Local well-posedness for the gKdV equation on the background of a bounded function

2021/04/30 by José Manuel Palacios, Palacios, José Manuel · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2104.15126

openalex publication_date 2021/04/30 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We prove the local well-posedness for the generalized Korteweg-de Vries equation in H^s(ℝ), s>1/2, under general assumptions on the nonlinearity f(x), on the background of an L^∞_t,x-function Ψ(t,x), with Ψ(t,x) satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in H^s(ℝ). This result not only gives us a framework to solve the gKdV equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of a periodic solution. As a direct corollary, we obtain the unconditional uniqueness of the gKdV equation in H^s(ℝ) for s>1/2. We also prove global existence in the energy space H¹(ℝ), in the case where the nonlinearity satisfies that \vert f''(x)\vert\lesssim 1.

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• Improved refined bilinear estimates and well-posedness for generalized KdV type equations on ℝ

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