Geodesic
A~direct algorithm for constructing recursion operators and Lax pairs for integrable models
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 294-312

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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 $R=L_1^{-1}L_2$, where the linear differential operators $L_1$ and $L_2$ are chosen such that the ordinary differential equation $(L_2-\lambda~L_1)U=0$ is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter $\lambda\in\mathbb{C}$. To construct the operator $L_1$, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek $L_2$, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation $L_1\widetilde U=L_2U$ defines a Bäcklund transformation mapping a solution $U$ of the linearized equation to another solution $\widetilde U$ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.
Mots-clés : Lax pair
Keywords: integrable chain, higher symmetry, invariant manifold, recursion operator. 
@article{TMF_2018_196_2_a6,
     author = {I. T. Habibullin and A. R. Khakimova},
     title = {A~direct algorithm for constructing recursion operators and {Lax} pairs for integrable models},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {294--312},
     publisher = {mathdoc},
     volume = {196},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a6/}
}
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I. T. Habibullin; A. R. Khakimova. A~direct algorithm for constructing recursion operators and Lax pairs for integrable models. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 294-312. http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a6/