Nonclassical approximate symmetries of evolution equations with a small parameter.
Noncommutative root space Witt, Ricci flow, and Poisson bracket continual Lie algebras.
Nondegenerate systems and generic properties of the integrable Hamiltonian systems
Nonlinear oscillations of completely resonant wave equations
Nonlinear Schrödinger equation on four-dimensional compact manifolds
We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness for small energy...
Nonlinear vibrations of completely resonant wave equations
We present recent existence results of small amplitude periodic and quasi-periodic solutions of completely resonant nonlinear wave equations. Both infinite-dimensional bifurcation phenomena and small divisors difficulties occur. The proofs rely on bifurcation theory, Nash-Moser implicit function theorems, dynamical systems techniques and variational methods.
Notes on symplectic non-squeezing of the KdV flow
We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle . The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our...
On a direct construction of inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension.
On a family of operators and their Lie algebras.
On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation.
On a nonlocal Ostrovsky-Whitham type dynamical system, its Riemann type inhomogeneous regularizations and their integrability.
On an infinite sequence of invariant measures for the cubic nonlinear Schrödinger equation.
On approximation of solutions to some -dimensional integrable systems by Bäcklund transformations.
On asymptotic stability of solitary waves for nonlinear Schrödinger equations
On classical series expansions for quasi-periodic motions.
On commuting differential operators.
On commuting matrix differential operators.
On conservation laws in affine Toda systems.
On dissipative phenomena of the interaction of Hamiltonian systems.
On foundations of the Conley index theory
The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the critical...