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  "body": "\n## Definition\n\nConsider a [[partial differential equation|GC5BDZ6QBC]] for a function $u \\colon \\R \\times \\R \\to \\R^m$ of a space variable $x$ and a time variable $t$.\nA *traveling wave* solution is a solution of the form\n$$u(x, t) = U(x - c t),$$\nwhere $c \\in \\R$ is the *speed* and $U \\colon \\R \\to \\R^m$ is the *profile*.\nWhen the limits $U(\\pm\\infty)$ exist and differ, the wave is called a *front*.\nWhen they exist and are equal and $U$ is not constant, it is called a *pulse*.\n\n## Remarks\n\nIn $\\R^n$ one also considers plane traveling waves $u(x, t) = U(x \\cdot e - ct)$ for a unit vector $e$.\nSubstituting $\\xi = x - ct$ turns the equation into an ordinary differential equation for $U(\\xi)$, in which $c$ is an unknown parameter.\n\nExample: the viscous [[Burgers' equation|NEGHXN8RE1]] $u_t + u u_x = \\nu u_{xx}$, with $\\nu \u003e 0$, has the front $u = c\\left(1 - \\tanh\\frac{c(x - ct)}{2\\nu}\\right)$ for every $c \u003e 0$.\nCheck: with $U(\\xi) = c(1 - \\tanh \\tfrac{c\\xi}{2\\nu})$, the equation becomes $-cU' + UU' = \\nu U''$.\nIntegrating once with $U(+\\infty) = 0$ gives $\\nu U' = \\tfrac12 U^2 - cU$, which this $U$ satisfies.\nThe front goes from $2c$ to $0$, and its speed $c$ is the average of the two states, as for a shock.\n\nNon-example: $u(x, t) = e^{-t} \\sin x$ solves the heat equation $u_t = u_{xx}$, but it is not a traveling wave.\nFor a traveling wave, $\\sup_x \\abs{u(x, t)} = \\sup \\abs{U}$ does not depend on $t$, while here it equals $e^{-t}$.\n\n## Explanations\n\n### L1 human\n\nA traveling wave is a fixed shape that slides along the line at constant speed.\n",
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