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 L̂ʳ space where L̂ʳ = \{f ∈ S^{}(R)|{Vmatrix} f {Vmatrix}_{L̂ʳ} = {Vmatrix} f̂ {Vmatrix}_{L^{r^{}}} < ∞ \} . We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to L̂ʳ -framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.