Two new reductions methods for polynomial differential equations and applications to nonlinear PDEs.

For ordinary differential equations of the form P ( y , v , v ′ , … , v ( n ) ) = 0 which are polynomial in the variables v , v ′ , … , v ( n ) two new reduction methods to first-order equations are considered. The reduced equations are of the forms v ′ = Q 1 ( y , v 1 ∕ q ) and v ′ = ( Q 2 ( y , v ) ) 1 ∕ q , where Q 1 and Q 2 are two polynomials of degree p in v 1 ∕ q and v , respectively, whose coefficients depend on y . In contrast to most of the known reduction methods of these types, which use either q = 1 or q = 2 , in this paper the values of the positive integers p , q are not predetermined. A procedure to obtain the possible values of the integers for which a reduction of any of these types may exist is provided. As a consequence, new reductions that cannot be obtained by other known methods may be found. The new methods have been applied to obtain some reductions and, consequently, new solutions for three polynomial ordinary differential equations related to well-known equations in mathematical physics: the Kuramoto–Sivashinsky equation, a generalized Benney equation and a 5 th-order KdV equation. Some pieces of computer algebra code, written in Maple and implementing the underlying algorithms to derive the reductions, are also included. [ABSTRACT FROM AUTHOR]