off-diagonal entries of Haantjes tensor, are obtained as well.
The author finds two new integrable systems with two quadratic and one quartic invariants in Poisson involution. Since in four-dimensional and \(n\)-dimensional Euclidean spaces there are a few vanishing off-entries of Haantjes tensor, whereas other off-diagonal entries do not equal to zero, the problem is up to date open, and can be treated within the geometric framework of non-invariant Killing tensors.
Reviewer: Anatoliy K. Prykarpatsky (Kraków)Poincaré integral invariant and conserved quantity of Toda lattice systemshttps://zbmath.org/1507.371142023-05-08T18:47:08.967005Z"Sasa, Narimasa"https://zbmath.org/authors/?q=ai:sasa.narimasaSummary: Numerical properties of the Poincaré integral invariant of the Toda lattice systems are investigated based on the discrete Fourier interpolation method. In the 1D Toda lattice, we show that the Poincaré integral invariant is conserved in a finite time interval in a symplectic time integration. In contrast, a conserved quantity of the 2D Toda lattice is conserved for a long time interval because of the interaction perpendicular to the lattice direction.On the mean density of states of some matrices related to the beta ensembles and an application to the Toda latticehttps://zbmath.org/1507.600162023-05-08T18:47:08.967005Z"Mazzuca, G."https://zbmath.org/authors/?q=ai:mazzuca.guidoSummary: In this paper, we study tridiagonal random matrix models related to the classical \(\beta\)-ensembles (Gaussian, Laguerre, and Jacobi) in the high-temperature regime, i.e., when the size \(N\) of the matrix tends to infinity with the constraint that \(\beta N = 2\alpha\) constant, \(\alpha > 0\). We call these ensembles the Gaussian, Laguerre, and Jacobi \(\alpha\)-ensembles, and we prove the convergence of their empirical spectral distributions to their mean densities of states, and we compute them explicitly. As an application, we explicitly compute the mean density of states of the Lax matrix of the Toda lattice with periodic boundary conditions with respect to the Gibbs ensemble.
{\copyright 2022 American Institute of Physics}Algebraic derivation of the energy eigenvalues for the quantum oscillator defined on the sphere and the hyperbolic planehttps://zbmath.org/1507.810832023-05-08T18:47:08.967005Z"Srivastava, Atulit"https://zbmath.org/authors/?q=ai:srivastava.atulit"Soni, S. K."https://zbmath.org/authors/?q=ai:soni.surendra-kumarSummary: We give an algebraic derivation of eigenvalues of energy of a quantum harmonic oscillator on the surface of constant curvature, i.e., on the sphere or on the hyperbolic plane. We use the method proposed by \textit{C. Daskaloyannis} [J. Math. Phys. 42, No. 3, 1100--1119 (2001; Zbl 1053.37033)] for fixing the energy eigenvalues of two-dimensional quadratically superintegrable systems by assuming that they are determined by the existence of a finite-dimensional representation of the polynomial algebra of motion integral operators. The tool for realizing representations is the deformed parafermionic oscillator. The eigenvalues of energy are calculated, and the result derived by us algebraically agrees with the known energy eigenvalues calculated by using classical analytical methods. This assertion, which is the main result of this article, is demonstrated by a detailed presentation. We also discuss the qualitative difference of the energy spectra on the sphere and on the hyperbolic plane.
{\copyright 2022 American Institute of Physics}Universal chain structure of quadratic algebras for superintegrable systems with coalgebra symmetryhttps://zbmath.org/1507.811162023-05-08T18:47:08.967005Z"Latini, Danilo"https://zbmath.org/authors/?q=ai:latini.daniloSummary: In this paper we show that the \textit{chain structure of} (\textit{overlapping}) \textit{quadratic algebras}, recently introduced in [\textit{Y. Liao} et al., J. Phys. A, Math. Theor. 51, No. 25, Article ID 255201, 13 p. (2018; Zbl 1395.81110)] in the analysis of the \(n\)D quantum quasi-generalized Kepler-Coulomb system, naturally arises for \(n\)D Hamiltonian systems endowed with an \(\mathfrak{sl}(2, \mathbb{R})\) coalgebra symmetry. As a consequence of this hidden symmetry, in fact, such systems are automatically endowed with \(2n - 3\) (second-order) functionally/algebraically independent classical/quantum conserved quantities arising as the image, through a given symplectic/differential representation, of the so-called left and right Casimirs of the coalgebra. These integrals, which are said to be \textsf{universal} being in common to the entire coalgebraic family of Hamiltonians, are shown to be the building blocks of the overlapping quadratic algebras mentioned above. For this reason a subalgebra of these quadratic structures turns out to be, as a matter of fact, universal in the sense that it is shared by any Hamiltonian belonging to this class. As a new specific result arising from this observation, we present the chain structure of quadratic algebras for the \(n\)D quasi-generalized Kepler-Coulomb system on the \(n\)-sphere \(\mathbb{S}^n_\kappa\) and on the hyperbolic \(n\)-space \(\mathbb{H}^n_\kappa\). Both the classical and the quantum frameworks will be considered.Two-dimensional superintegrable systems from operator algebras in one dimensionhttps://zbmath.org/1507.811172023-05-08T18:47:08.967005Z"Marquette, Ian"https://zbmath.org/authors/?q=ai:marquette.ian"Sajedi, Masoumeh"https://zbmath.org/authors/?q=ai:sajedi.masoumeh"Winternitz, Pavel"https://zbmath.org/authors/?q=ai:winternitz.pavelSummary: We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian \(H\) and a polynomial of order \(N\) in the momentum \(p\). We assume that their Poisson commutator \(\{H, K\}\) vanishes, is a constant, a constant times \(H\), or a constant times \(K\). In the quantum case \(H\) and \(K\) are operators and their Lie commutator has one of the above properties. We use two copies of such \((H, K)\) pairs to generate two-dimensional superintegrable systems in the Euclidean space \(E_2\), allowing the separation of variables in Cartesian coordinates. Nearly all known separable superintegrable systems in \(E_2\) can be obtained in this manner and we obtain new ones for \(N = 4\) .Integrable local and non-local vector non-linear Schrödinger equation with balanced loss and gainhttps://zbmath.org/1507.811802023-05-08T18:47:08.967005Z"Sinha, Debdeep"https://zbmath.org/authors/?q=ai:sinha.debdeepSummary: The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a general cubic non-linearity are considered in presence of a linear term characterized, in general, by a non-hermitian matrix which under certain condition incorporates balanced loss and gain and a linear coupling between the complex fields of the governing non-linear equations. It is shown that the systems posses a Lax pair and an infinite number of conserved quantities and hence integrable. Apart from the particular form of the local and non-local reductions, the systems are integrable when the matrix representing the linear term is pseudo hermitian with respect to the hermitian matrix comprising the generic cubic non-linearity. The inverse scattering transformation method is employed to find exact soliton solutions for both the local and non-local cases. Further, it is shown that the presence of the linear term restricts the possible form of the norming constants and hence the polarization vector.