Related papers: On integrability of the differential constraints a…
Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different…
The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations…
We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the…
Obstacles to integrability in perturbed evolution equations are overcome by allowing higher-order terms in the expansion of the solution to depend explicitly on time and position. With a special expansion algorithm, obstacles vanish…
A new integrable system of two symmetrically coupled derivative nonlinear Schroedinger equations is detected by means of the singularity analysis. A nonlinear transformation is proposed which uncouples the equations of the new system.
We address the problem of classification of integrable differential-difference equations in 2+1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalisation to dispersive…
We give a definition of integration by quadratures of first-order ordinary differential equations, and recover a little known result by Maximovic which states that a first-order ordinary differential equation can be integrated by…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering…
The (2+1)-dimensional integrable Zakharov equations and their reductions are considered
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained and the…
We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such…
The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in…
This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
We describe the various types of singularities that can arise for second order rational mappings and we discuss the historical and present-day, practical, role the singularity confinement property plays as an integrability detector. In…
The question of defining unique, generally applicable constrained second, and higher-order, derivatives is investigated. It is shown that second-order constrained derivatives obtained via two successive constrained differentiations provide…
We study the exponential stability of evolutionary equations. The focus is laid on second order problems and we provide a way to rewrite them as a suitable first order evolutionary equation, for which the stability can be proved by using…
Two integrable systems are constructed in a 2 + 1-dimensional space. Every of these systems involve two evolutions with negative numbers.