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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

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

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

DOI: https://doi.org/10.4213/faa109

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

Bibliographic databases:

UDC: 517.9

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
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• http://mi.mathnet.ru/eng/faa109
• https://doi.org/10.4213/faa109
• http://mi.mathnet.ru/eng/faa/v38/i2/p71

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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. E. V. Gudkova, I. T. Habibullin, “Kadomtsev–Petviashvili Equation on the Half-Plane”, Theoret. and Math. Phys., 140:2 (2004), 1086–1094
2. I. T. Habibullin, “Truncations of Toda chains and the reduction problem”, Theoret. and Math. Phys., 143:1 (2005), 515–528
3. Gudkova E., “Finite Reductions of the Two Dimensional Toda Chain”, J. Nonlinear Math. Phys., 12:2 (2005), 197–205
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
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
6. Vereschagin V.L., “Integrable boundary problems for 2D Toda lattice”, Phys Lett A, 374:46 (2010), 4653–4657
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
8. V. L. Vereshchagin, “Integrable boundary conditions for $(2+1)$-dimensional models of mathematical physics”, Theoret. and Math. Phys., 171:3 (2012), 792–799
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.
10. V. L. Vereshchagin, “Explicit Solutions of Boundary-Value Problems for $(2+1)$-Dimensional Integrable Systems”, Math. Notes, 93:3 (2013), 360–372
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