Differential Equations - Geometry, Symmetries and Integrability

Kruglikov, Boris; Lychagin, Valentin; Straume, Eldar

doi:10.1007/978-3-642-00873-3

2009

DOI: 10.1007/978-3-642-00873-3

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Differential Equations - Geometry, Symmetries and Integrability

Boris Kruglikov¹,

Valentin Lychagin²,

Eldar Straume³

Abstract: In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the second order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.

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Mixed hierarchy of soliton equations

Zhou

2009

Journal of Mathematical Physics

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The mixed hierarchy of soliton equations in (1+1) dimensions is introduced. It contains nonholonomic deformations of soliton equations such the KdV6 equation and the Kupershmidt deformations of soliton equations as special members. Based on the commutator representation method, a recipe for constructing zero curvature representations of mixed hierarchy is proposed. As applications, we obtain the mixed hierarchies and their zero curvature representations for the Korteweg–de Vries hierarchy, the Ablowitz–Kaup–Newell–Segur hierarchy, the modified Korteweg–de Vries hierarchy, the Toda lattice hierarchy, and the Volterra lattice hierarchy.

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Mixed hierarchy of soliton equations

Zhou

2009

Journal of Mathematical Physics

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Classification of degenerate non-homogeneous Hamiltonian operators

Marta

Vergallo

2023

Journal of Mathematical Physics

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We investigate non-homogeneous Hamiltonian operators composed of a first order Dubrovin–Novikov operator and an ultralocal one. The study of such operators turns out to be fundamental for the inverted system of equations associated with a class of Hamiltonian scalar equations. Often, the involved operators are degenerate in the first order term. For this reason, a complete classification of the operators with a degenerate leading coefficient in systems with two and three components is presented.

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On Differential Invariants and Classification of Ordinary Differential Equations of the Form y'' = A(x, y)y' + B(x, y)

Bibikov

2018

Math Notes

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Differential Equations - Geometry, Symmetries and Integrability

Cited by 5 publications

References 1 publication

Mixed hierarchy of soliton equations

Mixed hierarchy of soliton equations

Classification of degenerate non-homogeneous Hamiltonian operators

On Differential Invariants and Classification of Ordinary Differential Equations of the Form y'' = A(x, y)y' + B(x, y)

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