Nonlinear Schrodinger-Airy system

The nonlinear Schrodinger-Airy system

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

on 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 -1/2.

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