Existence of a minimal non-scattering solution to the mass-subcritical   generalized Korteweg-de Vries equation
  In this article, we prove existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation in the scale critical ^L^r space. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to ^L^r-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schr"odinger equation. 
Satoshi Masaki and Jun-ichi Segata
math.AP