= Solution
For a single spectral mass, the algebraic equation gives
$$
\Phi=\frac{E}{1+cE/(2\kappa)},\qquad E=e^{-\kappa x+\kappa^3t}.
$$
Differentiate the reconstruction, writing $r=cE/(2\kappa)$:
$$
q=\frac{\kappa cE}{[1+cE/(2\kappa)]^2}
=\frac{2\kappa^2r}{(1+r)^2}.
$$
With $x_0=\kappa^{-1}\log[c/(2\kappa)]$, the concise one-<soliton> answer is
$$
\boxed{q(x,t)=\frac{\kappa^2}{2}\operatorname{sech}^2\left[\frac\kappa2(x-\kappa^2t-x_0)\right].}
$$
Its speed is $\kappa^2$ and its maximum is $\kappa^2/2$. Equivalently, setting $\eta=\kappa/2$ gives $q=2\eta^2\operatorname{sech}^2[\eta(x-4\eta^2t-x_0)]$. To check the sign and speed independently, this profile obeys $q_{xx}=\kappa^2q-3q^2$ and $q_t=-\kappa^2q_x$; differentiation gives the stated <KdV equation> exactly.
Back to article page
