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

Abstract

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 ${\hat{L}}^r$ space where ${\hat{L}}^r=\{f\in {\mathcal{S}}^{\prime }(\mathbb{R})|{\Vert \begin{array}{c}f\end{array}\Vert }_{{\hat{L}}^r}={\Vert \begin{array}{c}\hat{f}\end{array}\Vert }_{L^{r^{\prime }}}<\infty \}$. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to ${\hat{L}}^r$-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.