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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nConsider a linear evolution [[partial differential equation|GC5BDZ6QBC]] with constant coefficients for $u \\colon \\R^n \\times \\R \\to \\C$.\nSuppose that its plane-wave solutions $u(x, t) = e^{i(\\xi \\cdot x - \\omega t)}$ are exactly those with $\\omega = \\omega_j(\\xi)$ for finitely many functions $\\omega_j \\colon \\R^n \\to \\C$.\nThese functions are the branches of the dispersion relation.\nThe equation is *dispersive* if:\n\n1. each $\\omega_j(\\xi)$ is real for all real $\\xi$, so plane waves neither grow nor decay;\n2. for some $j$, the [[Hessian matrix|0693Y2METT]] of $\\omega_j$ is not identically zero, so the group velocity $\\nabla \\omega_j(\\xi)$ depends on $\\xi$.\n\nA nonlinear equation is called dispersive when its linear part is.\n\n## Remarks\n\nExample: the Schrödinger equation $i u_t + \\Delta u = 0$ has $\\omega(\\xi) = \\abs{\\xi}^2$, which is real, with Hessian $2I \\ne 0$, so it is dispersive.\nThe Airy equation $u_t + u_{xxx} = 0$, the linear part of the Korteweg–De Vries equation, has $\\omega(\\xi) = -\\xi^3$, with $\\omega''(\\xi) = -6\\xi$, so it is dispersive.\nNon-example: the transport equation $u_t + c u_x = 0$ has $\\omega(\\xi) = c\\xi$, whose second derivative vanishes, so clause 2 fails; solutions move without changing shape.\nThe heat equation $u_t = \\Delta u$ has $\\omega(\\xi) = -i\\abs{\\xi}^2$, which is not real, so clause 1 fails; it is dissipative rather than dispersive.\nThe one-dimensional wave equation, with $\\omega = \\pm c\\xi$, is not dispersive.\nIn dimension $n \\ge 2$ the branches $\\pm c\\abs{\\xi}$ have nonzero Hessian, so the wave equation meets clause 2, although it is often not called dispersive.\nBy the [[Fourier transform|DPVM34ADZV]], the solution is $\\hat u(\\xi, t) = e^{-i\\omega(\\xi) t}\\hat u(\\xi, 0)$ for a single branch, and a nondegenerate Hessian, as for the Schrödinger equation, is what yields decay estimates in $L^\\infty$.\n",
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