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The integrability has been playing an essential role in the field of differential equations. This property may better help us obtain the topological structure and even the global dynamics for the considered system. A system is called…
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
Liouville's theorem in a grand ensemble, that is for situations where a system is in equilibrium with a reservoir of energy and particles, is a subject that, to our knowledge, has not been explicitly treated in literature related to…
We consider a natural generalisation of the Painlev\'e property and use it to identify the known integrable cases of the Lane-Emden equation with a real positive index. We classify certain first-order ordinary differential equations with…
This paper studies the existence and global stability of generalized Ornstein-Uhlenbeck process for affine stochastic functional differential equations. Various very basic and important properties are established. In the applications, we…
In this paper, for an operator defined by the action of an M-th order differential operator with rational-type coefficients on the function space L_k^2(R):={f: measurable | \|f\|_k <\infty} with norm \|f\|_k^2:= \int |f(x)|^2 (x^2+1)^k dx…
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…
We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators…
The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of…
We study two families of (matrix versions of) generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differential-difference equation in one continuous and two discrete…
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
Liouville theorems for scaling invariant nonlinear elliptic systems (saying that the system does not possess nontrivial entire solutions) guarantee a priori estimates of solutions of related, more general systems. Assume that $p=2q+3>1$ is…
We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the…
Liouvillian systems were initially introduced within the framework of differential algebra. They can be seen as a natural extension of differential flat systems. Many physical non flat systems seem to be Liouvillian. We present in this…
For given non-consistent initial conditions, we study the stability of a class of generalised linear systems of difference equations with constant coefficients and taking into account that the leading coefficient can be a singular matrix.…
We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call solvable intitial value…
We consider natural complex Hamiltonian systems with $n$ degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$. The well known…
In the example of the Schr\"odinger/KdV equation we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's…
We have already dealt with the problem of solving First Order Differential Equations (1ODEs) presenting elementary functions before in [1, 2]. In this present paper, we have established solid theoretical basis through a relation between the…
Necessary and sufficient conditions for the internal stability of formations whose dynamics are obtained is determined by linear differential equations.