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Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 509--518 | DOI:10.5890/DNC.2020.12.003

Ya. Dymarskii

Department of Higher Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Russia

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Abstract

We give an analytic and topological description of the Uhlenbeck manifold, that is a manifold of triples (a symmetric operator, an eigenvector, an eigenvalue), for the finite-dimensional symmetric matrices and the family of stationary periodic Schrodinger operators. Then, we describe an uplifting of Lax vector fields to these manifolds.

References

1. [1]	Uhlenbeck, K. (1976), Generic properties of eigenfunctions, Amer. J. Math., 98(4), 1059-1078.

2. [2]	Vishik, M.I. and Kuksin, S.B. (1985), Quasilinesr elliptic equations and Fredholm manifolds, Vestn. MGU, Ser. 1 Mat. Meckh., (6), 23-30.

3. [3]	Dymarskii, Y.M. (1985), Existence, oscillatory properties, and asymptotics of normed eigenfunctions of nonlinear boundary-value problems, Qualitative and Approximate Methods for Studying Operator Equations, Yaroslavl' state university, 133--139.

4. [4]	Dymarskii, Y.M. (2008), Manifold method in eigenvector theory of nonlinear operators, Journal of Mathematical Sciences, 154(5), 655--815.

5. [5]	Dymarskii, Y.M. and Evtushenko, Y.A. (2016), Foliation of the space of periodic boundary-value problems by hypersurfaces corresponding to fixed lengths of the nth spectral lacuna, Sbornik: Mathematics, 207(5), 678--701.

6. [6]	Lax, P. (1968), Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Applied Math., 21(5), 467--490.

7. [7]	Ablowitz, P. and Segur, H. (1981), Solitons and the inverse scattering transform, SIAM: Philadelphia.

8. [8]	Arnold, V.I. (1972), Modes and quasimodes, Funkts. Anal. Pril., 6(2), 94--101.

9. [9]	Coddington, E.A. and Levinson, N. (1955), Theory og Ordinary Differential Equations, McGraw-Hill: New York.