Selection of real regular finite gap-solutions of soliton equations - I

Lecture 1: “Easy case” 1. The problem of selecting real and real regular solutions. Real algebraic curves. Real ovals. M-curves. 2. “Easy case” and “difficult case”. 3. Real solutions of the Korteweg–de Vries equation. Real regular solutions. Regular and singular real components. 4. Defocusing Nonlinear Schr¨odinger equation. Real regular solutions. Regular and singular real components. 5. Real regular Kadomtsev-Petviashvili II solutions and M-curves.