Norming constants of embedded bound states and bounded positon solutions of the Korteweg-de Vries equation

TL;DR

This paper investigates the impact of embedded eigenvalues on KdV solutions, revealing bounded positon solutions and their relation to multi-soliton and multi-positon solutions, using the binary Darboux transformation.

Contribution

It demonstrates that embedded eigenvalues lead to bounded positon solutions in the KdV equation, extending understanding of spectral effects on nonlinear wave solutions.

Findings

Embedded eigenvalues produce additional explicit terms in KdV solutions.

Bounded positon solutions exist, answering Matveev's question.

The solutions resemble multi-soliton and multi-positon solutions but remain bounded.

Abstract

In the context of the full line Schrodinger equation, we revisit the binary Darboux transformation (double commutation method) which inserts or removes any number of positive eigenvalues embedded into the absolutely continuous spectrum without altering the rest of scattering data. We then show that embedded eigenvalues produce an additional explicit term in the KdV solution. This term looks similar to multi-soliton solution and describes waves traveling in the direction opposite to solitons. It also resembles the known formula for (singular) multi-positon solutions but remains bounded, which answers in the affirmative Matveev's question about existence of bounded positons.