On the asymptotic dynamics for the $L^2$-supercritical gKDV equation

TL;DR

This paper investigates the long-time behavior of solutions to the $L^2$-supercritical gKdV equation with nonlinearities greater than 5, revealing how the growth of the gradient norm influences asymptotic decay and blow-up phenomena.

Contribution

It introduces a unified framework for analyzing the non-solitonic dynamics of arbitrary $H^1$ solutions, including global and blow-up cases, using a novel virial method.

Findings

Established sharp decay rates on both half-lines.

Proved normalized local vanishing along sequences of times.

Developed a virial method that handles unbounded $H^1$ norms.

Abstract

We study the $L^2$-supercritical generalized Korteweg-de Vries equation (gKdV) with nonlinearities $p>5$. While local well-posedness in $H^1$ is classical, the long-time dynamics in the supercritical regime remains largely unexplored beyond small data global solutions, the construction of multi-solitons for any power and self-similar blow-up near the critical power $p=5$. We develop a unified description of the non-solitonic region for arbitrary $H^1$ solutions, both global and blowing up. Our analysis shows that the asymptotic $L^2$ and $L^p$ dynamics in this region is completely determined by the growth rate of the $L^2$ norm of the gradient (or, equivalently, the critical $H^{s_p}$ norm). In particular, we prove sharp far-field decay on both half-lines and establish normalized local vanishing along sequences of times, with improved estimates in the case of even-power nonlinearities. A key ingredient is a new virial method that compensates for the possible unboundedness of the $H^1$ norm by exploiting the conservation of mass and a careful localization of the nonlinear flux. This yields quantitative versions of decay phenomena previously known only in subcritical settings, and it applies without any smallness or proximity-to-soliton assumptions.