Related papers: Infinite-dimensional Hamilton-Jacobi theory and $L…
The change of variable theorem is proved under the sole hypothesis of differentiability of the transformation. Specifically, it is shown under this hypothesis that the transformed integral equals the given one over every measurable subset…
We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)^mu=|x|^av^p,\; (-\Delta)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions…
This article is interested in internality to the constants of systems of autonomous algebraic ordinary differential equations. Roughly, this means determining when can all solutions of such a system be written as a rational function of…
Consider a general linear Hamiltonian system $\partial_{t}u=JLu$ in a Hilbert space $X$. We assume that$\ L: X \to X^{*}$ induces a bounded and symmetric bi-linear form $\left\langle L\cdot,\cdot\right\rangle $ on $X$, which has only…
It is well known that a generic small perturbation of a Liouville-integrable Hamiltonian system causes breakup of resonant and near-resonant invariant tori. A general approach to the simple resonance case in the convex real-analytic setting…
The quantization method based on the quantum Hamiltonian Jacobi equation, is extended to two-dimensional non-separable but integrable Hamiltonians. It is shown that each wave function for those systems corresponds to a well-defined family…
The Kowalevski top in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to…
We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations with homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which…
Using the example of Liouville theory, we show how the separation into left- and rightmoving degrees of freedom of a nonrational conformal field theory can be made explicit in terms of its integrable structure. The key observation is that…
We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue…
In this paper we study the equation $$ w^{(4)} = 5 w" (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma, $$ which is one of the higher-order Painlev\'e equations (i.e., equations in the polynomial class having the Painlev\'e…
A Liouville function is a complex analytic function H with a Taylor series \sum_{n=1}^{\infty} x^n/a_n such the a_n's form a ``very fast growing'' sequence of integers. In this paper we exhibit the complete first-order theory of the complex…
We review some essential aspects of classically integrable systems. The detailed outline of the lectures consists of: 1. Introduction and motivation, with historical remarks; 2. Liouville theorem and action-angle variables, with examples…
In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V(q)$, $q\in\C^n$, of degree $k\in\Z^\star$. The known results of Morales and Ramis give necessary conditions for the complete integrability of…
There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. The third problem, however, should be added to the list: separation…
This paper is concerned with the polynomial integrability of the two-dimensional Hamiltonian systems associated to complex homogeneous polynomial potentials of degree $k$ of type $V_{k,l}=\alpha (q_2-i q_1)^l (q_2+iq_1)^{k-l}$ with…
We investigate the problem of the existence of first integrals for multidimensional and ordinary linear differential systems with constant coefficients. The spectral method of the first integrals basis construction for these systems of…
The contributions of Sophya Kowalewski to the integrability theory of the equations for the heavy top extend to a larger class of Hamiltonian systems on Lie groups; this paper explains these extensions, and along the way reveals further…
We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the…
In this letter we show that the results of degree growth (algebraic entropy) calculations for lattice equations strongly depend on the initial value problem that one chooses. We consider two problematic types of initial value…