On the method of strained parameters for a KdV type of equation with exact dispersion property

DOI10.1093/IMAMAT/HXU020zbMATH Open1333.35200arXiv1609.00781OpenAlexW2105405372MaRDI QIDQ5744251

Natanael Karjanto

Publication date: 18 February 2016

Published in: IMA Journal of Applied Mathematics (Search for Journal in Brave)

Abstract: This paper provides an alternative methodology for analysis of three-wave interactions under the exact dispersion relation associated with gravity waves in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact dispersion property is adopted as the governing equation for unidirectional wave packet evolution. Following the idea from Zakharov's seminal paper (Zakharov, V. E. (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. extit{Journal of Applied Mechanics and Technical Physics}, {�f 9}, 190--194), the equation is transformed from the spatial--temporal domain to the wavenumber--temporal domain. The solution of the transformed equation is sought using the perturbation theory, for which the ansatz is expressed in the form of a regular expansion in the increasing order of a small parameter. After implementing the na"{i}ve perturbation method, due to nonlinear mode generation and particular combinations of wavenumbers, the third-order solution contains spurious secular growth terms which appear as a consequence of resonant interaction and nonlinear mode generation. These spurious secular growth terms can be prevented by implementing the method of strained parameters for which nonlinear dispersion relation terms are produced for particular combination of wavenumbers.

Full work available at URL: https://arxiv.org/abs/1609.00781

zbMATH Keywords

KdV equationmethod of strained parametersnon-linear dispersion relationnaïve perturbation methodnon-linear mode generation

Mathematics Subject Classification ID

PDEs in connection with fluid mechanics (35Q35) KdV equations (Korteweg-de Vries equations) (35Q53) Water waves, gravity waves; dispersion and scattering, nonlinear interaction (76B15) Perturbations in context of PDEs (35B20)

Recommendations

• Title not available (Why is that?) 👍 👎
• Well-posedness and ill-posedness of KdV equation with higher dispersion 👍 👎
• Two reliable methods for solving variants of the KdV equation with compact and noncompact structures 👍 👎
• Strichartz estimates for dispersive equations and solvability of the Kawahara equation 👍 👎
• Existence and stability of dispersive solutions to the Kadomtsev-Petviashvili equation in the presence of dispersion effect 👍 👎
• The modified KdV equation with higher dispersion in Sobolev and analytic spaces on the line 👍 👎
• Consistent KdV expansion method and its applications to the KP equation 👍 👎
• Dispersive schemes for the critical Korteweg-de Vries equation 👍 👎
• Exact solutions for the generalized KdV equation with time-dependent damping and dispersion 👍 👎
• The $\bar{\partial}$-approach to the dispersionless KP hierarchy 👍 👎

This page was built for publication: On the method of strained parameters for a KdV type of equation with exact dispersion property

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q5744251)