Related papers: Non-commutative generalization of integrable quadr…
A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot-Guldberg Lie algebra,…
Polynomials known as Multiple Orthogonal Polynomials in a single variable are polynomials that satisfy orthogonality conditions concerning multiple measures and play a significant role in several applications such as Hermite-Pad\'e…
For the 1+1 dimensional Lax pair with a symplectic symmetry and cyclic symmetries, it is shown that there is a natural finite dimensional Hamiltonian system related to it by presenting a unified Lax matrix. The Liouville integrability of…
Nonlinear Hamiltonian systems describing the abstract Vlasov and Hartree equations are considered in the framework of algebraic Poissonian theory. The concept of uniformization is introduced; it generalizes the method of second quantization…
In this paper we prove that there exists only one family of classical Hamiltonian systems of two degrees of freedom with invariant plane $\Gamma=\{q_2=p_2=0\}$ whose normal variational equation around integral curves in $\Gamma$ is…
A method for constructing homogeneous Lyapunov functions of degree 1 from polynomial invariant sets is presented for linear time varying systems, homogeneous dynamic systems and the class of nonlinear systems that can be represented as…
Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A…
We consider real analytic Hamiltonians whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q$. By a theorem of Moser, every polynomial Hamiltonian of degree 3 reduces to such…
The one-variable non-symmetric Wilson polynomials are shown to coincide with the Bannai-Ito polynomials. The isomorphism between the corresponding degenerate double affine Hecke algebra of type $(C_1^{\vee}, C_1)$ and the Bannai-Ito algebra…
Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian…
Hamiltonian flows on compact surfaces are characterized, and the topological invariants of such flows with finitely many singular points are constructed from the viewpoints of integrable systems, fluid mechanics, and dynamical systems.…
There exist two versions of the Kadomtsev-Petviashvili equation, related to the Cartesian and cylindrical geometries of the waves. In this paper we derive and study a new version, related to the elliptic cylindrical geometry. The derivation…
In the first part of the article we study Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$ which are generated by sub-quadratic Hamiltonians and prove a middle dimensional rigidity result for the image of coisotropic cylinders. The tools…
We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the flows of completely integrable Hamiltonian systems are investigated.…
We extend the result of D. Phillips (On one-homogeneous solutions to elliptic systems in two dimensions. C. R. Math. Acad. Sci. Paris 335 (2002), no. 1, 39-42) by showing that one-homogeneous solutions of certain elliptic systems in…
In this paper we analyze the normal forms of a general quadratic Hamiltonian system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ - skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric…
We review a recent generalization of Normal Form Theory to systems (Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate variable. The main difference with the standard case relies on the non…
In this article, we construct novel explicit solutions for nonlinear Schr\"odinger systems with spatially inhomogeneous nonlinearity by means of the Lie symmetry method. We focus the attention to solutions with non-trivial phase, which have…
We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential…
Complex-linearization of a class of systems of second order ordinary differential equations (ODEs) has already been studied with complex symmetry analysis. Linearization of this class has been achieved earlier by complex method, however,…