---
tag: QQ8M0T04Q9
kind: concept
name: Lie group method for PDEs
msc: [35B06, 58J70, 35A30]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_point_symmetry
---

## Summary

The Lie group method studies a differential equation through its symmetries: continuous groups of changes of variables that map solutions to solutions.
Sophus Lie showed that such symmetries can be found by solving a linear system of determining equations for their infinitesimal generators.
Once found, they produce new solutions from old ones and reduce the equation to one with fewer variables.
For ordinary differential equations it is the [[Lie symmetry method|X3S9XHX83H]].
[[Similarity solutions|579J5K9K5Z]] and [[traveling waves|M37M411KP7]] are the invariant solutions of scaling and translation symmetries.
See the [[definition|SYXNSN5JX0]].

## Explanations

### L1 human

Moving a solution of the heat equation to the right, or starting it later, again gives a solution.
Stretching space by $2$ and time by $4$ also gives a solution.
These are symmetries of the equation, and each comes in a family with a continuous parameter.
Lie's method finds all such families by a mechanical computation.
A solution that is unchanged by one family depends on fewer variables, so it solves a simpler equation.