This article is cited in 12 papers

Boundary Conditions for Multidimensional Integrable Equations

I. T. Habibullin^a, E. V. Gudkova<sup>b

a</sup> 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.

UDC: 517.9

Received: 11.11.2002

DOI: 10.4213/faa109

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

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