Related papers: The four-dimensional Martinez Alonso - Shabat equa…
We consider the four-dimensional integrable Martinez Alonso--Shabat equation, and list three integrable three-dimensional reductions thereof. We also present a four-dimensional integrable modified Martinez Alonso--Shabat equation together…
We present new recursion operators for (shadows of nonlocal) symmetries of the 4D Mart\'inez Alonso-Shabat equation $u_{ty} = u_z u_{xy} - u_y u_{xz}$, and we show that their actions can produce new symmetries which are not contained in the…
We study a Lagrangian extension of the 5d Mart\'inez Alonso--Shabat equation $\mathcal{E}$ \begin{equation*} u_{yz}=u_{tx}+u_y\,u_{xs}-u_x\,u_{ys} \end{equation*} that coincides with the cotangent equation $\mathcal{T^*E}$ to the latter. We…
In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are…
We apply Cartan's method of equivalence to find a covering for the modified Khokhlov - Zabolotskaya equation.
We apply Cartan's method of equivalence to find a B\"acklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.
We consider the local equivalence problem for the class of linear second order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. E. Cartan's method is used for finding the…
We present a recursion operator and an infinite hierarchy of nonlocal commuting symmetries for the modified Martinez Alonso-Shabat equation $u_y u_{xz} + \alpha u_x u_{ty} - (u_z + \alpha u_t)u_{xy} = 0$.
In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations…
Following the Cartans's original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete…
It is widely known that the recursion operator is a very important component of integrability. It allows one to describe in a compact form both hierarchies of the generalized symmetries and infinite series of the local conservation laws. In…
This is the first part of a series of papers. The whole series aims to develop the tools for the study of all almost Hermitian symmetric structures in a unified way. In particular, methods for the construction of invariant operators, their…
We derive two non-equivalent coverings for the r-th mdKP equation from Maurer--Cartan forms of its symmetry pseudo-group. Also we find B\"acklund transformations between the covering equations.
This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber--preserving transformation using the Cartan method of equivalence. We will do three versions of the…
In this paper, we carry out the equivalence problem for fifth-order differential operators on the line under general fiber-preserving transformation using the Cartan method of equivalence. Two versions of equivalence problems have been…
On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the…
We apply the technique of integrable extensions to the symmetry pseudo-group of the r-th mdKP equation. This gives another look on deriving known coverings and allows us to find new coverings for this equation.
We consistently develop a recently proposed scheme of matrix extension of dispersionless integrable systems for the general case of multidimensional hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present extended Lax…
Cartan's equivalence method is applied to explicitly construct invariant coframes for four branches, which are used to characterize all non-linearizable third-order ODEs with a four-dimensional Lie symmetry subalgebra under point…
In geometry of nonlinear partial differential equations, recursion operators that act on symmetries of an equation $\mathcal{E}$ are understood as B\"{a}cklund auto-transformations of the equation $\mathcal{TE}$ tangent to $\mathcal{E}$. We…