Lax pair and cubic nonlinear Schrödinger equation

Question

Motivation: I'm trying to understand the Section 4 in the this section 4 in paper" Low regularity conservation laws for integrable PDE by Killip-Visan-Zhang ö

It reads as follows: many completely integrable PDE admit a Lax pair of the following form: $$\frac{d}{dt} L(t,\kappa)=[P(t, \kappa), L(t, \kappa)]$$ with $$L(t,\kappa)= \begin{bmatrix} -\partial+ \kappa & iq(x)\\ \mp I \bar{q}(x) & -\partial-\kappa \end{bmatrix} $$

and some operator pencil $P(t, \kappa).$

My Question: Can we say following cubic NLS $$-i\frac{d}{dt}q= -\beta(t)q'' \pm 2|q|^2q$$ where $\beta(t+1)=\beta(t)$ for all $t\in \mathbb R$ and $\beta$ is piecewise constant with finitely many discontinuity in $[0,1],$ admit aLax pair? If so what is $L$ and $P$?

Answer 1

The existence of Bäcklund transformations and associated linear problems for generalised nonlinear Schrödinger equations of the form $z_{xx} +iz =f(z,z^*)$ has been analysed by

Harnad and Winternitz, Pseudopotentials and Lie symmetries for the generalised nonlinear Schrödinger equation, J. Math. Phys. 23(4), April 1982

It should be possible - at least formally - to extend their analysis to your case.

Because your PDE includes an independent variable in the coefficients, you may also wish to review the note by B.A. Kupershmidt, Nonautonomous form of the theory of Lax equations, Lettere al Nuovo Cimento 33 (1982) 103–107.