An astonishing fact was established by Lee A. Rubel in 81: there exists a fixed non-trivial fourthorder polynomial differential algebraic equation (DAE) such that for any positive continuous function φ on the reals, and for any positive continuous function (t), it has a C∞ solution with |y(t) − φ(t)| < (t) for all t. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other exam...

The modulational instability, a destabilization mechanism for plane waves, occurs ubiquitously in spatially extended dynamical systems arising in fluid dynamics, nonlinear optics, atomic physics, and other disciplines. In this paper, we investigate analytically, numerically, and experimentally the modulational instability in a layered cubically nonlinear (Kerr) optical medium that consists of a...

An impulsive boundary value problem with nonlinear boundary conditions for a second order ordinary differential equation is studied. In particular, sufficient conditions are provided so that a compression expansion cone theoretic fixed point theorem can be applied to imply the existence of positive solutions. The nonlinear forcing term is assumed to satisfy usual sublinear or superlinear growth...

An impulsive boundary value problem with nonlinear boundary conditions for a second order ordinary differential equation is studied. In particular, sufficient conditions are provided so that a compression expansion cone theoretic fixed point theorem can be applied to imply the existence of positive solutions. The nonlinear forcing term is assumed to satisfy usual sublinear or superlinear growth...

The studied model was suggested to design a perfect hedging strategy for a large trader. In this case the implementation of a hedging strategy affects the price of the underlying security. The feedback-effect leads to a nonlinear version of the Black-Scholes partial differential equation. Using the Lie group theory we reduce the partial differential equation in special cases to ordinary differe...

We consider a system of two coupled ordinary differential equations which appears as an envelope equation in Bose–Einstein Condensation. This system can be viewed as a nonlinear extension of the celebrated model introduced by Landau and Zener. We show how the nonlinear system may appear from different physical models. We focus our attention on the large time behavior of the solution. We show th...

With the development of modern society, research on properties of ordinary differential equation is becoming one of the hotspots in mathematical field. Neutral differential equation which is usually generated in natural science and engineering field is always extensively concerned by many scientific researchers for it can effectively describe multiple complex phenomena in natural world. In rece...

A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapping to a well-known ordinary differential equation, as for example Airy’s equation. In this paper we generalize the mapping procedure to the nonlinear Schrödinger equation or Gross-Pitaevskii equation describing the macroscopic wave function of a Bose-Einstein condensate. The nonlinear Schrödinger e...

Ill PREFACE This Memorandum is part of RAND's continuing search for new ways of utilizing the modern digital computer. The authors present a method for numerically Integrating nonlinear partial differential integral equations, which occur In such fields as radiative transfer and mathematical biology. The method Is then specifically applied to solving a basic equation of transport In a spherical...

By constructing the commutative operators chain, we derive conditions for solving the eigenfunctions of Dirac equation and Schrödinger type equation via separation of variables. Detailed calculation shows that, only a few cases can be completely reduced into ordinary differential equation system. So the effective perturbation or approximation methods for the resolution of the spinor equation ar...