Related papers: Nonlinear superpositions and Ermakov systems
The purpose of the paper is to review a variety of recent developments in the theory of positive solutions of general linear elliptic and parabolic equations of second-order on noncompact Riemannian manifolds, and to point out a number of…
The Ermakov-Milne-Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics,…
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…
First, we point out that the present applied superposition principle is linear, it must be developed into a generality. Next, the linear operators and equations should be developed nonlinearly. They will include nonlinear Klein-Gordon…
The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to…
We prove a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations on Euclidean spaces and their corresponding linearized first-order continuity equation over the space of Borel (sub-)probability measures. As a consequence,…
Moving boundary problems of Stefan-type for a novel third order nonlinear evolution equation with temporal modulation are here shown to be amenable to exact Airy-type solution via a classical Ermakov equation with its admitted nonlinear…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is explored. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy.…
In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise,…
There are two useful ways to extend nonlinear partial differential inequalities of second order: one uses viscosity theory and the other uses the theory of distributions. This paper considers the convex situation where both extensions can…
In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of…
We give sufficient conditions for a first order expansion of the real line to define the standard model of the monadic second order theory of one successor. Such an expansion does not satisfy any of the combinatorial tameness properties…
The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
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…
Group classification of systems of two coupled nonlinear reaction-diffusion equation with a diagonal diffusion matrix is carried out. Symmetries of diffusion systems with singular diffusion matrix and additional first order derivative terms…
This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of…
We obtain explicit formulas for the solutions of the system of second-order difference equations of the form $x_{n+ 1} = \frac{x_n y_{n-1}}{y_n (a_n + b_n x_n y_{n - 1})}, \quad y_{n+1} = \frac{x_{n - 1} y_n}{x_n (c_n+d_n x_{n-1} y_n)}$,…
Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit…