A note on decay rates of solutions to a system of cubic nonlinear Schrödinger equations in one space dimension

Abstract

Abstract:We consider the initial value problem for a system of cubic nonlinear Schrödinger equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution exists globally and decays of the rate O(t −1/2 (log t) −1/2 ) in L ∞ as t tends to infinity, if the system satisfies certain mass relations.

On Schrödinger systems with cubic dissipative nonlinearities of derivative type

“…(see Example 2.1 in [9] for the detail). The advantage of the method of [5] is that it does not rely on the explicit asymptotic profile at all.…”

“…This gain of additional logarithmic time decay should be interpreted as another kind of long-range effect. Among several extensions of this result (see e.g., [3], [9], [11], [12], [13] etc. and the references cited therein), let us focus on the following two cases: (i) the case where the nonlinearity depends also on ∂ x u, and (ii) the case of systems.…”

“…Strictly saying, two-dimensional quadratic nonlinear Schrödinger systems are treated in [5], but we can adopt the method of [5] directly to one-dimensional cubic nonlinear Schrödinger systems, as pointed in [9]. When we restrict ourselves to a two-component model…”

Long-Long or Long-Short Range Interactions of Nonlinear Schrödinger Systems in One Space Dimension

“…For the Schrödinger systems with long range interactions, asymptotic behavior of solutions was known on the final state problem (see, e.g., [21], [9]). There are some results on the initial value problem for the Schrödinger systems (see e.g., [8], [13], [14] and [18]) and time decay estimates of the solutions were studied in some critical cases. The existence of ground states for some nonlinear Schrödinger systems was investigated in [11].…”

Asymptotic behavior of solutions to the cubic coupled Schrödinger systems in one space dimension

“…Indeed this coupling effect can provide more nonlinear asymptotic behavior than a single equation. For example, Grébert, Paturel and Thomann exhibited some beating effects (see [3]), that is to say some energy exchanges between different modes of the solutions, although they were just dealing with the spatial domain T. Another example of study of Schrödinger systems is given by Kim in [9]. Considering relations on the masses in the equations, the author obtains L ∞ decay of the solution for the spatial domain R. The goal of this article is to present a self contained proof of nonlinear behavior in the spatial domain R. Following the method of Kato and Pusateri in [8], we will first prove a L ∞ decay of the solutions.…”