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Constructive Study of Modulational Instability in Higher Order Korteweg-de Vries Equations

1
Institute of Analysis, Johannes Kepler University Linz, 4040 Linz, Austria
2
National Research University—Higher School of Economics, Moscow 101000, Russia
3
Institute of Applied Physics, Russian Academy of Sciences, Nizhni Novgorod 603950, Russia
4
Nizhni Novgorod State Technical University n.a. R.Y. Alexeev, Nizhni Novgorod 603155, Russia
*
Author to whom correspondence should be addressed.
Fluids 2019, 4(1), 54; https://doi.org/10.3390/fluids4010054
Received: 2 February 2019 / Revised: 4 March 2019 / Accepted: 12 March 2019 / Published: 17 March 2019
(This article belongs to the Special Issue Nonlinear Wave Hydrodynamics)
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Abstract

Our present study is devoted to the constructive study of the modulational instability for the Korteweg-de Vries (KdV)-family of equations u t + s u p u x + u x x x (here s = ± 1 and p > 0 is an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear corrections to the frequency of the Stokes wave and then explored the coefficients of the corresponding modified nonlinear Schrödinger equations, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. A brief discussion of the results, open questions and further research directions completes the paper. View Full-Text
Keywords: Korteweg-de Vries equation; modulational instability; higher order KdV equations Korteweg-de Vries equation; modulational instability; higher order KdV equations
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Tobisch, E.; Pelinovsky, E. Constructive Study of Modulational Instability in Higher Order Korteweg-de Vries Equations. Fluids 2019, 4, 54.

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