Darboux Integrability Research Articles

An improvement to Darboux integrability theorem for systems having a center

We consider planar polynomial differential systems of degree m with a center at the origin and with an arbitrary linear part. We show that if the system has m(m + 1) 2 − [ (m + 1) 2 ] algebraic solutions or exponential factors then it has a Darboux integrating factor. This result is an improvement of the classical Darboux integrability theorem and other recent results about integrability.

Algebraic aspects of integrability for polynomial systems

We present an introductory survey to the Darboux integrability theory of planar complex and real polynomial differential systems. Our presentation contains some improvements to the classical theory.

Inverse problems in the theory of analytic planar vector

In this communication we state and analyze the new inverse problems in the theory of differential equations related to the construction of an analytic planar vector field from a given, finite number of solutions, trajectories or partial integrals. Likewise we study the problem of determining a stationary complex analytic vector field from a given finite subset of terms in the formal power series V(z, w) = \lambda (z^2 + w^2) + \sum^\infty_{k=3} H_k (z, w), H-k (az, aw) = a^kH_k(z,w), and from the subsidiary condition \Gamma (V) = \sum^\infty_{k=1} G_{2k}(z^2 + w^2)^{k+1}, where G_{2k} is the Liapunov constant. The particular case when V (z,w) = f_0(z,w) – f_0(0,0) and (f_0, D \subset \mathbb C^2) is a canonic element in the neigbourhood of the origin of the complex analytic first integral F is analyzed. The results are applied to the quadratic planar vector fields. In particular we constructed the all quadratic vector field tangent to the curve (y – q (x))^2 – p(x) = 0. where q and p are polynomials of degree k and m ≤ 2k respectively. We showed that the quadratic differential systems admits a limit cycle of this type only when the algebraic curve is of the fourth degree. For the case when k > 5 it proved that there exist an unique quadratic vector field tangent to the given curve and it is Darboux's integrable.

Factorization of linear partial differential operators and Darboux integrability of nonlinear PDEs

Using a new definition of generalized divisors we prove that the lattice of such divisors for a given linear partial differential operator is modular and obtain analogues of the well-known theorems of the Loewy-Ore theory of factorization of linear ordinary differential operators. Possible applications to factorized Gröbner bases computations in the commutative and non-commutative cases are discussed, an application to finding criterions of Darboux integrability of nonlinear PDEs is given.

On the Darboux integrable hyperbolic equations

We prove that if the sequence of Laplace invariants for the linearization operator of a nonlinear hyperbolic equation is finite then the equation is Darboux integrable.

On some geometry associated with a generalised Toda lattice

We define the notion of Darboux integrability for linear second order partial differential operators,.We then build on certain geometric results of E. Vessiot related to the theory of Monge characteristics to show that the Darboux integrable operators L can be used to obtain a solution of the A2 Toda field theory. This solution is parametrised by four arbitrary functions. The approach presented in this paper thereby represents an alternative means of linearising the A2 Toda equations and may be contrasted with the known linearisation via the Lax pair.

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras I

We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle — the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.