2019/11/01 by Andreia Chapouto, Chapouto, Andreia · 1 citation
Mathematics · Physics and Astronomy · #35Q53 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1911.00551
openalex publication_date 2019/11/01 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the\ncircle. We first consider the real-valued setting and show global\nwell-posedness of the (usual) renormalized mKdV equation in the\nFourier-Lebesgue spaces. In the complex-valued setting, we observe that the\nmomentum plays an important role in the well-posedness theory. In particular,\nwe prove that the complex-valued mKdV equation is ill-posed in the sense of\nnon-existence of solutions when the momentum is infinite, in the spirit of the\nwork on the nonlinear Schr "odinger equation by Guo-Oh (2018). This\nnon-existence result motivates the introduction of the second renormalized mKdV\nequation, which we propose as the correct model in the complex-valued setting\noutside of H^ frac12( mathbbT). Furthermore, imposing a new notion of\nfinite momentum for the initial data, at low regularity, we show existence of\nsolutions to the complex-valued mKdV equation. In particular, we require an\nenergy estimate, from which conservation of momentum follows.\n