{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Delta(x, u, \\partial u, \\dots, \\partial^k u) = 0$ be a system of [[partial differential equations|GC5BDZ6QBC]] for $u \\colon U \\to \\R^q$ with $U \\subseteq \\R^p$ open.\nA *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\n$$v = \\sum_{i=1}^p \\xi^i(x, u)\\, \\partial_{x^i} + \\sum_{\\alpha=1}^q \\phi^\\alpha(x, u)\\, \\partial_{u^\\alpha}.$$\nThe 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.\nIt is a *Lie point symmetry* of the system if $g_\\varepsilon \\cdot u$ is a solution for every solution $u$ and every small $\\varepsilon$.\n\nThe *Lie group method* consists of the following steps.\n\n1. Write the condition that the $k$-th prolongation of $v$ annihilates $\\Delta$ on the solution set.\n2. Split this condition into a linear system of determining equations for $\\xi^i$ and $\\phi^\\alpha$, and solve it.\n3. Use the resulting symmetries to map known solutions to new ones.\n   Find invariant solutions by reducing the system to fewer independent variables.\n\n## Remarks\n\nFor systems of maximal rank, step 1 is necessary and sufficient for $v$ to generate symmetries.\nThe set of all such $v$ is a Lie algebra of vector fields.\n\nExample: 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 $u \\mapsto e^{\\varepsilon} u$.\nIt also has the Galilean symmetry $u(x, t) \\mapsto e^{-\\varepsilon x + \\varepsilon^2 t}\\, u(x - 2\\varepsilon t, t)$.\nCheck 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.\nSolutions 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 $\\eta = x / \\sqrt t$.\n\nNon-example: the map $u(x, t) \\mapsto w(x, t) = u(x/\\lambda, t/\\lambda)$ with $\\lambda \u003e 0$, $\\lambda \\ne 1$, is not a symmetry of the heat equation.\nIndeed $w_t - w_{xx} = \\lambda^{-1} u_t - \\lambda^{-2} u_{xx}$ at $(x/\\lambda, t/\\lambda)$, which equals $(\\lambda^{-1} - \\lambda^{-2}) u_t$ there and is non-zero for most solutions.\n\n## Explanations\n\n### L1 human\n\nThe method first finds all continuous symmetries of an equation by solving linear equations.\nThen it uses them to build new solutions and to reduce the equation to simpler ones.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "QQ8M0T04Q9",
    "kind": "definition",
    "msc": [
      "35B06",
      "58J70",
      "35A30"
    ],
    "name": "Lie group method for PDEs",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lie_point_symmetry"
      }
    ],
    "tag": "SYXNSN5JX0",
    "uses": [
      "GC5BDZ6QBC",
      "06738GGWWK"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/SYXNSN5JX0/",
  "links": [
    {
      "field": "defines",
      "target": "QQ8M0T04Q9"
    },
    {
      "field": "uses",
      "target": "GC5BDZ6QBC"
    },
    {
      "field": "uses",
      "target": "06738GGWWK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/SYXNSN5JX0.md",
  "msc": [
    "35B06",
    "58J70",
    "35A30"
  ],
  "name": "Lie group method for PDEs",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SYX/SYXNSN5JX0.md",
  "status": null,
  "tag": "SYXNSN5JX0",
  "uri": "m2://SYXNSN5JX0",
  "url": "https://tamnd.github.io/math2/wiki/lie-group-method-for-pdes-definition/",
  "writing": {
    "tag": "SYXNSN5JX0",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}