Large time asymptotics of solutions to the periodic problem for the quadratic nonlinear Schrödinger equation

Abstract

We study the large time asymptotics of solutions to the periodic problem for the quadratic nonlinear Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{c} u_{t}+iu_{xx}=-u^{2},\text { }x\in \left[ -\pi ,\pi \right] ,t>0, \\ u(0,x)=\phi \left( x\right) ,\text { }x\in \left[ -\pi ,\pi \right] . \end{array} \right. \end{aligned}$$

We assume that the initial data $\phi \left( x\right)$ are $2\pi$ - periodic and have small amplitude. Then we show that the periodic solutions satisfy the following asymptotics

$$\begin{aligned} \left\| u\left( t\right) -\frac{\varepsilon }{1+\varepsilon t}\right\| _{{\textbf{L}}^{\infty }\left( -\pi ,\pi \right) }=O\left( \left( 1+\varepsilon t\right) ^{-2}\right) \end{aligned}$$

as $t\rightarrow \infty .$