Asymptotic Solutions of the Korteweg‐deVries Equation

The long‐time asymptotic solution of the Korteweg‐deVries equation, corresponding to initial data which decay rapidly as | x |→∞ and produce no solitons, is found to be considerably more complicated than previously reported. In general, the asymptotic solution consists of exponential decay, similarity, rapid oscillations and a “collisionless shock” layer. The wave amplitude in this layer decays as [(ln t )/ t ] 2/3 . Only for very special initial conditions is the shock layer absent from the solution.