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Funktsional. Anal. i Prilozhen., 1986, Volume 20, Issue 2, Pages 14–24 (Mi faa1268)  

This article is cited in 24 scientific papers (total in 24 papers)

Inverse scattering problem for the two-dimensional Schrödinger operator, the $\bar\partial$-method and nonlinear equations

P. G. Grinevich, S. V. Manakov


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English version:
Functional Analysis and Its Applications, 1986, 20:2, 94–103

Bibliographic databases:

UDC: 517.597+517.984.54
Received: 21.06.1985

Citation: P. G. Grinevich, S. V. Manakov, “Inverse scattering problem for the two-dimensional Schrödinger operator, the $\bar\partial$-method and nonlinear equations”, Funktsional. Anal. i Prilozhen., 20:2 (1986), 14–24; Funct. Anal. Appl., 20:2 (1986), 94–103

Citation in format AMSBIB
\Bibitem{GriMan86}
\by P.~G.~Grinevich, S.~V.~Manakov
\paper Inverse scattering problem for the two-dimensional Schr\"odinger operator, the $\bar\partial$-method and nonlinear equations
\jour Funktsional. Anal. i Prilozhen.
\yr 1986
\vol 20
\issue 2
\pages 14--24
\mathnet{http://mi.mathnet.ru/faa1268}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=847135}
\zmath{https://zbmath.org/?q=an:0617.35031}
\transl
\jour Funct. Anal. Appl.
\yr 1986
\vol 20
\issue 2
\pages 94--103
\crossref{https://doi.org/10.1007/BF01077263}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
    2. R. G. Novikov, “Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude for fixed energy”, Funct. Anal. Appl., 20:3 (1986), 246–248  mathnet  crossref  mathscinet  zmath  isi
    3. 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
    4. 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
    5. L. V. Bogdanov, “On the two-dimensional Zakharov–Shabat problem”, Theoret. and Math. Phys., 72:1 (1987), 790–793  mathnet  crossref  mathscinet  zmath  isi
    6. 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
    7. R. G. Novikov, “Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$”, Funct. Anal. Appl., 22:4 (1988), 263–272  mathnet  crossref  mathscinet  zmath  isi
    8. V. D. Lipovskii, A. V. Shirokov, “$2+1$ Toda chain. I. Inverse scattering method”, Theoret. and Math. Phys., 75:3 (1988), 555–566  mathnet  crossref  mathscinet  isi
    9. 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
    10. R. G. Novikov, “Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2”, Proc. Steklov Inst. Math., 225 (1999), 285–302  mathnet  mathscinet  zmath
    11. 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
    12. O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230  mathnet  crossref  mathscinet  zmath  elib
    13. I. A. Taimanov, S. P. Tsarev, “Two-dimensional Schrödinger operators with fast decaying potential and multidimensional $L_2$-kernel”, Russian Math. Surveys, 62:3 (2007), 631–633  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. 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
    15. Burov V.A., Alekseenko N.V., Rumyantseva O.D., “Mnogochastotnoe obobschenie algoritma novikova dlya resheniya obratnoi dvumernoi zadachi rasseyaniya”, Akusticheskii zhurnal, 55:6 (2009), 784–798
    16. 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
    17. 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
    18. V. G. Dubrovsky, A. V. Topovsky, M. Yu. Basalaev, “New exact solutions of two-dimensional integrable equations using the $\bar\partial$-dressing method”, Theoret. and Math. Phys., 167:3 (2011), 725–739  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    19. Burov V.A., Shurup A.S., Rumyantseva O.D., Zotov D.I., “Funktsionalno-analiticheskoe reshenie zadachi akusticheskoi tomografii po dannym ot tochechnykh preobrazovatelei”, Izvestiya rossiiskoi akademii nauk. seriya fizicheskaya, 76:12 (2012), 1524–1524  elib
    20. Burov V.A., Shurup A.S., Zotov D.I., Rumyantseva O.D., “Modelirovanie funktsionalnogo resheniya zadachi akusticheskoi tomografii dlya dannykh ot kvazitochechnykh preobrazovatelei”, Akusticheskii zhurnal, 59:3 (2013), 391–391  elib
    21. 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
    22. Klein Ch. Saut J.-C., “IST Versus PDE: A Comparative Study”, Hamiltonian Partial Differential Equations and Applications, Fields Institute Communications, ed. Guyenne P. Nicholls D. Sulem C., Springer, 2015, 383–449  crossref  mathscinet  zmath  isi
    23. E. L. Lakshtanov, B. R. Vainberg, “A test for the existence of exceptional points in the Faddeev scattering problem”, Theoret. and Math. Phys., 190:1 (2017), 77–90  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    24. P. G. Grinevich, S. P. Novikov, “Singular solitons and spectral meromorphy”, Russian Math. Surveys, 72:6 (2017), 1083–1107  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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