Related papers: Le th\'eor\`eme d'Andr\'e-Chudnovsky-Katz

We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying…

In this paper we prove the existence and the stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for the 1-dimensional forced Kirchoff equation with periodic boundary conditions. This is the first KAM result for a…

A two-dimensional Gauss-Kuzmin theorem for $N$-continued fraction expansions is shown. More exactly, we obtain a Gauss-Kuzmin theorem related to the natural extension of the measure-dynamical system corresponding to these expansions. Then,…

This paper is concerned with the following fractional Schr\"{o}dinger equations involving critical exponents: \begin{eqnarray*} (-\Delta)^{\alpha}u+V(x)u=k(x)f(u)+\lambda|u|^{2_{\alpha}^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N},…

We study the Gross-Pitaevskii equation involving a nonlocal interaction potential. Our aim is to give sufficient conditions that cover a variety of nonlocal interactions such that the associated Cauchy problem is globally well-posed with…

If the semigroup is slowly non-dissipative, i.e., its solutions can diverge to infinity as time tends to infinity, one still can study its dynamics via the approach by the unbounded attractors - the counterpart of the classical notion of…

The purpose of this work is to investigate root finding problems defined on (quasi-)metric spaces, and ranging in Euclidean spaces. The motivation for this line of inquiry stems from recent models in biology and phylogenetics, where…

Fitzpatrick's variational representation of maximal monotone operators is here extended to a class of pseudo-monotone operators in Banach spaces. On this basis, the initial-value problem associated with the first-order flow of such an…

The stationary Gross-Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the PT (parity-time reversal) symmetry. Under rather general assumptions on the potentials we prove…

Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for…

This paper is devoted to the eigenvalue problem for the quantum Gaudin system. We prove the universal correspondence between eigenvalues of Gaudin Hamiltonians and the so-called G-opers without monodromy in general gl(n) case modulo a…

We consider the $d$-dimensional nonlinear Schr\"odinger equation under periodic boundary conditions: $-i\dot u=-\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $ where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is…

We make the elementary observation that the differential equation associated with Newton's second law $m\ddot\gamma(t)=-D V(\gamma(t))$ always has a solution for given initial conditions provided that the potential energy $V$ is semiconvex.…

It is known that for every Banach space X and every proper WOT-closed subalgebra A of L(X), if A contains a compact operator then it is not transitive. That is, there exist non-zero x in X and f in X* such that f(Tx)=0 for all T in A. In…

The noncommutative Gurarij space $\mathbb{\mathbb{\mathbb{NG}}}$, initially defined by Oikhberg, is a canonical object in the theory of operator spaces. As the Fra\"{\i}ss\'{e} limit of the class of finite-dimensional nuclear operator…

This paper is mainly devoted to describing the entire solutions of nonlinear partial differential equation $$ u_{z_1}u_{z_2}\cdots u_{z_n}=e^g, $$ with the eikonal equation as a prototype, where $g$ is a polynomial in $\mathbb{C}^n$.…

In this paper, we consider the global well-posedness to the non-cutoff Boltzmann equation with soft potential in the $L^\infty$ setting. We show that when the initial data is close to equilibrium and the perturbation is small in $L^2 \cap…

In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in previous work by the present author. We discuss solutions of the Einstein vacuum (EV) equations. We…

This paper focuses on a class of zero-norm composite optimization problems. For this class of nonconvex nonsmooth problems, we establish the Kurdyka-Lojasiewicz property of exponent being a half for its objective function under a suitable…

This paper is the continuation of our work with Victor Guillemin; Victor and I proved that the Taylor expansion of the potential at a generic non degenerate critical point is determined by the semi-classical spectrum of the associated…