Long-wavelength limit for the Green–Naghdi equations

Min Li; Min Li

doi:10.3934/era.2022138

Electronic Research Archive

2022, Volume 30, Issue 7: 2700-2718. doi: 10.3934/era.2022138

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Research article

Long-wavelength limit for the Green–Naghdi equations

Min Li ^,

Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China

Received: 29 January 2022 Revised: 02 May 2022 Accepted: 06 May 2022 Published: 23 May 2022

This paper studies the long-wavelength limit for the one-dimensional Green–Naghdi (GN) equations, which are often used to describe the propagation of fully nonlinear waves in coastal oceanography. We prove that, under the long-wavelength, small-amplitude approximation, the formal Korteweg–de Vries (KdV) equation for the GN equations is mathematically valid in the time interval for which the KdV dynamics survive. The main idea in the proof is to apply the Gardner–Morikawa transform, the reductive perturbation method, and some error energy estimates. The main novelties of this paper are the construction of valid approximate solutions of the GN equations with respect to the small wave amplitude parameter and global uniform energy estimates for the error system.
- Green–Naghdi equations,
- long-wavelength limit,
- Korteweg–de Vries equation
Citation: Min Li. Long-wavelength limit for the Green–Naghdi equations[J]. Electronic Research Archive, 2022, 30(7): 2700-2718. doi: 10.3934/era.2022138

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Abstract

This paper studies the long-wavelength limit for the one-dimensional Green–Naghdi (GN) equations, which are often used to describe the propagation of fully nonlinear waves in coastal oceanography. We prove that, under the long-wavelength, small-amplitude approximation, the formal Korteweg–de Vries (KdV) equation for the GN equations is mathematically valid in the time interval for which the KdV dynamics survive. The main idea in the proof is to apply the Gardner–Morikawa transform, the reductive perturbation method, and some error energy estimates. The main novelties of this paper are the construction of valid approximate solutions of the GN equations with respect to the small wave amplitude parameter and global uniform energy estimates for the error system.

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