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  "body": "\n## Definition\n\nLet $E$ be an evolution equation for $u(x, t)$ with $x, t \\in \\R$.\nA *solitary wave* of $E$ is a [[traveling wave|M37M411KP7]] solution $u(x, t) = \\varphi(x - ct)$ with $c \\in \\R$ and a profile $\\varphi$ that tends to constant limits as $x \\to \\pm \\infty$.\nA *soliton* is a solitary wave of $E$ that survives interaction with other solitary waves of $E$.\nThat is, after each such interaction it emerges with the same shape and speed, shifted only in position.\n\n## Remarks\n\nThe second clause is often checked by the existence of exact multi-soliton solutions, which split for $t \\to \\pm \\infty$ into separated solitary waves with the same speeds.\nThere is no single formal definition used by all authors.\n\nExample: for the [[Korteweg–De Vries equation|KTT75QC72Z]] $u_t + 6 u u_x + u_{xxx} = 0$ and $c \u003e 0$, put $\\varphi(\\xi) = \\frac{c}{2} \\operatorname{sech}^2\\big(\\frac{\\sqrt{c}}{2} \\xi\\big)$, using the [[hyperbolic secant|01PPDSDP2N]].\nFor $u = \\varphi(x - ct)$ the equation becomes $-c\\varphi' + 6\\varphi\\varphi' + \\varphi''' = 0$, which integrates to $\\varphi'' = c\\varphi - 3\\varphi^2$.\nWith $\\varphi = A \\operatorname{sech}^2(k\\xi)$ one has $\\varphi'' = 4k^2 A \\operatorname{sech}^2(k\\xi) - 6k^2 A \\operatorname{sech}^4(k\\xi)$, so the equation holds with $4k^2 = c$ and $A = 2k^2 = c/2$.\nThe KdV equation also has explicit two-soliton solutions, so this solitary wave is a soliton.\n\nNon-example: the linear equation $u_t + u_{xxx} = 0$ has no solitary wave with $c \\ne 0$ that decays at infinity.\nIndeed, $\\varphi''' = c\\varphi'$ gives $\\varphi'' = c\\varphi$, whose decaying solutions on all of $\\R$ are only $\\varphi = 0$.\n\n## Explanations\n\n### L1 human\n\nA solitary wave is a single hump that moves without changing shape.\nIt is a soliton when it also survives collisions with other such humps, keeping its size and speed.\n",
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