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Funktsional. Anal. i Prilozhen.:

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Funktsional. Anal. i Prilozhen., 2004, Volume 38, Issue 2, Pages 71–83 (Mi faa109)  

This article is cited in 10 scientific papers (total in 10 papers)

Boundary Conditions for Multidimensional Integrable Equations

I. T. Habibullina, E. V. Gudkovab

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
b Ufa State University of Oil and Technology

Abstract: We suggest an efficient method for finding boundary conditions compatible with integrability for multidimensional integrable equations of Kadomtsev–Petviashvili type. It is observed in all known examples that imposing an integrable boundary condition at a point results in an additional involution for the $t$-operator of the Lax pair. The converse is also likely to be true: if constraints imposed on the coefficients of the $t$-operator of the $L$-$A$ pair result in a broader group of involutions of the $t$-operator, then these constraints determine integrable boundary conditions.
New examples of boundary conditions are found for the Kadomtsev–Petviashvili and modified Kadomtsev–Petviashvili equations.

Keywords: integrable equation, Hamiltonian structure, Kadomtsev–Petviashvili equation, Lax pair


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English version:
Functional Analysis and Its Applications, 2004, 38:2, 138–148

Bibliographic databases:

UDC: 517.9
Received: 11.11.2002

Citation: I. T. Habibullin, E. V. Gudkova, “Boundary Conditions for Multidimensional Integrable Equations”, Funktsional. Anal. i Prilozhen., 38:2 (2004), 71–83; Funct. Anal. Appl., 38:2 (2004), 138–148

Citation in format AMSBIB
\by I.~T.~Habibullin, E.~V.~Gudkova
\paper Boundary Conditions for Multidimensional Integrable Equations
\jour Funktsional. Anal. i Prilozhen.
\yr 2004
\vol 38
\issue 2
\pages 71--83
\jour Funct. Anal. Appl.
\yr 2004
\vol 38
\issue 2
\pages 138--148

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    This publication is cited in the following articles:
    1. E. V. Gudkova, I. T. Habibullin, “Kadomtsev–Petviashvili Equation on the Half-Plane”, Theoret. and Math. Phys., 140:2 (2004), 1086–1094  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. I. T. Habibullin, “Truncations of Toda chains and the reduction problem”, Theoret. and Math. Phys., 143:1 (2005), 515–528  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Gudkova E., “Finite Reductions of the Two Dimensional Toda Chain”, J. Nonlinear Math. Phys., 12:2 (2005), 197–205  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Guerses M., Habibullin I., Zheltukhin K., “Integrable Boundary Value Problems for Elliptic Type Toda Lattice in a Disk”, J. Math. Phys., 48:10 (2007), 102702  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. V. L. Vereshchagin, “Explicit solutions of an integrable boundary value problem for the two-dimensional Toda lattice”, Theoret. and Math. Phys., 165:1 (2010), 1256–1261  mathnet  crossref  crossref  adsnasa  isi
    6. Vereschagin V.L., “Integrable boundary problems for 2D Toda lattice”, Phys Lett A, 374:46 (2010), 4653–4657  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Habibullin I., Zheltukhin K., Yangubaeva M., “Cartan matrices and integrable lattice Toda field equations”, Journal of Physics a-Mathematical and Theoretical, 44:46 (2011), 465202  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. V. L. Vereshchagin, “Integrable boundary conditions for $(2+1)$-dimensional models of mathematical physics”, Theoret. and Math. Phys., 171:3 (2012), 792–799  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    9. Rustem Garifullin, Ismagil Habibullin, Marina Yangubaeva, “Affine and finite Lie algebras and integrable Toda field equations on discrete space-time”, SIGMA, 8 (2012), 062, 33 pp.  mathnet  crossref  mathscinet
    10. V. L. Vereshchagin, “Explicit Solutions of Boundary-Value Problems for $(2+1)$-Dimensional Integrable Systems”, Math. Notes, 93:3 (2013), 360–372  mathnet  crossref  crossref  mathscinet  isi  elib  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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