74 citations · 309 across the 33 of their papers we have counts for
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The \textsc{ABC}-equation revisited: missed integrable cases. Part I: classification and (quasi) integrable structures
I. S. Krasil'shchik
We consider the \textsc{ABC}-equation , , , , which was proved to be integrable in the case (the Ver…
Integrability structures of the -dimensional Euler equation
I. S. Krasil'shchik, O. I. Morozov
We construct local and nonlocal Hamiltonian structures and variational symplectic structures for the -dimensional Euler equation in the vorticity form and study the action o…
The Calogero--Bogoyavlenskii--Schiff breaking soliton equation: recursion operators and higher symmetries
I. S. Krasil'shchik, O. I. Morozov
We find two one-parametric families of recursion operators and use them to construct higher symmetries for the Calogero--Bogoyavlenskii--Schiff breaking soliton equation. Then we p…
Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Mart{\'ı}nez Alonso--Shabat equation
I. S. Krasil'shchik, O. I. Morozov
We study a Lagrangian extension of the 5d Martínez Alonso--Shabat equation \begin{equation*} u_{yz}=u_{tx}+u_y\,u_{xs}-u_x\,u_{ys} \end{equation*} that coincides with…
On recursion operators for symmetries of the Pavlov-Mikhalev equation
I. S. Krasil'shchik
In geometry of nonlinear partial differential equations, recursion operators that act on symmetries of an equation are understood as Bäcklund auto-transformations of…
Recursion operators in the cotangent covering of the rdDym equation
I. S. Krasil'shchik, A. M. Verbovetsky
We describe a general method of constructing nonlocal recursion operators for symmetries of PDEs. As an example, the cotangent equation to the 3D rdDym equation $u_{yt} = u_xu_{xy}…
Nonlocal conservation laws of PDEs possessing differential coverings
I. Krasil'shchik
In his 1892 paper [L. Bianchi, Sulla trasformazione di Bäcklund per le superfici pseudosferiche, Rend. Mat. Acc. Lincei, s. 5, v. 1 (1892) 2, 3--12], L. Bianchi noticed, among othe…
On the algebra of nonlocal symmetries for the 4D Mart\'ınez Alonso-Shabat equation
I. S. Krasil'shchik, P. Vojcak
We consider the 4D Mart\'ınez Alonso-Shabat equation (also referred to as the universal hierarchy equation) and using its known Lax pair construc…
Geometric study of gas behavior in a one-dimensional nozzle (the case of the van der Waals gas
I. S. Krasil'shchik, V. V. Lychagin
We construct a three-component system of PDEs describing dynamics of van der Walls gas in one-dimensional nozzle. The group of conservation laws for this system is described. We al…
Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation
I. S. Krasil'shchik, O. I. Morozov, P. Vojčák
We study the Veronese web equation and using its isospectral Lax pair construct two infinite series of nonlocal conservation laws. In t…
Integrability of anti-self-dual vacuum Einstein equations with nonzero cosmological constant: an infinite hierarchy of nonlocal conservation laws
I. Krasil'shchik, A. Sergyeyev
We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equa…
On symmetries of the Gibbons-Tsarev equation
H. Baran, P. Blaschke, I. S. Krasil'shchik +1
We study the Gibbons-Tsarev equation and, using the known Lax pair, we construct infinite series of conservation laws and the algebra of…
Reductions of the universal hierarchy and rdDym equations and their symmetry properties
P. Holba, I. S. Krasil'shchik, O. I. Morozov +1
We consider the equations and that arise as reductions of the universal hierarchy and…
Nonlocal symmetries of Lax integrable equations: a comparative study
H. Baran, I. S. Krasil'shchik, O. I. Morozov +1
We continue here the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation , (2) the 3D Pavlov e…
Infinitely many nonlocal conservation laws for the equation with
I. S. Krasil'shchik, A. Sergyeyev, O. I. Morozov
We construct an infinite hierarchy of nonlocal conservation laws for the equation , where are constants and $A+B+C\neq…
Coverings over Lax integrable equations and their nonlocal symmetries
H. Baran, I. S. Krasil'shchik, O. I. Morozov +1
Using the Lax representation with non-removable parameter, we construct two hierarchies of nonlocal conservation laws for the 3D rdDym equation and…
Integrability of S-deformable surfaces: conservation laws, Hamiltonian structures and more
I. S. Krasil'shchik, A. Sergyeyev
We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dim…
Integrability properties of some symmetry reductions
H. Baran, I. S. Krasil'shchik, O. I. Morozov +1
In our recent paper [H. Baran, I.S. Krasil'shchik, O.I. Morozov, P. Voj{č}{á}k, Symmetry reductions and exact solutions of Lax integrable -dimensional systems, Journal of Nonlin…
Symmetry reductions and exact solutions of Lax integrable -dimensional systems
H. Baran, I. S. Krasil'shchik, O. I. Morozov +1
We present a complete description of -dimensional equations that arise as symmetry reductions of fourf -dimensional Lax-integrable equations: (1) the universal hierarchy equa…
A natural geometric construction underlying a class of Lax pairs
Joseph Krasil'shchik
In the framework of the theory of differential coverings \cite{KV}, we discuss a general geometric construction that serves the base for the so-called Lax pairs containing differen…
Higher symmetries of cotangent coverings for Lax-integrable multi-dimensional partial differential equations and Lagrangian deformations
H. Baran, I. S. Krasil'shchik, O. I. Morozov +1
We present examples of Lax-integrable multi-dimensional systems of partial differential equations with higher local symmetries. We also consider Lagrangian deformations of these eq…
A unified approach to computation of integrable structures
Iosif Krasil'shchik, Alexander Verbovetsky, Raffaele Vitolo
We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of par…
On integrable structures for a generalized Monge-Ampere equation
Paul Kersten, Iosif Krasil'shchik, Alexander Verbovetsky +1
We consider a 3rd-order generalized Monge-Ampere equation u_yyy - u_xxy^2 + u_xxx u_xyy = 0 (which is closely related to the associativity equation in the 2-d topological field the…
On integrability of the Camassa-Holm equation and its invariants. A geometrical approach
Valentina Golovko, Paul Kersten, Iosif Krasil'shchik +1
Using geometrical approach exposed in arXiv:math/0304245 and arXiv:nlin/0511012, we explore the Camassa-Holm equation (both in its initial scalar form, and in the form of 2x2-syste…