Nonlinear Schrodinger-Airy system

The nonlinear Schrodinger-Airy system

$\partial_t u + i c \partial_x^2 u + \partial_x^3 u = i \gamma |u|^2 u + \delta |u|^2 \partial_x u + \epsilon u^2 \partial_x u$

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 $c=\delta=\epsilon = 0\,$ , 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 $s\geq1/4 \,$ 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