{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\sigma \\in \\set{1, -1}$.\nThe *Kadomtsev–Petviashvili equation* is the [[partial differential equation|GC5BDZ6QBC]]\n$$\\partial_x \\big( u_t + 6 u u_x + u_{xxx} \\big) + 3 \\sigma u_{yy} = 0$$\nfor an unknown real function $u(x, y, t)$ of $(x, y) \\in \\R^2$ and $t \\in \\R$, where subscripts denote [[partial derivatives|09EGC6F8PT]].\nThe case $\\sigma = 1$ is called *KP-II* and the case $\\sigma = -1$ is called *KP-I*.\n\n## Remarks\n\nExample: let $v(x, t)$ be a solution of the [[Korteweg–De Vries equation|KTT75QC72Z]] $v_t + 6 v v_x + v_{xxx} = 0$, and put $u(x, y, t) = v(x, t)$.\nThen the bracket vanishes and $u_{yy} = 0$, so $u$ solves both KP-I and KP-II.\n\nNon-example: $u(x, y, t) = y^2$ is not a solution, because the first term is $0$ while $3 \\sigma u_{yy} = 6 \\sigma \\ne 0$.\n\nOther normalisations are used, for example $\\partial_x(u_t + u u_x + \\varepsilon^2 u_{xxx}) + \\lambda u_{yy} = 0$ with $\\lambda = \\pm 1$.\nThey reduce to the form above by scaling the variables, and the sign of the $u_{yy}$ term relative to the dispersive term decides between KP-I and KP-II.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "P93R13XP0P",
    "kind": "definition",
    "msc": [
      "35Q53",
      "35Q51"
    ],
    "name": "Kadomtsev–Petviashvili equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kadomtsev–Petviashvili_equation"
      }
    ],
    "tag": "YBBS082ZY1",
    "uses": [
      "GC5BDZ6QBC",
      "KTT75QC72Z",
      "09EGC6F8PT"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YBBS082ZY1/",
  "links": [
    {
      "field": "defines",
      "target": "P93R13XP0P"
    },
    {
      "field": "uses",
      "target": "GC5BDZ6QBC"
    },
    {
      "field": "uses",
      "target": "KTT75QC72Z"
    },
    {
      "field": "uses",
      "target": "09EGC6F8PT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YBBS082ZY1.md",
  "msc": [
    "35Q53",
    "35Q51"
  ],
  "name": "Kadomtsev–Petviashvili equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YBB/YBBS082ZY1.md",
  "status": null,
  "tag": "YBBS082ZY1",
  "uri": "m2://YBBS082ZY1",
  "url": "https://tamnd.github.io/math2/wiki/kadomtsev-petviashvili-equation-definition/",
  "writing": {
    "tag": "YBBS082ZY1",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}