New Reductions and Nonlinear Systems for 2D Schrodinger Operators

Details New Reductions and Nonlinear Systems for 2D Schrodinger Operators New Reductions and Nonlinear Systems for 2D Schrodinger Operators

2010-01-25

arxiv.org/abs/1001.4300v3

Nonlinear Sciences

Exactly Solvable and Integrable Systems

LaTeX, 13 pages, 2 fugures

Scientific paper

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 $\Gamma\subset W$ in the full complex 2D manifold of Bloch-Floquet functions W for the periodic elliptic 2D operator H where $H\psi(x,y,P)=\lambda(P)\psi(x,y,P),P\in \Gamma$. 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 $\Gamma$ except maybe one selected level $H\psi=const$ 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.

Affiliated with

Also associated with

No associations

Rating

New Reductions and Nonlinear Systems for 2D Schrodinger Operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with New Reductions and Nonlinear Systems for 2D Schrodinger Operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and New Reductions and Nonlinear Systems for 2D Schrodinger Operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-130142