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A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

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Abstract

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L1−1 L2, where the linear differential operators L1 and L2 are chosen such that the ordinary differential equation (L2 −λL1)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L2, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L1\(\tilde U\) = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution \(\tilde U\) of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.

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Authors and Affiliations

  1. Institute of Mathematics, Ufa Federal Research Centre, RAS, Ufa, Russia

    I. T. Habibullin & A. R. Khakimova

  2. Bashkir State University, Ufa, Russia

    I. T. Habibullin & A. R. Khakimova

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  1. I. T. Habibullin
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  2. A. R. Khakimova
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Correspondence to I. T. Habibullin.

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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 196, No. 2, pp. 294–312, August, 2018.

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Habibullin, I.T., Khakimova, A.R. A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models. Theor Math Phys 196, 1200–1216 (2018). https://doi.org/10.1134/S004057791808007X

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