On dispersive decay for the generalized Korteweg--de Vries equation

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This is a Quantitative Study study.

Matthew Kowalski; Minjie Shan

doi:10.48550/arxiv.2510.01728

October 2, 2025Open Access

On dispersive decay for the generalized Korteweg--de Vries equation

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Key Points

Solutions to the gKdV equation decay in L^∞ like |t|^(-1/3) under certain initial conditions.
The mass-critical model only requires initial data in the space dotH^{1/4} ∩ dotH^{-1/12}.
Development of persistence of negative regularity improves understanding of gKdV solutions.
Extending Lorentz--Strichartz estimates to mixed norms broadens the framework of analysis.

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Implication

This analysis provides dispersive estimates for solutions to the generalized Korteweg--de Vries equation, suggesting insights into initial data and decay patterns.

Abstract

We prove pointwise-in-time dispersive estimates for solutions to the generalized Korteweg--de Vries (gKdV) equation. In particular, for solutions to the mass-critical model, we assume only that initial data lie in Ḣ1/4 ∩ Ḣ-1/12 and show that solutions decay in L^∞ like |t|-1/3. To accomplish this, we develop a persistence of negative regularity for solutions to gKdV and extend Lorentz--Strichartz estimates to the mixed norm case.

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Kowalski et al. (2025) studied this question.

synapsesocial.com/papers/68e25385d6d66a53c2474d72 https://doi.org/10.48550/arxiv.2510.01728

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On dispersive decay for the generalized Korteweg--de Vries equation

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On dispersive decay for the generalized Korteweg--de Vries equation

Key Points

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