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

Abstract

Abstract We study the solution u ( x, t ) of the Korteweg‐de Vries equation u t − 6 uu x + u xxx = 0 with arbitrary initial conditions u ( x , 0) = u 0 ( x ), u 0 ( 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.