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A fractional extension of the integrable Toda lattice with decaying data on the line is obtained. Completeness of squared eigenfunctions of a linear discrete real tridiagonal eigenvalue problem is derived. This completeness relation allows nonlinear evolution equations expressed in terms of operators to be written in terms of underlying squared eigenfunctions and is related to a discretization of the continuous Schrödinger equation. The methods are discrete counterparts of continuous ones recently used to find fractional integrable extensions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger (NLS) equations. Inverse scattering transform (IST) methods are used to linearize and find explicit soliton solutions to the integrable fractional Toda (fToda) lattice equation. The methodology can also be used to find and solve fractional extensions of a Toda lattice hierarchy.
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Symmetric Toda, gradient flows, and tridiagonalization
The Toda lattice (1967) is a Hamiltonian system given by n points on a line governed by an exponential potential. Flaschka (1974) showed that the Toda lattice is integrable by interpreting it as a flow on the space of symmetric tridiagonal n × n matrices, while Moser (1975) showed that it is a gradient flow on a projective space. The symmetric Toda flow of Deift, Li, Nanda, and Tomei (1986) generalizes the Toda lattice flow from tridiagonal to all symmetric matrices. They showed the flow is integrable, in the classical sense of having d integrals in involution on its 2d-dimensional phase space. The system may be viewed as integrable in other ways as well. Firstly, Symes (1980, 1982) solved it explicitly via QR-factorization and conjugation. Secondly, Deift, Li, Nanda, and Tomei (1986) ‘tridiagonalized’ the system into a family of tridiagonal Toda lattices which are solvable and integrable. In this paper we derive their tridiagonalization procedure in a natural way using the fact that the symmetric Toda flow is diffeomorphic to a twisted gradient flow on a flag variety, which may then be decomposed into flows on a product of Grassmannians. These flows may in turn be embedded into projective spaces via Plücker embeddings, and mapped back to tridiagonal Toda lattice flows using Moser’s construction. In addition, we study the tridiagonalized flows projected onto a product of permutohedra, using the twisted moment map of Bloch, Flaschka, and Ratiu (1990). These ideas are facilitated in a natural way by the theory of total positivity, building on our previous work (2023).
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- Award ID(s):
- 2103026
- PAR ID:
- 10515504
- Publisher / Repository:
- ElsevierToda flows Integrability Gradient flows
- Date Published:
- Journal Name:
- Physcia D
- Volume:
- 450
- ISSN:
- 1550-2368
- Page Range / eLocation ID:
- 133766
- Subject(s) / Keyword(s):
- Toda flows Integrability Gradient flows
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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