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 TMF, 2005, Volume 144, Number 2, Pages 410–422 (Mi tmf1866)

Perturbative Analysis of Wave Interaction in Nonlinear Systems

A. Vekslera, Y. Zarmiab

a Ben-Gurion University of the Negev
b Jacob Blaustein Institute for Desert Research

Abstract: We propose a new way to handle obstacles to asymptotic integrability in perturbed nonlinear PDEs in the method of normal forms (NFs) in the case of multiwave solutions. Instead of including the whole obstacle in the NF, we include only its resonant part (if it exists) in the NF and assign the remainder to the homological equation. This leaves the NF integrable, and its solutions retain the character of the solutions of the unperturbed equation. We use the freedom in the expansion to construct canonical obstacles that are confined to the interaction region of the waves. For soliton solutions (e. g., of the KdV equation), the interaction region is a finite domain around the origin; the canonical obstacles then do not generate secular terms in the homological equation. When the interaction region is infinite (or semi-infinite, e.g., in wave-front solutions of the Burgers equation), the obstacles may contain resonant terms. The obstacles generate waves of a new type that cannot be written as functionals of the solutions of the NF. When the obstacle contributes a resonant term to the NF, this leads to a nonstandard update of the wave velocity.

Keywords: nonlinear evolution equations, wave interaction, obstacles to asymptotic integrability, perturbed KdV equation, perturbed Burgers equation

DOI: https://doi.org/10.4213/tmf1866

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English version:
Theoretical and Mathematical Physics, 2005, 144:2, 1227–1237

Bibliographic databases:

Citation: A. Veksler, Y. Zarmi, “Perturbative Analysis of Wave Interaction in Nonlinear Systems”, TMF, 144:2 (2005), 410–422; Theoret. and Math. Phys., 144:2 (2005), 1227–1237

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf1866
• https://doi.org/10.4213/tmf1866
• http://mi.mathnet.ru/eng/tmf/v144/i2/p410

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Veksler, A, “Wave interactions and the analysis of the perturbed Burgers equation”, Physica D-Nonlinear Phenomena, 211:1–2 (2005), 57
2. Veksler A, Zarmi Y, “Spontaneously generated waves in perturbed evolution equations”, Nonlinearity, 20:3 (2007), 523–536
3. Zarmi, Y, “Two-component description of dynamical systems that can be approximated by solitons: The case of the ion acoustic wave equations of plasma physics”, Physica D-Nonlinear Phenomena, 238:14 (2009), 1274
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