Related papers: On the structure of some typical singularities for…
Integrability of the differential constraints arising from the singularity analysis of two (1+1)-dimensional second-order evolution equations is studied. Two nonlinear ordinary differential equations are obtained in this way, which are…
Variable order differential equations with non-integrable singularities are considered on spatial networks. Properties of the spectrum are established, and the solution of the inverse spectral problem is obtained.
The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order…
We consider equations of the type: \[\partial_t \omega = \omega R(\omega),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized…
The paper is devoted to the implicit function theorem involving singular mappings.We also discuss the form of the tangent cone to the solution set of the generalized equations in singular case and give some examples of applications to…
The discrete KdV (dKdV) equation, the pinnacle of discrete integrability, is often thought to possess the singularity confinement property because it confines on an elementary quadrilateral. Here we investigate the singularity structure of…
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…
Apparently, all partial differential equations that describe physical phenomena in space-time can be cast into a universal quasilinear, first-order form. In this paper, we do two things. First, we describe some broad features of systems of…
The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to algebraic curves.
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…
The reliability is of the most importance when employing a numerical method to solve the eigenvalue integral equations. In this paper, we present one type of particular singularities (pseudosingularities) existing in eigenvalue integral…
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena…
An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati…
We determine the position and the type of spontaneous singularities of solutions of generic analytic nonlinear differential systems in the complex plane, arising along antistokes directions towards irregular singular points of the system.…
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…
This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the…
The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can…
We apply a novel method for the equivalence group and its infinitesimal generators to the investigation of invariants of linear ordinary differential equations. First, a comparative study of this method is illustrated by an example. Next,…
Ordinary differential equations (ODEs) and ordinary difference systems (O$\Delta$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…