The construction of two-dimensional optimal systems for the invariant solutions

TL;DR

This paper develops a systematic method to construct two-dimensional optimal systems of invariant solutions for differential equations, demonstrated on heat and Navier-Stokes equations, leading to new solutions and classifications.

Contribution

A new algorithm for constructing two-dimensional optimal systems based on invariants, applied to heat and Navier-Stokes equations, providing comprehensive classifications and explicit solutions.

Findings

Constructed 2D optimal systems for heat and Navier-Stokes equations.

Discovered 11 two-parameter elements for the heat equation.

Generated new reduced ODEs and explicit solutions for Navier-Stokes.

Abstract

To search for inequivalent group invariant solutions, a general and systematic approach is established to construct two-dimensional optimal systems, which is based on commutator relations, adjoint matrix and the invariants. The details of computing all the invariants for two-dimensional subalgebras is presented and the optimality of twodimensional optimal systems is shown clearly under different values of invariants, with no further proof. Applying the algorithm to (1+1)-dimensional heat equation and (2+1)-dimensional Navier-Stokes (NS) equation, their twodimensional optimal systems are obtained, respectively. For the heat equation, eleven two-parameter elements in the optimal system are found one by one, which are discovered more comprehensive. The two-dimensional optimal system of NS equations is used to generate intrinsically different reduced ordinary differential equations and some interesting explicit solutions are provided.