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TMF, 2008, Volume 157, Number 2, Pages 188–207 (Mi tmf6274)  

This article is cited in 20 scientific papers (total in 20 papers)

Two-dimensional rational solitons and their blowup via the Moutard transformation

I. A. Taimanova, S. P. Tsarevb

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b Krasnoyarsk State Pedagogical University named after V. P. Astaf'ev

Abstract: We construct a family of two-dimensional stationary Schrödinger operators with rapidly decaying smooth rational potentials and nontrivial $L_2$ kernels. We show that some of the constructed potentials generate solutions of the Veselov–Novikov equation that decay rapidly at infinity, are nonsingular at $t=0$, and have singularities at finite times $t\ge t_0>0$.

Keywords: two-dimensional Schrödinger operator, Moutard transformation, Veselov–Novikov equation, solution blowup

DOI: https://doi.org/10.4213/tmf6274

Full text: PDF file (1590 kB)
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English version:
Theoretical and Mathematical Physics, 2008, 157:2, 1525–1541

Bibliographic databases:

Received: 21.01.2008

Citation: I. A. Taimanov, S. P. Tsarev, “Two-dimensional rational solitons and their blowup via the Moutard transformation”, TMF, 157:2 (2008), 188–207; Theoret. and Math. Phys., 157:2 (2008), 1525–1541

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. P. Tsarev, E. S. Shemyakova, “Differential Transformations of Parabolic Second-Order Operators in the Plane”, Proc. Steklov Inst. Math., 266 (2009), 219–227  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. V. G. Marikhin, “The dressing method and separation of variables: The two-dimensional case”, Theoret. and Math. Phys., 161:3 (2009), 1599–1603  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. 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
    4. V. G. Marikhin, “Solutions of two-dimensional Schrödinger-type equations in a magnetic field”, Theoret. and Math. Phys., 168:2 (2011), 1041–1047  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    5. Chang J.-H., “The Gould-Hopper polynomials in the Novikov-Veselov equation”, J. Math. Phys., 52:9 (2011), 092703  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Schulze-Halberg A., “Darboux Transformations for (1+2)-Dimensional Fokker-Planck Equations with Constant Diffusion Matrix”, J. Math. Phys., 53:10 (2012), 103519  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    7. Ganzha E.I., “On Laplace and Dini Transformations for Multidimensional Equations with a Decomposable Principal Symbol”, Program. Comput. Softw., 38:3 (2012), 150–155  crossref  mathscinet  zmath  isi  elib  elib  scopus
    8. Jen-Hsu Chang, “On the $N$-Solitons Solutions in the Novikov–Veselov Equation”, SIGMA, 9 (2013), 006, 13 pp.  mathnet  crossref  mathscinet
    9. R. G. Novikov, I. A. Taimanov, “The Moutard transformation and two-dimensional multipoint delta-type potentials”, Russian Math. Surveys, 68:5 (2013), 957–959  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  elib  elib
    10. 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
    11. Kudryavtsev A.G., “Exactly Solvable Two-Dimensional Stationary Schrodinger Operators Obtained by the Nonlocal Darboux Transformation”, Phys. Lett. A, 377:38 (2013), 2477–2480  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    12. Music M., Perry P., Siltanen S., “Exceptional Circles of Radial Potentials”, Inverse Probl., 29:4 (2013), 045004  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    13. R. G. Novikov, I. A. Taimanov, S. P. Tsarev, “Two-Dimensional von Neumann–Wigner Potentials with a Multiple Positive Eigenvalue”, Funct. Anal. Appl., 48:4 (2014), 295–297  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. Perry P.A., “Miura Maps and Inverse Scattering For the Novikov-Veselov Equation”, Anal. PDE, 7:2 (2014), 311–343  crossref  mathscinet  zmath  isi  scopus  scopus
    15. I. A. Taimanov, “The Moutard Transformation of Two-Dimensional Dirac Operators and Möbius Geometry”, Math. Notes, 97:1 (2015), 124–135  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. 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
    17. A. G. Kudryavtsev, “Nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation and its relation to the Moutard transformation”, Theoret. and Math. Phys., 187:1 (2016), 455–462  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    18. R. G. Novikov, I. A. Taimanov, “Moutard type transformation for matrix generalized analytic functions and gauge transformations”, Russian Math. Surveys, 71:5 (2016), 970–972  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. Adilkhanov A.N. Taimanov I.A., “On numerical study of the discrete spectrum of a two-dimensional Schrödinger operator with soliton potential”, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 83–92  crossref  mathscinet  isi  elib  scopus
    20. R. G. Novikov, I. A. Taimanov, “Darboux–Moutard transformations and Poincaré–Steklov operators”, Proc. Steklov Inst. Math., 302 (2018), 315–324  mathnet  crossref  crossref  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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