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  "body": "\n## Definition\n\nLet $n \\ge 1$ and consider a [[partial differential equation|GC5BDZ6QBC]] for a function $u \\colon \\R^n \\times (0, \\infty) \\to \\R$.\nLet $\\alpha, \\beta \\in \\R$, and for $\\lambda \u003e 0$ put $u_\\lambda(x, t) = \\lambda^{\\alpha} u(\\lambda^{\\beta} x, \\lambda t)$.\nAssume that the equation is invariant under these scalings: if $u$ is a solution, then so is $u_\\lambda$ for every $\\lambda \u003e 0$.\nA solution $u$ is a *similarity solution*, or *self-similar solution*, with exponents $(\\alpha, \\beta)$ if $u_\\lambda = u$ for every $\\lambda \u003e 0$.\n\n## Remarks\n\nTaking $\\lambda = 1/t$ shows that $u$ is self-similar exactly when\n$$u(x, t) = t^{-\\alpha}\\, v\\!\\left( \\frac{x}{t^{\\beta}} \\right)$$\nfor a function $v$ on $\\R^n$, namely $v = u(\\cdot, 1)$.\nSubstituting this form turns the equation into an equation for $v$ in the variable $\\eta = x / t^\\beta$.\n\nExample: the [[heat equation|VP2HN86VAG]] $u_t = \\Delta u$ is invariant for $\\beta = 1/2$ and every $\\alpha$.\nThe heat kernel $\\Phi(x, t) = (4\\pi t)^{-n/2} e^{-\\abs{x}^2 / (4t)}$ is self-similar with $\\alpha = n/2$, $\\beta = 1/2$.\nCheck: $\\lambda^{n/2} \\Phi(\\lambda^{1/2} x, \\lambda t) = \\lambda^{n/2} (4\\pi\\lambda t)^{-n/2} e^{-\\lambda \\abs{x}^2/(4\\lambda t)} = \\Phi(x, t)$.\n\nNon-example: $u(x, t) = e^{-t} \\sin x$ solves $u_t = u_{xx}$ for $n = 1$, but it is not self-similar.\nA self-similar solution would have zero set $\\sqrt{t}\\, Z$ at time $t$, with $Z$ the zero set of $v$.\nBut the zero set of $u(\\cdot, t)$ is $\\pi \\Z$ for every $t$.\nAt $t = 1$ this forces $Z = \\pi \\Z$, and then at $t = 4$ the zero set would be $2 \\pi \\Z \\ne \\pi \\Z$.\n\nFor a scalar conservation law $u_t + F(u)_x = 0$, the scalings with $\\alpha = 0$, $\\beta = 1$ are symmetries.\nThe corresponding similarity solutions $u = v(x/t)$ include the [[rarefaction waves|MFN4Q5SH8H]] and the solutions of the [[Riemann problem|GJWKX22VVE]].\n\n## Explanations\n\n### L1 human\n\nA similarity solution looks the same at every time after the right change of scale.\nIts whole history is encoded in the profile $v$ at time $1$.\n",
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    "defines": "579J5K9K5Z",
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    "name": "Similarity solution",
    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Self-similar_solution"
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  "msc": [
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  "name": "Similarity solution",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/3DJ/3DJMFT0SQM.md",
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  "url": "https://tamnd.github.io/math2/wiki/similarity-solution-definition/",
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        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
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