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Notes on Multi-parameters Perturbation Method for Dispersive and Nonlinear Partial Differential Equations

Edi Cahyono, La Ode Ngkoimani, Marwan Ramli
Waves develop amplitude change during the propagation. For the simplest case, this has been shown by bichromatic waves, waves that contain two frequencies. The waves of larger amplitudes and smaller frequency differences showed more significant amplitude deformation. The amplitude deformation can be explained from the so-called side band interactions. Perturbation method for diffusive and nonlinear equation is applied to show the amplitude deformation. The solution is in the form of a series expansion of the amplitude as Stokes’ approach for single frequency waves. Stokes’ approach shows that the dispersion relation involves the amplitude. However, this does not explain the amplitude deformation. Therefore, this paper discusses bichromatic wave solution of diffusive and nonlinear equation. Moreover, the focus is on a hidden parameter appears in the series expansion, namely the wave number difference. The parameter of wave number difference may affect the convergence of the series expansion. The relation among the parameters of amplitude, wave number and wave number difference for the convergence of the series is presented. Keywords: Amplitude deformation, dispersive and nonlinear equation, perturbation method, side band interactions

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Diterbitkan pada: International Journal of Mathematical Analysis Vol. 9, 2015, no. 43, 2121 - 2132 ISSN 1312-8876 (print), ISSN 1314-7579 (online)

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