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Dokl. Akad. Nauk SSSR, 1984, Volume 279, Number 1, Pages 20–24 (Mi dan9374)  

This article is cited in 58 scientific papers (total in 59 papers)

MATHEMATICS

Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations

A. P. Veselov, S. P. Novikov

Landau Institute for Theoretical Physics, USSR Academy of Sciences, Chernogolovka Moscow region

Full text: PDF file (644 kB)

Bibliographic databases:
UDC: 517.984+512.77
Received: 06.06.1984

Citation: A. P. Veselov, S. P. Novikov, “Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations”, Dokl. Akad. Nauk SSSR, 279:1 (1984), 20–24

Citation in format AMSBIB
\Bibitem{VesNov84}
\by A.~P.~Veselov, S.~P.~Novikov
\paper Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations
\jour Dokl. Akad. Nauk SSSR
\yr 1984
\vol 279
\issue 1
\pages 20--24
\mathnet{http://mi.mathnet.ru/dan9374}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=0769198}
\zmath{https://zbmath.org/?q=an:0613.35020}


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    This publication is cited in the following articles:
    1. P. G. Grinevich, R. G. Novikov, “Analogs of multisoliton potentials for the two-dimensional Schrödinger operator”, Funct. Anal. Appl., 19:4 (1985), 276–285  mathnet  crossref  mathscinet  zmath  isi
    2. P. G. Grinevich, “Rational solitons of the Veselov–Novikov equations are reflectionless two-dimensional potentials at fixed energy”, Theoret. and Math. Phys., 69:2 (1986), 1170–1172  mathnet  crossref  mathscinet  zmath  isi
    3. P. G. Grinevich, S. V. Manakov, “Inverse scattering problem for the two-dimensional Schrödinger operator, the $\bar\partial$-method and nonlinear equations”, Funct. Anal. Appl., 20:2 (1986), 94–103  mathnet  crossref  mathscinet  zmath
    4. A. P. Veselov, “Integration of the stationary problem for a classical spin chain”, Theoret. and Math. Phys., 71:1 (1987), 446–450  mathnet  crossref  mathscinet  isi
    5. P. G. Grinevich, S. P. Novikov, “Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions. I. Energies below the ground state”, Funct. Anal. Appl., 22:1 (1988), 19–27  mathnet  crossref  mathscinet  zmath  isi
    6. S. M. Natanzon, “Nonsingular finite-zone two-dimensional Schrödinger operators and prymians of real curves”, Funct. Anal. Appl., 22:1 (1988), 68–70  mathnet  crossref  mathscinet  zmath  isi
    7. I. M. Krichever, “Method of averaging for two-dimensional “integrable” equations”, Funct. Anal. Appl., 22:3 (1988), 200–213  mathnet  crossref  mathscinet  zmath  isi
    8. A. P. Veselov, “Integrable discrete-time systems and difference operators”, Funct. Anal. Appl., 22:2 (1988), 83–93  mathnet  crossref  mathscinet  zmath  isi
    9. I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications”, Russian Math. Surveys, 44:2 (1989), 145–225  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. S. M. Natanzon, “Prymians of real curves and their applications to the effectivization of Schrödinger operators”, Funct. Anal. Appl., 23:1 (1989), 33–45  mathnet  crossref  mathscinet  zmath  isi
    11. S. M. Natanzon, “Klein surfaces”, Russian Math. Surveys, 45:5 (1990), 53–108  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. I. A. Taimanov, “Prym varieties of branched coverings and nonlinear equations”, Math. USSR-Sb., 70:2 (1991), 367–384  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. I. Yu. Cherdantsev, R. A. Sharipov, “Finite-gap solutions of the Bullough–Dodd–Zhiber–Shabat equation”, Theoret. and Math. Phys., 82:1 (1990), 108–11  mathnet  crossref  mathscinet  zmath  isi
    14. S. M. Natanzon, “Differential equations on the Prym theta function. a realness criterion for two-dimensional, finite-zone, potential Schrödinger operators”, Funct. Anal. Appl., 26:1 (1992), 13–20  mathnet  crossref  mathscinet  zmath  isi
    15. L. V. Ryzhik, E. I. Shulman, “Symmetry algebra of nonlinear integrable equations”, Theoret. and Math. Phys., 95:1 (1993), 387–392  mathnet  crossref  mathscinet  zmath
    16. I. M. Krichever, “Two-Dimensional Algebraic-Geometric Operators with Self-Consistent Potentials”, Funct. Anal. Appl., 28:1 (1994), 21–32  mathnet  crossref  mathscinet  zmath  isi
    17. T. I. Garagash, A. K. Pogrebkov, “Scattering problem for the differential operator $\partial_x\partial_y+1+a(x,y)\partial_y+ b(x,y)$”, Theoret. and Math. Phys., 102:2 (1995), 117–132  mathnet  crossref  mathscinet  zmath  isi
    18. V. M. Buchstaber, V. Z. Ènol'skii, “Abelian Bloch solutions of the two-dimensional Schrödinger equation”, Russian Math. Surveys, 50:1 (1995), 195–197  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    19. V. I. Vakulenko, “Solution of Virasoro constraints for DKP hierarhy”, Theoret. and Math. Phys., 107:1 (1996), 435–440  mathnet  crossref  crossref  mathscinet  zmath  isi
