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

TL;DR

This paper proves the existence of a minimal non-scattering solution to the mass-subcritical gKdV equation using concentration compactness, profile decomposition, and Schrödinger equation approximations.

Contribution

It establishes the existence of a minimal non-scattering solution in the scale critical space for the mass-subcritical gKdV equation, employing novel analytical techniques.

Findings

Existence of a minimal non-scattering solution proven.

Development of a linear profile decomposition in ^L^r space.

Approximation of gKdV solutions via nonlinear Schrödinger solutions.

Abstract

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.