Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1001.4300

New Reductions and Nonlinear Systems for 2D Schrodinger Operators

P. Grinevich·A. Mironov·S. Novikov
View original source ↗Read source document ↗Export metadata ↓

Abstract

New Completely Integrable (2+1)-System is studied. It is based on the so-called L-A-B-triples $L_t=[H,L]-fL$ where L is a 2D Schrodinger Operator. This approach was invented by S.Manakov and B.Dubrovin, I.Krichever, S.Novikov(DKN) in the works published in 1976. A nonstandard reduction for the 2D Schrodinger Operator (completely different from the one found by S.Novikov and A.Veselov in 1984) compatible with time dynamics of the new Nonlinear System, is studied here. It can be naturally treated as a 2D extension of the famous Burgers System. The Algebro-Geometric (AG) Periodic Solutions here are very specific and unusual (for general and reduced cases). The reduced system is linearizable like Burgers. However, the general one (and probably the reduced one also) certainly lead in the stationary AG case to the nonstandard examples of algebraic curves $Γ\subset W$ in the full complex 2D manifold of Bloch-Floquet functions W for the periodic elliptic 2D operator H where $Hψ(x,y,P)=λ(P)ψ(x,y,P),P\in Γ$. However, in the nontrivial case the operators are nonselfadjoint. A Conjecture is formulated that for the nontrivial selfadjoint elliptic 2D Schrodinger operators H with periodic coefficients The Whole 2D Complex Manifolds W cannot not contain any Zariski open part of algebraic curve $Γ$ except maybe one selected level $Hψ=const$ found in 1976 by DKN. Version 2 contains new results. It also corrects some non-accurate claims; in particular, the non-reduced system is non-linearizable in any trivial sense. In version 3 a misprint in the formulation of Theorem 1 was corrected.

Explore related subjects

nlin.SImath-phmath.MPmath.SPquant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

P. Grinevich, A. Mironov, S. Novikov. 2010-02-09. New Reductions and Nonlinear Systems for 2D Schrodinger Operators. https://arxiv.org/abs/1001.4300

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On vector Schwarz-KdV equation

A collection of miscellaneous continuous, semi-discrete, and discrete integrable systems can be associated with each integrable evolution equation of the KdV type. We give them for the Schwarz--KdV equation and generalize to the vector case. The existence of these vector generalizations is a non-trivial experimental fact for which no mathematical explanation is yet known.

nlin.SI↗
arXiv2026

An integrable $\mathbb{Z}_2^2$-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure

By constructing Lax operators in the loop algebra of the $\mathbb{Z}_2^2$-graded extension of the Lie superalgebra $\mathfrak{osp}(1|2)$, we derive a $\mathbb{Z}_2^2$-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four $\mathbb{Z}_2^2$-graded commutative functions, each associated with a distinct $\mathbb{Z}_2^2$-degree. We further show that the $\mathbb{Z}_2^2$-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial $\mathbb{Z}_2^2$-degree, which are mutually in involution with respect to the $\mathbb{Z}_2^2$-graded Poisson brackets.

nlin.SI↗
arXiv2026

Integrability of the deformed Toda systems

In 2020 M. Mucciconi and L. Petrov introduced a long-range deformation of the quantum open non-relativistic Toda system. We prove the integrability of the deformed Toda system by constructing a $2 \times 2$ Lax operator, which produces the commutative family of differential operators containing the Hamiltonian of the deformed Toda system. Moreover, we show that the same integrable deformation exists on both classical and quantum levels and can be applied to both non-relativistic and relativistic Toda systems. For the open non-relativistic deformed Toda systems we also present an $n \times n$ Lax matrix and prove that it produces the same family of Hamiltonians. We also show how to obtain the van Diejen-type deformed Toda system. Lastly, we show that on the quantum level the algebraic Bethe ansatz technique can be applied to the deformed Toda system.

nlin.SI↗