Bounded Solutions of KdV and Non-Periodic One-Gap Potentials in Quantum Mechanics
- 126 Downloads
We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrödinger operators have the same structure as those of N-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by 2N real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.
Keywordsintegrable systems Schrödinger operator soliton solutions
Mathematics Subject Classification70H06 81U15
Unable to display preview. Download preview PDF.
- 4.Belokolos, E.D., Gesztesy, F., Makarov, K.A., Sakhnovich, L.A.: Matrix-valued generalizations of the theorems of Borg and Hochstadt. In: Evolution equations, Lecture Notes in Pure and Appl. Math., vol. 234, pp. 1–34. Dekker, New York (2003)Google Scholar
- 5.Marchenko, V.: Nonlinear equations and operator algebras. Math. and its Appl. (Sov. Ser.), vol. 17. D. Reidel Publishing Co., Dordrecht (1988)Google Scholar
- 10.Shabat, A.B.: On potentials with zero reflection coefficient. Dinamika sploshnoi sredy, vol. 5, p. 130. Novosibirsk, 1970Google Scholar
- 11.Krichever, I.: Private communicationGoogle Scholar
- 15.Its, A.R.: Liouville’s theorem and the method of the inverse problem. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 133, 113–125 (1984)Google Scholar
- 16.Deift, P.A., Its, A.R., Zhou, X.: A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. (2) 146(1), 149–235 (1997)Google Scholar