On a direct algorithm for constructing recursion operators and Lax pairs for integrable models

2017/10/24 by I. T. Habibullin, Habibullin, I. T., A R Khakimova +1
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1710.08626

openalex publication_date 2017/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We suggested an algorithm for searching the recursion operators for nonlinear integrable equations. It was observed that the recursion operator R can be represented as a ratio of the form R=L₁^-1L₂ where the linear differential operators L₁ and L₂ are chosen in such a way 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. For constructing the operator L₁ we use the concept of the invariant manifold which is a generalization of the symmetry. Then for searching L₂ we take an auxiliary linear equation connected with the linearized equation by the Darboux transformation. Connection of the invariant manifold with the Lax pairs and the Dubrovin-Weierstrass equations is discussed.

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