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For one bound state at $-{\kappa }^2$ and no reflection, writeThe rank-one Gelfand-Levitan-Marchenko kernel has the form $K(x,y;t)=-A(x,t)e^{-\kappa y}$. Substitution into the integral equation givesand henceChoose $x_0$ by $C_0/(2\kappa )=e^{2\kappa x_0}$. ThenDifferentiating the reconstruction formula gives the one-soliton solutionIt is a negative soliton of amplitude $2{\kappa }^2$ travelling to the right with speed $4{\kappa }^2$, as required by this sign convention for KdV.

Putand seek the rank-one Gelfand-Levitan-Marchenko kernel in the separable formSubstitution into the Gelfand-Levitan-Marchenko equation givessoand hence

WriteThe reconstruction formula for this inverse scattering transform convention yieldsIf $q=e^{-2s}$, then $4q/(1+q{)}^2={\mathrm{sech}}^{2}s$. Takingtherefore produces the required one-soliton of the Korteweg-De Vries equation, withHere the usual bound-state norming constant has ${\beta }_0>0$ and $\chi >0$, so $\varphi$ is real.