= Sech-squared solitary wave of the KdV equation
{title2=$u=-2\kappa^2\operatorname{sech}^2[\kappa(x-4\kappa^2t-x_0)]$}
For the sign convention $u_t-6uu_x+u_{xxx}=0$ and any $\kappa>0$, the displayed negative <solitary wave> travels at <speed> $4\kappa^2$. To verify it, put $\xi=x-ct$ and write $u=U(\xi)$. The equation becomes $(-cU-3U^2+U'')'=0$. A decaying profile has zero integration constant, so $U''=cU+3U^2$. Direct differentiation of the displayed profile gives $U''=4\kappa^2U+3U^2$, proving the result. Its depth is $2\kappa^2$, so narrower and deeper pulses travel faster. Changing the sign of the nonlinear term reverses the amplitude sign. This wave belongs to the <soliton> family of the <Korteweg-De Vries equation>.
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