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

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Teoreticheskaya i Matematicheskaya Fizika, 2008, Volume 157, Number 2, Pages 188–207
DOI: https://doi.org/10.4213/tmf6274 (Mi tmf6274)

This article is cited in 30 scientific papers (total in 31 papers)

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

I. A. Taimanov^a, S. P. Tsarev^b

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

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

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.

Received: 21.01.2008

English version:
Theoretical and Mathematical Physics, 2008, Volume 157, Issue 2, Pages 1525–1541
DOI: https://doi.org/10.1007/s11232-008-0127-3

Bibliographic databases:

Language: Russian

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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Linking options:

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This publication is cited in the following 31 articles:

Allal Ghanmi, Khalil Lamsaf, “Two-dimensional (p,q)-heat polynomials of Gould–Hopper type”, Journal of Mathematical Analysis and Applications, 530:1 (2024), 127682
Iskander A. Taimanov, “On a Formation of Singularities of Solutions to Soliton Equations Represented by L, A, B-triples”, Acta. Math. Sin.-English Ser., 40:1 (2024), 406
A. G. Kudryavtsev, “On the twofold Moutard transformation of the stationary Schrгöinger equation with axial symmetry”, JETP Letters, 119:7 (2024), 534–537
M. M. Malamud, V. V. Marchenko, “On Kernels of Invariant Schrödinger Operators with Point Interactions. Grinevich–Novikov Conjecture”, Dokl. Math., 109:2 (2024), 125
M. M. Malamud, V. V. Marchenko, “On kernels of invariant Schrödinger operators with point interactions. Grinevich–Novikov problem”, Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, 516 (2024), 31
P. G. Grinevich, “Riemann Surfaces Close to Degenerate Ones in the Theory of Rogue Waves”, Proc. Steklov Inst. Math., 325 (2024), 86–110
Joseph Adams, Axel Grünrock, “Low Regularity Local Well-Posedness for the Zero Energy Novikov–Veselov Equation”, SIAM J. Math. Anal., 55:1 (2023), 19
A. V. Bolsinov, V. M. Buchstaber, A. P. Veselov, P. G. Grinevich, I. A. Dynnikov, V. V. Kozlov, Yu. A. Kordyukov, D. V. Millionshchikov, A. E. Mironov, R. G. Novikov, S. P. Novikov, A. A. Yakovlev, “Iskander Asanovich Taimanov (on his 60th birthday)”, Russian Math. Surveys, 77:6 (2022), 1159–1168
A. G. Kudryavtsev, “On the nonlocal darboux transformation for time-independent axially symmetric Schrödinger and Helmholtz equations”, JETP Letters, 113:6 (2021), 409–412
A. G. Kudryavtsev, “Exact solutions of the time-independent axially symmetric Schrödinger equation”, JETP Letters, 111:2 (2020), 126–128
R. G. Novikov, I. A. Taimanov, “Darboux–Moutard transformations and Poincaré–Steklov operators”, Proc. Steklov Inst. Math., 302 (2018), 315–324
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
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
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
I. A. Taimanov, “The Moutard Transformation of Two-Dimensional Dirac Operators and Möbius Geometry”, Math. Notes, 97:1 (2015), 124–135
I. A. Taimanov, “Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces”, Theoret. and Math. Phys., 182:2 (2015), 173–181
I. A. Taimanov, “A fast decaying solution to the modified Novikov-Veselov equation with a one-point singularity”, Dokl. Math., 91:1 (2015), 35
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
Perry P.A., “Miura Maps and Inverse Scattering For the Novikov-Veselov Equation”, Anal. PDE, 7:2 (2014), 311–343
Jen-Hsu Chang, “On the $N$-Solitons Solutions in the Novikov–Veselov Equation”, SIGMA, 9 (2013), 006, 13 pp.

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

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