    20. I. M. Krichever, “Algebraic-Geometric $n$-Orthogonal Curvilinear Coordinate Systems and Solutions of the Associativity Equations”, Funct. Anal. Appl., 31:1 (1997), 25–39  mathnet  crossref  crossref  mathscinet  zmath  isi
    21. I. A. Taimanov, “Secants of Abelian varieties, theta functions, and soliton equations”, Russian Math. Surveys, 52:1 (1997), 147–218  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    22. I. A. Taimanov, “The Weierstrass Representation of Closed Surfaces in $\mathbb{R}^3$”, Funct. Anal. Appl., 32:4 (1998), 258–267  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    23. R. G. Novikov, “Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2”, Proc. Steklov Inst. Math., 225 (1999), 285–302  mathnet  mathscinet  zmath
    24. P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Russian Math. Surveys, 55:6 (2000), 1015–1083  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    25. V. E. Adler, A. B. Shabat, R. I. Yamilov, “Symmetry approach to the integrability problem”, Theoret. and Math. Phys., 125:3 (2000), 1603–1661  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    26. A. A. Oblomkov, “Isoenergy Spectral Problem for Multidimensional Difference Operators”, Funct. Anal. Appl., 36:2 (2002), 120–133  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    27. I. A. Taimanov, “On two-dimensional finite-gap potential Schrödinger and Dirac operators with singular spectral curves”, Siberian Math. J., 44:4 (2003), 686–694  mathnet  crossref  mathscinet  zmath  isi  elib
    28. P. R. Gordoa, “Algebraic and Differential Nonlinear Superposition Formulas”, Theoret. and Math. Phys., 137:1 (2003), 1430–1438  mathnet  crossref  crossref  mathscinet  zmath  isi
    29. L. V. Bogdanov, B. G. Konopelchenko, A. Moro, “Symmetry constraints for real dispersionless Veselov–Novikov equation”, J. Math. Sci., 136:6 (2006), 4411–4418  mathnet  crossref  mathscinet  zmath  elib  elib
    30. A. E. Mironov, “Ierarkhiya uravnenii Veselova–Novikova i integriruemye deformatsii minimalnykh lagranzhevykh torov v $\mathbb CP^2$”, Sib. elektron. matem. izv., 1 (2004), 38–46  mathnet  mathscinet  zmath
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    32. A. E. Mironov, “Relationship Between Symmetries of the Tzizeica Equation and the Novikov–Veselov Hierarchy”, Math. Notes, 82:4 (2007), 569–572  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    33. Wang Hong-Yan, “The Nizhnik–Veselov–Novikov equation with self-consistent sources”, Theoret. and Math. Phys., 157:1 (2008), 1474–1483  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    34. I. A. Taimanov, S. P. Tsarev, “Two-dimensional rational solitons and their blowup via the Moutard transformation”, Theoret. and Math. Phys., 157:2 (2008), 1525–1541  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    35. I. M. Krichever, “Abelian solutions of the soliton equations and Riemann–Schottky problems”, Russian Math. Surveys, 63:6 (2008), 1011–1022  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    37. V. G. Dubrovskii, A. V. Gramolin, “Gauge-invariant description of several $(2+1)$-dimensional integrable nonlinear evolution equations”, Theoret. and Math. Phys., 160:1 (2009), 905–916  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
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    42. E. Sh. Gutshabash, “Moutard transformation and its application to some physical problems. I. The case of two independent variables”, J. Math. Sci. (N. Y.), 192:1 (2013), 57–69  mathnet  crossref  mathscinet
    43. A. V. Kazeykina, “Absence of Conductivity-Type Solitons for the Novikov–Veselov Equation at Zero Energy”, Funct. Anal. Appl., 47:1 (2013), 64–66  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
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    45. I. A. Taimanov, S. P. Tsarev, “Faddeev eigenfunctions for two-dimensional Schrödinger operators via the Moutard transformation”, Theoret. and Math. Phys., 176:3 (2013), 1176–1183  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    46. B. O. Vasilevskiǐ, “The Green's function of a five-point discretization of a two-dimensional finite-gap Schrödinger operator: The case of four singular points on the spectral curve”, Siberian Math. J., 54:6 (2013), 994–1004  mathnet  crossref  mathscinet  isi
    47. A. V. Kazeykina, “Absence of Solitons with Sufficient Algebraic Localization for the Novikov–Veselov Equation at Nonzero Energy”, Funct. Anal. Appl., 48:1 (2014), 24–35  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    48. I. A. Taimanov, “Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces”, Theoret. and Math. Phys., 182:2 (2015), 173–181  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    49. P. G. Grinevich, A. E. Mironov, S. P. Novikov, “On the non-relativistic two-dimensional purely magnetic supersymmetric Pauli operator”, Russian Math. Surveys, 70:2 (2015), 299–329  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
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