KdV Equation in the Quarter–Plane: Evolution of the Weyl Functions and Unbounded Solutions

Abstract

Abstract. The matrix KdV equation with a negative dispersion term is considered in the right upper quarter-plane. The evolution law is derived for the Weyl function of a corresponding auxiliary linear system. Using the low energy asymptotics of the Weyl functions, the unboundedness of solutions is obtained for some classes of the initial-boundary conditions.

Evolution of Weyl Functions and Initial-Boundary Value Problems

“…Indeed, the solution constructed in Proposition 5.6 satisfies the conditions of Proposition 5.5 excluding, possibly, (5.16), and so (5.16) does not hold. Some classes of unbounded solutions of the KdV equation with the minus sign before the dispersion term are constructed in [52] using low energy asymptotics of the Weyl functions.…”

“…Some classes of unbounded solutions of the KdV equation with the minus sign before the dispersion term are constructed in [52] using low energy asymptotics of the Weyl functions.…”