Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

In this article, we prove the 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 Lˆ^r space where Lˆ^r={f∈S^′(R)|‖f‖_Lˆ^r=‖fˆ‖_L^r′<∞}. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to Lˆ^r-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.