# Nonlinear Schrodinger-Airy system

The nonlinear Schrodinger-Airy system

${\displaystyle \partial _{t}u+ic\partial _{x}^{2}u+\partial _{x}^{3}u=i\gamma |u|^{2}u+\delta |u|^{2}\partial _{x}u+\epsilon u^{2}\partial _{x}u}$

R is a combination of the cubic NLS equation, the derivative cubic NLS equation, complex mKdV, and a cubic nonlinear Airy equation. This equation is a general model for propogation of pulses in an optical fiber Kod1985, HasKod1987

When ${\displaystyle c=\delta =\epsilon =0}$, scaling is ${\displaystyle s=-1}$.When ${\displaystyle c=\gamma =0}$, scaling is \u20131/2.

LWP is known when ${\displaystyle s\geq 1/4}$. St1997d

For ${\displaystyle s>3/4}$ this is in Lau1997, Lau2001

The ${\displaystyle s\geq 1/4}$ result is also known when ${\displaystyle c}$ is a time-dependent function [Cv2002], [CvLi2003]

For ${\displaystyle s<-1/4}$ and ${\displaystyle \delta }$ or ${\displaystyle \epsilon }$ non-zero, the solution map is not ${\displaystyle C^{3}}$.

When delta = epsilon = 0 LWP is known for s > -1/4 Cv2004

For ${\displaystyle s<-1/4}$ the solution map is not C^3 [CvLi-p]