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  "body": "\n## Summary\n\nA soliton is a localised wave that moves at constant speed without changing its shape.\nIt keeps its shape and speed after it collides with other solitons; the only trace of the collision is a shift in position.\nSolitons arise when nonlinear steepening and dispersion balance, in equations such as the [[Korteweg–De Vries equation|KTT75QC72Z]], the [[nonlinear Schrödinger equation|GEJHRXC176]] and the [[sine-Gordon equation|802QSF9NM3]].\nThey are central in the theory of integrable equations and in models of water waves and of light pulses in optical fibres.\n\n## Explanations\n\n### L1 human\n\nA soliton is a single hump of water, or of light, that travels far without spreading out or breaking.\nUsually a wave packet spreads because its parts travel at different speeds, and a tall wave steepens and breaks because its top moves faster.\nIn a soliton these two effects cancel.\nFor the KdV equation $u_t + 6 u u_x + u_{xxx} = 0$ the soliton is $u = \\frac{c}{2} \\operatorname{sech}^2\\big(\\frac{\\sqrt{c}}{2}(x - ct)\\big)$: taller solitons are narrower and faster.\nWhen a tall fast soliton overtakes a small slow one, both come out of the collision unchanged.\n",
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