[en] The author presents a rigorous demonstration of the emergence of solitons from the KdV initial value problem with arbitrary initial function. Studying multisoliton solutions of the KdV in the general case of a nonzero reflection coefficient, he derives a new phase shift formula. He derives an estimate which indicates how well a real potential in the Zakharov-Shabat system is approximated by its reflectionless part. Moreover, the associated inverse scattering formalism is simplified considerably. He presents an asymptotic analysis of the sine-Gordon equation on right half lines almost linearly moving leftward. (Auth.)