Modulation theory for the steady forced KdV-Burgers equation and the construction of periodic solutions

Laura-Jane Louise Hattam; Simon Rex Clarke

doi:10.1016/j.wavemoti.2015.02.004

Modulation theory for the steady forced KdV-Burgers equation and the construction of periodic solutions

Laura-Jane Louise Hattam, Simon Rex Clarke

School of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

9 Citations (Scopus)

Abstract

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg-de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).

Original language	English
Pages (from-to)	67-84
Number of pages	18
Journal	Wave Motion
Volume	56
DOIs	https://doi.org/10.1016/j.wavemoti.2015.02.004
Publication status	Published - 2015

Keywords

Korteweg–de Vries
Modulation theory
Periodic solutions

Access to Document

10.1016/j.wavemoti.2015.02.004

RM sElocation

Cite this

@article{7b4f0ae16b9f4dbe9a8c9b99be52cd71,

title = "Modulation theory for the steady forced KdV-Burgers equation and the construction of periodic solutions",

abstract = "We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg-de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).",

keywords = "Korteweg–de Vries, Modulation theory, Periodic solutions",

author = "Hattam, {Laura-Jane Louise} and Clarke, {Simon Rex}",

year = "2015",

doi = "10.1016/j.wavemoti.2015.02.004",

language = "English",

volume = "56",

pages = "67--84",

journal = "Wave Motion",

issn = "0165-2125",

publisher = "Elsevier",

}

TY - JOUR

T1 - Modulation theory for the steady forced KdV-Burgers equation and the construction of periodic solutions

AU - Hattam, Laura-Jane Louise

AU - Clarke, Simon Rex

PY - 2015

Y1 - 2015

N2 - We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg-de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).

AB - We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg-de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).

KW - Korteweg–de Vries

KW - Modulation theory

KW - Periodic solutions

UR - http://www.sciencedirect.com/science/article/pii/S0165212515000219/pdfft?md5=fbe6793d4851fccf2c181e0263254ed4&pid=1-s2.0-S0165212515000219-main.pdf

U2 - 10.1016/j.wavemoti.2015.02.004

DO - 10.1016/j.wavemoti.2015.02.004

M3 - Article

VL - 56

SP - 67

EP - 84

JO - Wave Motion

JF - Wave Motion

SN - 0165-2125

ER -