290,496 research outputs found
Ermakov-Pinney and Emden-Fowler equations: new solutions from novel B\"acklund transformations
The class of nonlinear ordinary differential equations , where F is a smooth function, is studied. Various nonlinear ordinary
differential equations, whose applicative importance is well known, belong to
such a class of nonlinear ordinary differential equations. Indeed, the
Emden-Fowler equation, the Ermakov-Pinney equation and the generalized Ermakov
equations are among them. B\"acklund transformations and auto B\"acklund
transformations are constructed: these last transformations induce the
construction of a ladder of new solutions adimitted by the given differential
equations starting from a trivial solutions. Notably, the highly nonlinear
structure of this class of nonlinear ordinary differential equations implies
that numerical methods are very difficulty to apply
Painleve property and the first integrals of nonlinear differential equations
Link between the Painleve property and the first integrals of nonlinear
ordinary differential equations in polynomial form is discussed. The form of
the first integrals of the nonlinear differential equations is shown to
determine by the values of the Fuchs indices. Taking this idea into
consideration we present the algorithm to look for the first integrals of the
nonlinear differential equations in the polynomial form. The first integrals of
five nonlinear ordinary differential equations are found. The general solution
of one of the fourth ordinary differential equations is given.Comment: 22 page
Meromorphic solutions of nonlinear ordinary differential equations
Exact solutions of some popular nonlinear ordinary differential equations are
analyzed taking their Laurent series into account. Using the Laurent series for
solutions of nonlinear ordinary differential equations we discuss the nature of
many methods for finding exact solutions. We show that most of these methods
are conceptually identical to one another and they allow us to have only the
same solutions of nonlinear ordinary differential equations
Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations
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 implies the linearizability of systems of partial differential
equations corresponding to those complex ordinary differential equations. The
invertible complex transformations can be used to obtain invertible real
transformations that map a system of nonlinear partial differential equations
into a system of linear partial differential equation. Explicit invariant
criteria are given that provide procedures for writing down the solutions of
the linearized equations. A few non-trivial examples are mentioned.Comment: This paper along with its first part ODE-I were combined in a single
research paper "Linearizability criteria for systems of two second-order
differential equations by complex methods" which has been published in
Nonlinear Dynamics. Due to citations of both parts I and II these are not
replaced with the above published articl
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