A method for studying waves with spatially localized envelopes in a class of nonlinear partial differential equations

Melvin E. King, Alexander F. Vakakis

Research output: Contribution to journalArticlepeer-review

Abstract

A methodology for investigating stationary and travelling waves with spatially localized envelopes is presented. The nonlinear governing partial differential equations considered possess a constant first integral of motion, and are separable in space and time when the small parameter of the problem is set to zero. To study stationary waves, a coordinate transformation on the governing nonlinear partial differential equation is imposed which eliminates the time dependence from the problem. An amplitude modulation function is then introduced to express the response of the system at an arbitrary point as a nonlinear function of a reference response. Analytic approximations to the amplitude modulation function are developed by expressing it in power series, and asymptotically solving sets of singular functional equations at the various orders of approximation. Travelling solutions may be computed from stationary ones, by imposing appropriate Lorentz transformations. As an application of the methodology, stationary and travelling breathers of a nonlinear partial differential equation are analytically computed.

Original languageEnglish (US)
Pages (from-to)391-405
Number of pages15
JournalWave Motion
Volume19
Issue number4
DOIs
StatePublished - Jun 1994

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Physics and Astronomy
  • Computational Mathematics
  • Applied Mathematics

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