The adiabatic complex WKB method (joint with A. Fedotov):

Consider the equation


Problem : how to decouple slow and fast variable?

Buslaev's method : go over to semi-classical PDE : adiabatic real WKB method.

Our method : introduce a new parameter  to get


A priori  is free variable; for consistent solutions, it is not so as

The new parameter will play the role of the slow variable; and we let it be complex.