Related papers: On differential equations invariant under a projec…
We give a Lie-algebraic classification of third order quasilinear equations which admit non-trivial Lie point symmetries.
The survey provides classification results for integrable one-field evolution equations of orders 2, 3 and 5 with the constant separant. The classification is based on necessary integrability conditions following from the existence of the…
Let a differential 4D-manifold with a smooth coframe field be given. Consider the operators on it that are linear in the second order derivatives or quadratic in the first order derivatives of the coframe, both with coefficients that depend…
Third order nonlinear evolution equations, that is the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other ones are considered: they all are connected via Baecklund transformations. These links can be depicted in a…
We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions, can be generalized to arbitrary order linear…
We prove the invariance of homogeneous second-order Hamiltonian operators under the action of projective reciprocal transformations. We establish a correspondence between such operators in dimension $n$ and $3$-forms in dimension $n + 1$.…
We study the symmetry reduction of nonlinear partial differential equations which are used for describing diffusion processes in nonhomogeneous medium. We find ansatzes reducing partial differential equations to systems of ordinary…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. We have proved, in particular, that the above class contains no…
We propose a general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations. This allows us to give the general integrable evolution corresponding to an arbitrary time and space dependence of the…
All non-equivalent integrable evolution equations of third order of the form $u_t=D_x\frac{\delta H}{\delta u}$ are found.
We consider evolutionary equations of the form $u_t=F(u, w)$ where $w=D_x^{-1}D_yu$ is the nonlocality, and the right hand side $F$ is polynomial in the derivatives of $u$ and $w$. The recent paper \cite{FMN} provides a complete list of…
We examine the reductions of the order of certain third- and second-order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations. We use the folding transformation which enables one to…
A differential calculus is set up on a deformation of the oscillator algebra. It is uniquely determined by the requirement of invariance under a seven-dimensional quantum group. The quantum space and its associated differential calculus are…
We consider evolution equations for curves in the 3-dimensional sphere $S^3$ that are invariant under the group $SU(2,1)$ of pseudoconformal transformations, which preserves the standard contact structure on the sphere. In particular, we…
Linearization of coupled second order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper we describe a simple and straightforward method to…
A genus one curve of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our…
We generalize the notion of semi-normalized classes of systems of differential equations, study properties of such classes and extend the algebraic method of group classification to them. In particular, we prove the important theorems on…
In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow can not be computed exactly. Instead, we use a numerical…
We consider differential-difference equations that determine the continuous symmetries of discrete equations on the triangular lattice. It is shown that a certain combination of continuous flows can be represented as a scalar evolution…