Scaling, stability and singularities for nonlinear, dispersive wave equations: the critical case

For a class of generalized Korteweg-de Vries equations of the form u(t) + (u(p))(x) - D(beta)u(x) = 0 (*) posed in R and for the focusing nonlinear Schrodinger equations iu(t) + Deltau + \u\(p)u = 0 (**) posed on R-n, it is well known that the initial-value problem is globally in time well posed provided the exponent p is less than a critical power p(crit). For p greater than or equal to p(crit), it is known for equation (**) and suspected for equation (*) (known for p = 5 and beta = 2) that large initial data need not lead to globally defined solutions. It is our purpose here to investigate the critical case p = p(crit) in more detail than heretofore. Building on previous work of Weinstein, Laedke, Spatschek and their collaborators, earlier work of the present authors and others, a stability result is formulated for small perturbations of ground-state solutions of (**) and solitary-wave solutions of (*). This theorem features a scaling that is natural in the critical case. When interpreted in the contexts in view, our general result provides information about singularity formation in the case the solution blows up in finite time and about large-time asymptotics in the case the solution is globally defined.