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Uspekhi Mat. Nauk, 1976, Volume 31, Issue 5(191), Pages 245–246 (Mi umn3978)  

This article is cited in 46 scientific papers (total in 46 papers)

In the Moscow Mathematical Society
Communications of the Moscow Mathematical Society

The method of the inverse scattering problem, and two-dimensional evolution equations

S. V. Manakov

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Received: 28.01.1976

Citation: S. V. Manakov, “The method of the inverse scattering problem, and two-dimensional evolution equations”, Uspekhi Mat. Nauk, 31:5(191) (1976), 245–246

Citation in format AMSBIB
\by S.~V.~Manakov
\paper The method of the inverse scattering problem, and two-dimensional evolution equations
\jour Uspekhi Mat. Nauk
\yr 1976
\vol 31
\issue 5(191)
\pages 245--246

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    This publication is cited in the following articles:
    1. I. M. Krichever, S. P. Novikov, “Holomorphic bundles over algebraic curves and non-linear equations”, Russian Math. Surveys, 35:6 (1980), 53–79  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. V. E. Zakharov, S. V. Manakov, “Construction of higher-dimensional nonlinear integrable systems and of their solutions”, Funct. Anal. Appl., 19:2 (1985), 89–101  mathnet  crossref  mathscinet  zmath  isi
    3. 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
    4. O. I. Bogoyavlenskii, “Nekotorye konstruktsii integriruemykh dinamicheskikh sistem”, Izv. AN SSSR. Ser. matem., 51:4 (1987), 737–766  mathnet  mathscinet  zmath; O. I. Bogoyavlenskii, “Some constructions of integrable dynamical systems”, Math. USSR-Izv., 31:1 (1988), 47–75  crossref
    5. R. G. Novikov, G. M. Henkin, “The $\bar\partial$-equation in the multidimensional inverse scattering problem”, Russian Math. Surveys, 42:3 (1987), 109–180  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. L. V. Bogdanov, “Veselov–Novikov equation as a natural two-dimensional generalization of the Korteweg–de Vries equation”, Theoret. and Math. Phys., 70:2 (1987), 219–223  mathnet  crossref  mathscinet  zmath  isi
    7. O. I. Bogoyavlenskii, “Algebraicheskie konstruktsii nekotorykh integriruemykh uravnenii”, Izv. AN SSSR. Ser. matem., 52:4 (1988), 712–739  mathnet  mathscinet  zmath; O. I. Bogoyavlenskii, “Algebraic constructions of certain integrable equations”, Math. USSR-Izv., 33:1 (1989), 39–65  crossref
    8. 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
    9. I. M. Krichever, “Method of averaging for two-dimensional “integrable” equations”, Funct. Anal. Appl., 22:3 (1988), 200–213  mathnet  crossref  mathscinet  zmath  isi
    10. O. I. Bogoyavlenskii, “Oprokidyvayuschiesya solitony. III”, Izv. AN SSSR. Ser. matem., 54:1 (1990), 123–131  mathnet  mathscinet  zmath  adsnasa; O. I. Bogoyavlenskii, “Breaking solitons. III”, Math. USSR-Izv., 36:1 (1991), 129–137  crossref
    11. O. I. Bogoyavlenskii, “Breaking solitons in $2+1$-dimensional integrable equations”, Russian Math. Surveys, 45:4 (1990), 1–89  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. I. A. Taimanov, “Mnogoobraziya Prima razvetvlennykh nakrytii i nelineinye uravneniya”, Matem. sb., 181:7 (1990), 934–950  mathnet  mathscinet  zmath  adsnasa; I. A. Taimanov, “Prym varieties of branched coverings and nonlinear equations”, Math. USSR-Sb., 70:2 (1991), 367–384  crossref  isi
    13. O. I. Bogoyavlenskii, “Algebraic constructions of integrable dynamical systems-extensions of the Volterra system”, Russian Math. Surveys, 46:3 (1991), 1–64  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    14. D. Fofana, “Integriruemaya sistema, rasshiryayuschaya uravnenie Kortevega–de Friza”, Izv. AN SSSR. Ser. matem., 55:6 (1991), 1287–1299  mathnet  mathscinet  zmath  adsnasa; D. Fofana, “An integrable system extending the Korteweg–de Vries equation”, Math. USSR-Izv., 39:3 (1992), 1239–1250  crossref  isi
    15. Theoret. and Math. Phys., 99:2 (1994), 599–605  mathnet  crossref  mathscinet  zmath  isi
    16. A. I. Zenchuk, “Some generalizations of the 2-dimensional Toda chain and $\operatorname{sh}$-Gordon equation”, Theoret. and Math. Phys., 110:2 (1997), 183–189  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. 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
    18. I. M. Krichever, S. P. Novikov, “Trivalent graphs and solitons”, Russian Math. Surveys, 54:6 (1999), 1248–1249  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    19. R. G. Novikov, “Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2”, Proc. Steklov Inst. Math., 225 (1999), 285–302  mathnet  mathscinet  zmath
    20. 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
