On the scattering theory for the cubic nonlinear Schrödinger and Hartree type equations in one space dimension

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

“…The results similar to that of Theorems 1.1-1.2 in the particular case λ 2 = λ 3 = λ 4 = λ 5 = λ 6 = 0 were obtained in papers [9], [12].…”

“…The aim of the present work is to prove the results of paper [15] for the more difficult subcritical cases. As far as we know there are no results for the scattering problem in subcritical cases except the case of the nonlinear Schrödinger equation without derivatives of unknown function in the nonlinear term (see [5], [6], [7], [9], [10], [12]). The main point in the proof of our results is the application of the transformation for the solution u (t) = M (t) D (t) v (t) (see [16]), where…”

“…Analytic function spaces were used by many authors to overcome the derivative loss in the nonlinear evolution equations (e.g., see [1], [10], [9], [18]). In what follows we consider the positive time t only, since for the negative one the results are analogous.…”

Long Range Scattering and Modified Wave Operators for some Hartree Type Equations, III. Gevrey Spaces and Low Dimensions

“…The method is an extension of the energy method used in [6,9,10], and uses in particular an auxiliary system of equations introduced in [9] to study the asymptotic behaviour of small solutions. The spaces of initial data, namely in the present case of asymptotic states, are Sobolev spaces of finite order.…”

Modified Scattering States for Subcritical Derivative Nonlinear Schrödinger Equations

“…In the region 0 < S 5 4 the value of the phase in the asymptotic formula (5) is determined with accuracy of the growing as t'-2%ummand. For the region 6 E (4, l) the phase in the asymptotic formula (5) …”