V. E. Adler, A. B. Shabat, “Model equation of the theory of solitons”, TMF, 153:1 (2007), 29–45; Theoret. and Math. Phys., 153:1 (2007), 1373

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Teoreticheskaya i Matematicheskaya Fizika, 2007, Volume 153, Number 1, Pages 29–45
DOI: https://doi.org/10.4213/tmf6119 (Mi tmf6119)

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

Model equation of the theory of solitons

V. E. Adler, A. B. Shabat

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Full-text PDF (492 kB) Citations (20)

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DOI: https://doi.org/10.4213/tmf6119

Abstract: We consider the hierarchy of integrable

(1 + 2)

$(1+2)$ -dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain

n

$n$ arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows developing a rather general theory of integrable

(1 + 1)

$(1+1)$ -dimensional equations.

Keywords: hierarchy of commuting vector fields, Riemann invariant, Dubrovin equations.

Received: 29.01.2007

English version:
Theoretical and Mathematical Physics, 2007, Volume 153, Issue 1, Pages 1373–1387
DOI: https://doi.org/10.1007/s11232-007-0121-1

Bibliographic databases:

Language: Russian

Citation: V. E. Adler, A. B. Shabat, “Model equation of the theory of solitons”, TMF, 153:1 (2007), 29–45; Theoret. and Math. Phys., 153:1 (2007), 1373–1387

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This publication is cited in the following 20 articles:

V. E. Adler, “3D consistency of negative flows”, Theoret. and Math. Phys., 221:2 (2024), 1836–1851
I. S. Krasil'shchik, O. I. Morozov, “Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Martínez Alonso–Shabat equation”, Anal.Math.Phys., 13:1 (2023)
Makridin V Z., Pavlov V M., “Multidimensional Conservation Laws and Integrable Systems II”, Stud. Appl. Math., 148:2 (2022), 813–824
O. I. Morozov, “Higher Symmetries of the Cotangent Covering for the Modified Veronese Web Equation”, Lobachevskii J Math, 43:10 (2022), 2797
Ferapontov E., Kruglikov B., “Dispersionless Integrable Hierarchies and Gl(2, Double-Struck Capital R) Geometry”, Math. Proc. Camb. Philos. Soc., 170:1 (2021), PII S0305004119000355, 129–154
Marvan M., Pavlov M.V., “Integrable Dispersive Chains and Their Multi-Phase Solutions”, Lett. Math. Phys., 109:5 (2019), 1219–1245
Oksana Ye. Hentosh, Yarema A. Prikarpatsky, Denis Blackmore, Anatolij K. Prikarpatski, “Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics”, SIGMA, 15 (2019), 079, 20 pp.
Lelito A., Morozov O.I., “Three-Component Nonlocal Conservation Laws For Lax-Integrable 3D Partial Differential Equations”, J. Geom. Phys., 131 (2018), 89–100
H. Baran, I. S. Krasil'shchik, O. I. Morozov, P. Vojčák, “Nonlocal symmetries of integrable linearly degenerate equations: A comparative study”, Theoret. and Math. Phys., 196:2 (2018), 1089–1110
F. Calogero, “Zeros of entire functions and related systems of infinitely many nonlinearly coupled evolution equations”, Theoret. and Math. Phys., 196:2 (2018), 1111–1128
Morozov O.I., Pavlov M.V., “Backlund Transformations Between Four Lax-Integrable 3D Equations”, J. Nonlinear Math. Phys., 24:4 (2017), 465–468
Ferapontov E.V., Moss J., “Linearly Degenerate Partial Differential Equations and Quadratic Line Complexes”, Commun. Anal. Geom., 23:1 (2015), 91–127
A. B. Shabat, “Rational interpolation and solitons”, Theoret. and Math. Phys., 179:3 (2014), 637–648
Morozov O.I., Sergyeyev A., “The Four-Dimensional Martinez Alonso-Shabat Equation: Reductions and Nonlocal Symmetries”, J. Geom. Phys., 85 (2014), 40–45
Pavlov M.V., “Integrable Dispersive Chains and Energy Dependent Schrodinger Operator”, J. Phys. A-Math. Theor., 47:29 (2014), 295204
Ferapontov E.V., Moss J., “Characteristic Integrals in 3D and Linear Degeneracy”, J. Nonlinear Math. Phys., 21:2 (2014), 214–224
Baran H., Krasil'shchik I.S., Morozov O.I., Vojcak P., “Symmetry Reductions and Exact Solutions of Lax Integrable 3-Dimensional Systems”, J. Nonlinear Math. Phys., 21:4 (2014), 643–671
A. B. Shabat, “Simmetricheskie mnogochleny i zakony sokhraneniya”, Vladikavk. matem. zhurn., 14:4 (2012), 83–94
Burovskiy P.A., Ferapontov E.V., Tsarev S.P., “Second-order quasilinear PDEs and conformal structures in projective space”, Internat. J. Math., 21:6 (2010), 799–841
A. Shabat, Lecture Notes in Physics, 767, Integrability, 2009, 139

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

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