Darboux transformations and recursion operators for differential-difference equations. (English. Russian original) Zbl 1301.37041
Theor. Math. Phys. 177, No. 3, 1606-1654 (2013); translation from Teor. Mat. Fiz. 177, No. 3, 387-440 (2013).
The aim of this article is to make an extensive contribution in multi-Hamiltonian structures, recursion operators and Darboux-Lax representations for a wide class of integrable differential-difference equations. Some related results are known, but scattered in the literature. In many cases, the authors complete the picture by providing explicit expressions for the Hamiltonian, symplectic and recursion operators and the Darboux-Lax representations.
Firstly, the authors review two closely related concepts concerning Lax representations: the Darboux transformations of the Lax representation, from which the integrable differential-difference equations are derived, and the derivation of the recursion operator for the resulting equations with the usage of Darboux transformations. They are illustrated by two typical examples: the nonlinear Schrödinger equation (NLS) and the deformation of the derivative NLS equation corresponding to the dihedral reduction group \(\mathbb{D}_2\).
The article is completed by an extensive list of integrable differential-difference equations, containing except for the equations themselves, their Hamiltonian structures, recursion operators, nonlinear generalized symmetries and Lax representations, and also links with other equations and the weakly nonlocal inverses of recursion operators when they exist; partial results on the master symmetries are also included. However the list is far from complete. A similar list for \(1+1\) integrable evolutionary equations is presented in the article of J. P. Wang [J. Nonlinear Math. Phys. 9, Suppl. 1, 213–233 (2002)].
Firstly, the authors review two closely related concepts concerning Lax representations: the Darboux transformations of the Lax representation, from which the integrable differential-difference equations are derived, and the derivation of the recursion operator for the resulting equations with the usage of Darboux transformations. They are illustrated by two typical examples: the nonlinear Schrödinger equation (NLS) and the deformation of the derivative NLS equation corresponding to the dihedral reduction group \(\mathbb{D}_2\).
The article is completed by an extensive list of integrable differential-difference equations, containing except for the equations themselves, their Hamiltonian structures, recursion operators, nonlinear generalized symmetries and Lax representations, and also links with other equations and the weakly nonlocal inverses of recursion operators when they exist; partial results on the master symmetries are also included. However the list is far from complete. A similar list for \(1+1\) integrable evolutionary equations is presented in the article of J. P. Wang [J. Nonlinear Math. Phys. 9, Suppl. 1, 213–233 (2002)].
Reviewer: Irina V. Konopleva (Ul’yanovsk)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 39A14 | Partial difference equations |
Keywords:
symmetry; recursion operator; bi-Hamiltonian structure; Darboux transformation; Lax representation; integrable equationReferences:
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