Nonlinear Schrodinger-Airy system

The nonlinear Schrodinger-Airy system

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

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

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

For $s>3/4$ this is in Lau1997, Lau2001

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

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

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

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