Gradient Structures Associated with a Polynomial Differential Equation

Savin Treanţă

doi:10.3390/math8040535

Abstract

In this paper, by using the characteristic system method, the kernel of a polynomial differential equation involving a derivation in R n is described by solving the Cauchy Problem for the corresponding first order system of PDEs. Moreover, the kernel representation has a special significance on the space of solutions to the corresponding system of PDEs. As very important applications, it has been established that the mathematical framework developed in this work can be used for the study of some second-order PDEs involving a finite set of derivations.

Highlights

Throughout time, gradient-type representations for some solutions, gradient systems in a Lie algebra and the algebraic representation of gradient systems, have been investigated, with remarkable results, by Vârsan [1]
Sufficient conditions for linear subspaces of smooth vector fields in order to be written as a kernel of some linear first order partial differential equations are have been formulated and proved in Parveen and Akram [4]
In this paper, taking into account the results included in the quite recently work Treanţă [17], we investigate the kernel of a polynomial differential equation involving a derivation in Rn by solving the Cauchy Problem for the corresponding first order system of PDEs

Summary

Introduction

Throughout time, gradient-type representations for some solutions, gradient systems in a Lie algebra and the algebraic representation of gradient systems, have been investigated, with remarkable results, by Vârsan [1]. In this paper, taking into account the results included in the quite recently work Treanţă [17] (the kernel of a polynomial of scalar derivations is described by solving Cauchy Problems for the corresponding system of ODEs; a gradient representation for the associated Cauchy Problem solution is derived), we investigate the kernel of a polynomial differential equation involving a derivation in Rn by solving the Cauchy Problem for the corresponding first order system of PDEs. we extend a solution. Through the use of the characteristic system method and some results formulated, the associated Cauchy Problem solution is derived (see Theorem 1) This solution has a special significance on the space of solutions to the corresponding first order system of PDEs. Section 4 concludes the paper

Preliminary Results

Main Results

Conclusions and Further Developments