    21. I. A. Taimanov, “Two-dimensional Dirac operator and the theory of surfaces”, Russian Math. Surveys, 61:1 (2006), 79–159  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    22. Doliwa, A, “Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme”, Journal of Mathematical Physics, 48:1 (2007), 013513  crossref  mathscinet  zmath  adsnasa  isi
    23. Anatoly G.G. Meshkov, Maxim Ju. Balakhnev, “Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions”, SIGMA, 4 (2008), 018, 29 pp.  mathnet  crossref  mathscinet  zmath
    24. 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
    25. I. A. Taimanov, S. P. Tsarev, “On the Moutard transformation and its applications to spectral theory and soliton equations”, Journal of Mathematical Sciences, 170:3 (2010), 371–387  mathnet  crossref  mathscinet
    26. P. G. Grinevich, A. E. Mironov, S. P. Novikov, “2D-Schrödinger Operator, (2+1) evolution systems and new reductions, 2D-Burgers hierarchy and inverse problem data”, Russian Math. Surveys, 65:3 (2010), 580–582  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    27. P. G. Grinevich, A. E. Mironov, S. P. Novikov, “Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for spin-$1/2$ particles”, Theoret. and Math. Phys., 164:3 (2010), 1110–1127  mathnet  crossref  crossref  adsnasa  isi
    28. V. G. Dubrovskii, A. V. Topovsky, M. Yu. Basalaev, “New exact solutions with functional parameters of the Nizhnik–Veselov–Novikov equation with constant asymptotic values at infinity”, Theoret. and Math. Phys., 165:2 (2010), 1470–1489  mathnet  crossref  crossref  isi
    29. Dubrovsky V.G., Topovsky A.V., Basalaev M.Yu., “Two-dimensional stationary Schrodinger equation via the partial derivative-dressing method: New exactly solvable potentials, wave functions, and their physical interpretation”, Journal of Mathematical Physics, 51:9 (2010), 092106  crossref  isi
    30. Kazeykina A.V., Novikov R.G., “A Large Time Asymptotics for Transparent Potentials for the Novikov-Veselov Equation At Positive Energy”, J Nonlinear Math Phys, 18:3 (2011), 377–400  crossref  isi
    31. Kazeykina A.V., Novikov R.G., “Large time asymptotics for the Grinevich-Zakharov potentials”, Bull Sci Math, 135:4 (2011), 374–382  crossref  isi
    32. Kazeykina A.V., Novikov R.G., “Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy”, Nonlinearity, 24:6 (2011), 1821–1830  crossref  isi
    33. Novikov R.G., “Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy”, Phys Lett A, 375:9 (2011), 1233–1235  crossref  isi
    34. Kazeykina A.V., “A Large-Time Asymptotics for the Solution of the Cauchy Problem for the Novikov-Veselov Equation at Negative Energy with Non-Singular Scattering Data”, Inverse Probl., 28:5 (2012), 055017  crossref  isi
    35. Jen-Hsu Chang, “On the $N$-Solitons Solutions in the Novikov–Veselov Equation”, SIGMA, 9 (2013), 006, 13 pp.  mathnet  crossref  mathscinet
    36. 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
    37. 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
    38. Dubrovsky V.G., Topovsky A.V., “About Simple Nonlinear and Linear Superpositions of Special Exact Solutions of Veselov-Novikov Equation”, J. Math. Phys., 54:3 (2013), 033509  crossref  isi
    39. 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
    40. Jen-Hsu Chang, “Mach-Type Soliton in the Novikov–Veselov Equation”, SIGMA, 10 (2014), 111, 14 pp.  mathnet  crossref
    41. Dubrovsky V.G., Topovsky A.V., “About Linear Superpositions of Special Exact Solutions of Nizhnik-Veselov-Novikov Equation”, Physics and Mathematics of Nonlinear Phenomena 2013, Journal of Physics Conference Series, 482, IOP Publishing Ltd, 2014, 012011  crossref  isi
    42. 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
    43. Croke R. Mueller J.L. Music M. Perry P. Siltanen S. Stahel A., “the Novikov-Veselov Equation: Theory and Computation”, Nonlinear Wave Equations: Analytic and Computational Techniques, Contemporary Mathematics, 635, ed. Curtis C. Dzhamay A. Hereman W. Prinari B., Amer Mathematical Soc, 2015, 25–70, ISBN: 978-1-4704-1050-6  crossref  isi
    44. Klein Ch. Saut J.-C., “Ist Versus Pde: a Comparative Study”, Hamiltonian Partial Differential Equations and Applications, ed. Guyenne P. Nicholls D. Sulem C., Springer, 2015, 383–449, ISBN: 978-1-4939-2950-4; 978-1-4939-2949-8  crossref  isi
    45. Kazeykina A. Munoz C., “Dispersive Estimates For Rational Symbols and Local Well-Posedness of the Nonzero Energy Nv Equation”, 270, no. 5, 2016, 1744–1791  crossref  isi
    46. Chang J.-H., “the Interactions of Solitons in the Novikov-Veselov Equation”, Appl. Anal., 95:6 (2016), 1370–1388  crossref  isi
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