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  "body": "\n## Summary\n\nA dispersive partial differential equation is an evolution equation whose plane waves travel at speeds that depend on their frequency.\nA localized solution, made of many frequencies, therefore spreads out in space and decays in amplitude while its energy is conserved.\nThe main examples are the [[Schrödinger equation|EPZR6H2WJN]], the [[Korteweg–De Vries equation|KTT75QC72Z]] and the [[Klein–Gordon equation|AYCCDQWCJF]].\nDispersion competes with nonlinear effects, and the balance between the two produces phenomena such as [[solitons|63D4AXYCT4]].\n\n## Explanations\n\n### L1 human\n\nThrow a stone into a pond: the ring of waves spreads into a train of ripples, because long and short ripples travel at different speeds.\nThis separation by frequency is called dispersion.\nIn a dispersive equation, a bump does not keep its shape; it spreads out and gets lower.\nFor the Schrödinger equation $i u_t + u_{xx} = 0$, a plane wave $e^{i(kx - k^2 t)}$ has frequency $k^2$, and a packet of such waves near $k$ moves with speed $2k$, so fast and slow parts separate.\n",
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