Conservation Laws of Variable Coefficient Diffusion-Convection Equations

TL;DR

This paper classifies local conservation laws of variable coefficient diffusion-convection equations using equivalence groups, including nonlocal transformations, to systematically understand their structure and symmetries.

Contribution

It constructs and analyzes the extended equivalence group for these equations, enabling a comprehensive classification of conservation laws up to nonlocal equivalences.

Findings

Classification of conservation laws using extended equivalence group

Identification of gauge equivalence transformations within the symmetry structure

Simplified formulation of conservation laws through extended group analysis

Abstract

We study local conservation laws of variable coefficient diffusion-convection equations of the form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. The main tool of our investigation is the notion of equivalence of conservation laws with respect to the equivalence groups. That is why, for the class under consideration we first construct the usual equivalence group $G^{\sim}$ and the extended one $\hat G^{\sim}$ including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group $\hat G^{\sim}$ has interesting structure since it contains a non-trivial subgroup of gauge equivalence transformations. Then, using the most direct method, we carry out two classifications of local conservation laws up to equivalence relations generated by $G^{\sim}$ and $\hat G^{\sim}$, respectively. Equivalence with respect to $\hat G^{\sim}$ plays the major role for simple and clear formulation of the final results.