The effect of a positive bound state on the KdV solution: a case study ^*

Abstract

We consider a slowly decaying oscillatory potential such that the corresponding 1D Schrödinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial data for which the Cauchy problem for the Korteweg-de Vries (KdV) equation can be solved by the inverse scattering transform. We nevertheless show that the KdV equation with our potential does admit a closed form classical solution in terms of Hankel operators. Comparing with rapidly decaying initial data our solution gains a new term responsible for the positive eigenvalue. To some extent this term resembles a positon (singular) solution but remains bounded. Our approach is based upon certain limiting arguments and techniques of Hankel operators.

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

“…In the KdV context, Matveev posed in [24] the following question: "The interesting question whether nonsingular positon solutions exists in the continuous integrable models remains open as yet." Corollary 3.5 answers his question in the affirmative (for one positon it was answer in our recent [26]). Matveev also conjectured that there may exist bounded positon solutions with a trivial scattering matrix (i.e.…”

“…n is an embedded bound state. As we show in [26], the reflection coefficient R (k) alone can not tell if a resonance is a bound state or not. Therefore an extra condition is required.…”

“…Pelinovsky also asked "Does the "embedded solitons" disperse away in the time evolution?". Addressing this question amounts to understanding the behavior of ψ (x, t, ω n ) in the asymptotic regime around the "positon characteristic" x = −12ω 2 n t as t → ∞ (see our [26] for more detail). The main challenge is that |R (ω n )| = 1 and the powerful nonlinear steepest descend method due to Deift-Zhou needs a serious modification, which to the best of our knowledge is only available in the case when |R (0)| = 1 but less than 1 otherwise [7].…”

“…It is important that Wigner-von Neumann potentials are in L 2 and due to the seminal Bourgain's result [4] (1.1) remains well-posed. In our recent [26] we use L 2 well-posedness to treat a specific case of Wigner-von Neumann type of initial data (I.e. q (x) = (A/x) sin 2ωx+O x −2 ) that gives a hint for how IST may be extended shading some light on Vladimir Matveev's proposal [9]: "A very interesting unsolved problem is to study the large time behavior of the solutions to the KdV equation corresponding to the smooth initial data like cx −1 sin 2kx, c ∈ R", "The related inverse scattering problem is not yet solved and the study of the related large times evolution is a very challenging problem".…”

The binary Darboux transformation revisited and KdV solitons on arbitrary short‐range backgrounds

“…Note it holds true only for short-range potentials. (See, e.g., our recent work26 for explicit counterexamples. )…”