Symmetry Coefficients of Semilinear Partial Differential Equations

TL;DR

This paper characterizes the form of infinitesimal symmetries of semilinear PDEs, showing dependence constraints on variables and linearity conditions, with applications to important equations in analysis, geometry, and physics.

Contribution

It establishes new symmetry coefficient constraints for semilinear PDEs, especially relating to their order and linearity in derivatives.

Findings

Infinitesimals depend only on independent variables.

For m>1 and linearity in derivatives, infinitesimal of dependent variable is at most linear.

Results apply to key equations in analysis, geometry, and physics.

Abstract

We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m>1 and the equation is also linear in its derivatives of order m-1 of the dependent variable, then the infinitesimal of the dependent variable is at most linear on the dependent variable. Many examples of important partial differential equations in Analysis, Geometry and Mathematical - Physics are given in order to enlighten the main result.