= Solution
For a localized traveling wave put $q(x,t)=Q(\xi)$ with $\xi=x-ct-x_0$, and require $Q,Q',Q''\to0$ at infinity. The modified KdV equation integrates once to $Q''-cQ+2Q^3=0$. Multiplying by $Q'$ and integrating again gives
$$
(Q')^2=cQ^2-Q^4.
$$
A nonzero decaying real solution requires $c>0$. Separating on the rising or falling half of the wave and using the indicated inverse-hyperbolic-secant integral yields
$$
\boxed{q(x,t)=\pm\sqrt c\,\operatorname{sech}\big[\sqrt c(x-ct-x_0)\big],\qquad c>0.}
$$
The signs and translations are arbitrary. Differentiating the sech profile verifies $Q''=cQ-2Q^3$ directly. The relevant antiderivative is the inverse hyperbolic secant, since $d(\operatorname{arcsech}x)/dx=-1/[x\sqrt{1-x^2}]$ for $0<x<1$.
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