Exponential stability estimate for derivative nonlinear Schrödinger equation

Abstract

We consider the derivative nonlinear Schrödinger equations $i{u}_t=-u_{xx}+V\ast u+i{\partial }_x\left({\partial }_{\bar{u}}F\left(u,\bar{u}\right)\right),x\in T$with a nonlinearity $F(u,\bar{u})$ of order at least $3$ at the origin. We prove that for almost all $V$, if the initial data is $\varepsilon$-small in the modified Sobolev space, the solution is stable over time intervals of order ${\varepsilon }^{-\frac{1}{\varepsilon }}\cdot e^{e^{\frac{1}{\varepsilon }}}$. Our findings extend the stability time $e^{\frac{\left(ln\varepsilon \right)}{\varepsilon }}$ introduced by Cong (2022) to $e^{e^{\frac{1}{\varepsilon }}}$.