Integrability and Einstein's equations

Integrability of Klein - Gordon Equations *

Usin the Painlev test, it is shown that the only interablc nonlinear Klein-Gordon equations ux,=f(u) with f a linear combination of exponentials are the Liouville, sine-Gordon (or sinh-Gordon) and Mikhailov equations. In particular, the double sine-Gordon equation is not interable.

Preface: Symmetries and Integrability of Difference Equations

Preface: Symmetries and Integrability of Difference Equations

Integrability of Riccati equations and the stationary KdV equations

Using the S.Lie’s infinitesimal approach we establish the connection between integrability of a one-parameter family of the Riccati equations and the stationary KdV hierarchy. In this paper we will suggest a method for integrating a one-parameter family of the Riccati equations ux + u 2 = f(x, λ) (1) based on their Lie symmetries. Here f(x, λ) = λ + λVn−1(x) + · · ·+ λV1(x) + V0(x) and λ is an ...

Relationships between Darboux Integrability and Limit Cycles for a Class of Able Equations

We consider the class of polynomial differential equation x&= , 2(,)(,)(,)nnmnmPxyPxyPxy++++2(,)(,)(,)nnmnmyQxyQxyQxy++&=++. For where and are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditions we proved such Darboux integrable systems can have at most 1 limit cycle.