 Solution

For a localized traveling wave put $q(x,t)=Q(\xi )$ with $\xi =x-ct-x_0$, and require $Q,Q^{\prime },Q^{\prime \prime }\to 0$ at infinity. The modified KdV equation integrates once to $Q^{\prime \prime }-cQ+2Q^3=0$. Multiplying by $Q^{\prime }$ and integrating again givesA 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 yieldsThe signs and translations are arbitrary. Differentiating the sech profile verifies $Q^{\prime \prime }=cQ-2Q^3$ directly. The relevant antiderivative is the inverse hyperbolic secant, since $d(\mathrm{arcsech}x)/dx=-1/[x\sqrt{1-x^2}]$ for $0<x<1$.