2010/01/25 by P. G. Grinevich, P. Grinevich, Grinevich, P. +6
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Physics (quant-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #nlin.SI #quant-ph
paper · pdf · doi:10.48550/arxiv.1001.4300
LaTeX, 13 pages, 2 fugures
openalex publication_date 2010/01/25 · arxiv created 2010/02/09 · arxiv updated 2010/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
New Completely Integrable (2+1)-System is studied. It is based on the so-called L-A-B-triples Lt=[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⊂ 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∈ 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.