The emergence of solitons of the korteweg‐de vries equation from arbitrary initial conditions

We study the solution u(x, t) of the Korteweg‐de Vries equation ut − 6uux + uxxx = 0 with arbitrary initial conditions u(x, 0) = u0(x), u0(x) decaying sufficiently rapidly as |ξ|→∞. Using the method of the inverse scattering transformation we analyse the Gel'fand‐Levitan equation in all coordinate systems moving to the right and give a complete and rigorous description of the emergence of solitons.