Open AccessALGORITHM FOR CONSTRUCTING A LAX PAIR AND A RECURSION OPERATOR FOR INTEGRABLE EQUATIONS
Author(s) -
I. T. Habibullin,
A. R. Khakimova
Publication year - 2019
Publication title -
okeanologičeskie issledovaniâ
Language(s) - English
Resource type - Journals
eISSN - 2587-9634
pISSN - 1564-2291
DOI - 10.29006/1564-2291.jor-2019.47(1).38
Subject(s) - mathematics , lax pair , integrable system , partial differential equation , invariant (physics) , first order partial differential equation , linearization , nonlinear system , differential equation , recursion (computer science) , mathematical analysis , algebra over a field , pure mathematics , algorithm , mathematical physics , physics , quantum mechanics
The method of constructing particular solutions to nonlinear partial differential equations based on the notion of differential constraint (or invariant manifold) is well known in the literature, see (Yanenko, 1961; Sidorov et al., 1984). The matter of the method is to add a compatible equation to a given equation and as a rule, the compatible equation is simpler. Such technique allows one to find particular solutions to a studied equation. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) there was proposed a scheme for constructing the Lax pairs and recursion operators for integrable partial differential equations based on the use of similar idea. A suitable generalization is to impose a differential constraint not on the equation, but on its linearization. The resulting equation is referred to as a generalized invariant manifold. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) it is shown that generalized invariant varieties allow efficient construction of Lax pairs and recursion operators of integrable equations.The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».