Nearly linear dynamics of nonlinear dispersive waves

Dispersive averaging effects are used to show that the Korteweg–de Vries (KdV) equation with periodic boundary conditions possesses high frequency solutions, which behave nearly linearly. Numerical simulations are presented, which indicate the high accuracy of this approximation. Furthermore, this r...

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Bibliographic Details
Published in:Physica. D 2011-08, Vol.240 (17), p.1325-1333
Main Authors: Erdoğan, M.B., Tzirakis, N., Zharnitsky, V.
Format: Article
Language:English
Subjects:
Approximation
Asymptotic properties
Computer simulation
Dispersive averaging
Dynamic tests
Exact sciences and technology
KdV
Mathematical analysis
Mathematical models
Nonlinear dispersive waves
Nonlinear dynamics
Physics
Shallow water
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Summary:Dispersive averaging effects are used to show that the Korteweg–de Vries (KdV) equation with periodic boundary conditions possesses high frequency solutions, which behave nearly linearly. Numerical simulations are presented, which indicate the high accuracy of this approximation. Furthermore, this result is applied to shallow water wave dynamics in the limit of KdV approximation, which is obtained by asymptotic analysis in combination with numerical simulations of KdV. ► We consider the Korteweg–de Vries equation with periodic boundary conditions. ► We prove that high frequency solutions evolve almost linearly, as if in the Airy equation. ► The proof is based on differentiation by parts — which is a variant of the normal form procedure. ► The implications for shallow water waves are also discussed.
ISSN:0167-2789
1872-8022
DOI:10.1016/j.physd.2011.05.009