Related papers: Classification of discrete equations linearizable …
The Lie linearizability criteria are extended to complex functions for complex ordinary differential equations. The linearizability of complex ordinary differential equations is used to study the linearizability of corresponding systems of…
We present an algorithm for determining the Lie point symmetries of differential equations on fixed non transforming lattices, i.e. equations involving both continuous and discrete independent variables. The symmetries of a specific…
Algorithmic approach to the problem of linearization by point transformation of ordinary differential equation of arbitrary order is presented. Test-linearization is purely algorithmic.
The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable…
We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of…
Non-point invertible transformations are completely described for difference equations on the quad-graph and for their differential-difference analogues. As an illustration, these transformations are used to construct new examples of…
We conjecture an integrability and linearizability test for dispersive Z^2-lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscale expansion give a partial differential equation…
In this second of the set of two papers on Lie symmetry analysis of a class of Li\'enard type equation of the form $\ddot {x} + f(x)\dot {x} + g(x)= 0$, where over dot denotes differentiation with respect to time and $f(x)$ and $g(x)$ are…
Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for linearizability of a…
Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…
We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and…
A theorem providing necessary conditions enabling one to map a nonlinear system of first order partial differential equations to an equivalent first order autonomous and homogeneous quasilinear system is given. The reduction to quasilinear…
The classification of lattice equations that are integrable in the sense of higher-dimensional consistency is extended by allowing directed edges. We find two cases that are not transformable via the 'admissible transformations' to the…
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, \textit{i.e.}, classes of differential equations that are in correspondence with their Lie point symmetries. In particular,…
In this article we present the results obtained applying the multiple scale expansion up to the order $\varepsilon^6$ to a dispersive multilinear class of equations on a square lattice depending on 13 parameters. We show that the…
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.
We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general…
The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The…