Numerical Computations of Self - Similarblow - up Solutions of Thegeneralized

The structure of the blow-up in nite time of a solution of the Generalized Korteweg-de Vries equation arising from a perturbed unstable solitary wave is studied numerically. The computed solution is observed to blow-up in the L 1-norm in nite time by forming a spike of innnite height at x = x and at t = t. Scaled coordinates are introduced to examine the detailed structure of the solution in the immediate neighborhood of the blow-up. The appropriately rescaled solution is observed to converge in these coordinates as t ! t ? , indicating self-similar behavior. A best-t solution w() of the nonlinear ODE satissed by self-similar prooles is computed for the statistical data compiled from this convergence. The asymptotics at 1 of this solution of the ODE are studied, and found to coincide with those of solutions w () of the linearized ODE as ! 1. The self-similar part of the solution is also matched (numerically) to the part of the solution more removed from the blow-up point, showing how rapidly decaying initial data can give rise to self-similar blow-up. Heuristic explanations of how nonlinearity and dispersion cooperate to yield existence of a solution w() of the ODE with the desired asymptotics as ! 1 are discussed.