Abstract:
A survey is given of the spectral properties of matrix finite-zone operators. Conditions of the type of JJ-self-adjointness for such operators and explicit formulas expressing the coefficients of such operators in terms of theta functions are obtained. The simplest examples of such JJ-self-adjoint, finite-zone operators turn out to be connected with the theory of ovals of plane, real, algebraic curves.
This publication is cited in the following 51 articles:
Aleksandr O. Smirnov, Eugene A. Frolov, Lada L. Dmitrieva, “On a Hierarchy of Vector Derivative Nonlinear Schrödinger Equations”, Symmetry, 16:1 (2024), 60
A. O. Smirnov, I. V. Anisimov, “Finite-gap solutions of the real modified Korteweg–de Vries equation”, Theoret. and Math. Phys., 220:1 (2024), 1224–1240
A. O. Smirnov, S. D. Shilovsky, “On vector derivative nonlinear Schrödinger equation”, Ufa Math. J., 16:3 (2024), 92–106
A. O. Smirnov, A. A. Caplieva, “Vector form of Kundu–Eckhaus equation and its simplest solutions”, Ufa Math. J., 15:3 (2023), 148–163
Vladimir Stefanov Gerdjikov, Aleksander Aleksiev Stefanov, “Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions”, Symmetry, 15:10 (2023), 1933
Vladimir S. Gerdjikov, Aleksandr O. Smirnov, “On the elliptic null-phase solutions of the Kulish–Sklyanin model”, Chaos, Solitons & Fractals, 166 (2023), 112994
Aleksandr O. Smirnov, Eugeni A. Frolov, “On the Propagation Model of Two-Component Nonlinear Optical Waves”, Axioms, 12:10 (2023), 983
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Xianguo Geng, Jiao Wei, “Three-sheeted Riemann surface and solutions of the Itoh–Narita–Bogoyavlensky lattice hierarchy”, Rev. Math. Phys., 34:04 (2022)
Aleksandr O. Smirnov, “Spectral Curves for the Derivative Nonlinear Schrödinger Equations”, Symmetry, 13:7 (2021), 1203
A O Smirnov, V S Gerdjikov, E E Aman, “The Kulish-Sklyanin type hierarchy and spectral curves”, IOP Conf. Ser.: Mater. Sci. Eng., 1047:1 (2021), 012114
A O Smirnov, A S Kolesnikov, “Dubrovin's method and Ablowitz-Kaup-Newell-Segur hierarchy”, IOP Conf. Ser.: Mater. Sci. Eng., 1181:1 (2021), 012028
A O Smirnov, E G Filimonova, V B Matveev, “The spectral curve method for the Kaup-Newell hierarchy”, IOP Conf. Ser.: Mater. Sci. Eng., 919:5 (2020), 052051
V. S. Gerdjikov, A. O. Smirnov, V. B. Matveev, “From generalized Fourier transforms to spectral curves for the Manakov hierarchy. I. Generalized Fourier transforms”, Eur. Phys. J. Plus, 135:8 (2020)
Alexander Moll, “Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation”, Quart. Appl. Math., 78:4 (2020), 671
A O Smirnov, E A Frolov, V S Gerdjikov, “Spectral curves for the multi-phase solutions of Manakov system”, IOP Conf. Ser.: Mater. Sci. Eng., 862:5 (2020), 052041
A. O. Smirnov, V. S. Gerdjikov, V. B. Matveev, “From generalized Fourier transforms to spectral curves for the Manakov hierarchy. II. Spectral curves for the Manakov hierarchy”, Eur. Phys. J. Plus, 135:7 (2020)
A. B. Zheglov, “Surprising examples of nonrational smooth spectral surfaces”, Sb. Math., 209:8 (2018), 1131–1154
B. Dubrovin, T. Skrypnyk, “Separation of variables for linear Lax algebras and classical r-matrices”, Journal of Mathematical Physics, 59:9 (2018)
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