Nonlinear Schrodinger-Airy system

The nonlinear Schrodinger-Airy system

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

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 propagation 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 -1/2.

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 ${\displaystyle \delta =\epsilon =0\,}$ LWP is known for ${\displaystyle s>-1/4\,}$ Cv2004

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