This work is dedicated to putting on a solid analytic ground the theory of local well-posedness for the two dimensional Dysthe equation. This equation can be derived from the incompressible Navier-Stokes equation after performing an asymptotic expansion of a wavetrain modulation to the fourth order. Recently, this equation has been used to numerically study rare phenomena on large water bodies such as rogue waves. In order to study well-posedness, we use Strichartz, and improved smoothing and maximal function estimates. We follow ideas from the pioneering work of Kenig, Ponce and Vega, but since the equation is highly anisotropic, several technical challenges had to be resolved. We conclude our work by also presenting an ill-posedness result. (C) 2021 Elsevier Ltd. All rights reserved.

On the nonlinear Dysthe equation / Grande, Ricardo; Kurianski, Kristin M.; Staffilani, Gigliola. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 207:(2021). [10.1016/j.na.2021.112292]

On the nonlinear Dysthe equation

Grande, Ricardo
;
Staffilani, Gigliola
2021-01-01

Abstract

This work is dedicated to putting on a solid analytic ground the theory of local well-posedness for the two dimensional Dysthe equation. This equation can be derived from the incompressible Navier-Stokes equation after performing an asymptotic expansion of a wavetrain modulation to the fourth order. Recently, this equation has been used to numerically study rare phenomena on large water bodies such as rogue waves. In order to study well-posedness, we use Strichartz, and improved smoothing and maximal function estimates. We follow ideas from the pioneering work of Kenig, Ponce and Vega, but since the equation is highly anisotropic, several technical challenges had to be resolved. We conclude our work by also presenting an ill-posedness result. (C) 2021 Elsevier Ltd. All rights reserved.
2021
207
112292
https://arxiv.org/abs/2006.13392
Grande, Ricardo; Kurianski, Kristin M.; Staffilani, Gigliola
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135370
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