A basis of hierarchy of generalized symmetries and their conservation laws for the (3+1)-dimensional diffusion equation

TL;DR

This paper develops a hierarchical framework for understanding generalized symmetries and conservation laws of the (3+1)-dimensional diffusion equation, providing formulas for symmetry basis and generating infinite conservation laws.

Contribution

It introduces a hierarchy-based method to determine dependencies among symmetries and constructs a basis for conservation laws, advancing the analysis of symmetries in high-dimensional PDEs.

Findings

Derived a formula for the number of independent generalized symmetries.

Constructed a basis for conservation laws for the diffusion equation.

Generated infinite conservation laws within each symmetry class.

Abstract

We determine, by hierarchy, dependencies between higher order linear symmetries which occur when generating them using recursion operators. Thus, we deduce a formula which gives the number of independent generalized symmetries (basis) of several orders. We construct a basis for conservation laws (with respect to the group admitted by the system of differential equation) and hence generate infinite conservation laws in each equivalence class.