Large Time Behavior of Solutions for Derivative Cubic Nonlinear Schrödinger Equations

Abstract

We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrodinger equations of the following form

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

“…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 existence of modified scattering states was shown in papers [11], [15], [20] and modified wave operators were constructed in [2], [21]. In our previous paper [15] we considered the Cauchy problem (1.1) with cubic derivative nonlinearity (1.2), when the coefficient δ = 1. We proved that the solution of (1.1) with (1.2) and δ = 1 exists and satisfies the sharp time decay estimate u (t) L p ≤ C t …”

“…A nonlocal nonlinear Schrödinger equation with nonlinearity |x − y| −δ |u| 2 dy u, δ ∈ (0, 1), is also classified as subcritical (see paper [12]). Note that a cubic nonlinearity is not necessarily critical; for example, the nonlinearity ∂ x (u 3 ) is super critical due to its oscillating structure (see [13]). Also note that the quadratic nonlinearity (u x ) 2 is super critical (see [22]) and we do not know when a quadratic nonlinearity is critical or subcritical.…”

“…In the super critical case the scattering problem was studied in papers [3], [4], [8], [17], [19], [23], [24], [25] and in the critical case it was treated in [2], [11], [15], [20], [21]. The existence of modified scattering states was shown in papers [11], [15], [20] and modified wave operators were constructed in [2], [21]. In our previous paper [15] we considered the Cauchy problem (1.1) with cubic derivative nonlinearity (1.2), when the coefficient δ = 1.…”

Scattering Problem for the Supercritical Nonlinear Schrödinger Equation in 1d

“…When we di¤erentiate equation (4), then we encounter an undesirable time growth which appears from E jÀ1 , for j 0 1: Therefore we use the method originated by [6] and developed in [9].…”