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

TL;DR

The paper analyzes the L^2‑supercritical gKdV equation with p>5, addressing long‑time dynamics beyond small data or proximity to solitons. It develops a unified virial framework that uses a growth scale beta(t) to renormalize the evolving gradient, enabling quantitative decay and mass‑flow control in both global and finite‑time blow‑up regimes without smallness assumptions. Key contributions include sharp far‑field decay on both half‑lines, normalized local vanishing along time sequences, and stronger results in the even‑power case, all derived via localized virial identities and mass conservation. The work provides the first general non‑perturbative description of non‑solitonic dynamics for large‑data, L^2‑supercritical gKdV and offers techniques likely adaptable to other dispersive equations in similar regimes.

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.

Theorems & Definitions (16)

• Theorem 1.1: MMPP1
• Theorem 1.2: Blow up case
• Theorem 1.3: Globally defined case
• Lemma 2.1: Corollary 2.18 in KPV1
• proof : Sketch of proof
• Lemma 2.2
• proof : Sketch of proof
• Lemma 3.1
• Remark 3.2
• proof