THE GLOBAL CAUCHY PROBLEM FOR THE NLS WITH HIGHER ORDER ANISOTROPIC DISPERSION

We use a method developed by Strauss to obtain global well-posedness results in the mild sense and existence of asymptotic states for the small data Cauchy problem in modulation spaces ${M}^s_{p,q}(\mathbb{R}^d)$, where q = 1 and $s\geq0$ or $q\in(1,\infty]$ and $s>\frac{d}{q'}$ for a nonlinear Schrödinger equation with higher order anisotropic dispersion and algebraic nonlinearities.

Interpolation by sums of series of exponentials and global Cauchy problem for convolution operators

The study is made of the problem of multiple interpolation on an infinite nodes set by the sums of absolutely convergent series of exponentials whose exponents are from a given set. For entire function conditions on nodes and exponents are obtained that give solubility of the problem. A new approach is demonstrated that enable us, for the case of holomorphic function in a domain, to obtain criteria of solubility of the problem for some class of exponents set and for a special class of nodes set. Moreover the necessity of the conditions is proved in great generality namely for arbitrary nodes sets and in the setting of interpolation by functions that are represented as the Laplace transforms of the Radon measures over the exponents set. Solubility is obtained of the global Cauchy problem for convolution operator with data on the nodes set in domain, in the form of the series of exponentials whose exponents belong to a sparse subset of zero set of characteristic function of the operator. The results substantially strengthen the known results on the theme.

The Global Formulation of the Cauchy Problem

<p><span lang="EN-US">A Geometrical model for the global Cauchy problem, generalizing the traditional Cauchy problem is considered .The complete correspondence between the known<span>analytical formulation and the </span>geometrical interpretation is described, we have utilized the generalized Green's function and the open mapping theorem appropriate to the problem.</span></p>