---
tag: SYXNSN5JX0
kind: definition
name: Lie group method for PDEs
msc: [35B06, 58J70, 35A30]
defines: QQ8M0T04Q9
uses: [GC5BDZ6QBC, 06738GGWWK]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_point_symmetry
---

## Definition

Let $\Delta(x, u, \partial u, \dots, \partial^k u) = 0$ be a system of [[partial differential equations|GC5BDZ6QBC]] for $u \colon U \to \R^q$ with $U \subseteq \R^p$ open.
A *one-parameter group of point transformations* is a local [[flow|06738GGWWK]] $g_\varepsilon$ on an open subset of $\R^p \times \R^q$, generated by a smooth vector field
$$v = \sum_{i=1}^p \xi^i(x, u)\, \partial_{x^i} + \sum_{\alpha=1}^q \phi^\alpha(x, u)\, \partial_{u^\alpha}.$$
The group acts on functions by moving their graphs: $g_\varepsilon \cdot u$ is the function whose graph is $g_\varepsilon(\text{graph of } u)$, where this is again a graph.
It is a *Lie point symmetry* of the system if $g_\varepsilon \cdot u$ is a solution for every solution $u$ and every small $\varepsilon$.

The *Lie group method* consists of the following steps.

1. Write the condition that the $k$-th prolongation of $v$ annihilates $\Delta$ on the solution set.
2. Split this condition into a linear system of determining equations for $\xi^i$ and $\phi^\alpha$, and solve it.
3. Use the resulting symmetries to map known solutions to new ones.
   Find invariant solutions by reducing the system to fewer independent variables.

## Remarks

For systems of maximal rank, step 1 is necessary and sufficient for $v$ to generate symmetries.
The set of all such $v$ is a Lie algebra of vector fields.

Example: the heat equation $u_t = u_{xx}$ has the symmetries $u(x, t) \mapsto u(x - \varepsilon, t)$, $u(x, t) \mapsto u(e^{-\varepsilon} x, e^{-2\varepsilon} t)$ and $u \mapsto e^{\varepsilon} u$.
It also has the Galilean symmetry $u(x, t) \mapsto e^{-\varepsilon x + \varepsilon^2 t}\, u(x - 2\varepsilon t, t)$.
Check of the last one: if $w = e^{-\varepsilon x + \varepsilon^2 t} u(x - 2\varepsilon t, t)$, a direct computation gives $w_t - w_{xx} = e^{-\varepsilon x + \varepsilon^2 t} (u_t - u_{xx})$, which vanishes.
Solutions invariant under the scaling are the [[similarity solutions|579J5K9K5Z]] $u = v(x / \sqrt t)$, which solve the ordinary differential equation $v'' + \tfrac{\eta}{2} v' = 0$ with $\eta = x / \sqrt t$.

Non-example: the map $u(x, t) \mapsto w(x, t) = u(x/\lambda, t/\lambda)$ with $\lambda > 0$, $\lambda \ne 1$, is not a symmetry of the heat equation.
Indeed $w_t - w_{xx} = \lambda^{-1} u_t - \lambda^{-2} u_{xx}$ at $(x/\lambda, t/\lambda)$, which equals $(\lambda^{-1} - \lambda^{-2}) u_t$ there and is non-zero for most solutions.

## Explanations

### L1 human

The method first finds all continuous symmetries of an equation by solving linear equations.
Then it uses them to build new solutions and to reduce the equation to simpler ones.