Notes on multi-parameters perturbation method for dispersive and nonlinear partial differential equations

Abstract

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 2122 Edi Cahyono et al. 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.

The dynamics of surface wave propagation based on the Benjamin Bona Mahony equation

“…The amplification of wave in the framework of forced nonlinear Schrodinger equation is investigated by Slunyaev et al [20]. Multi-parameters perturbation method for dispersive wave equations is studied by Cahyono et al [21]. The third-order approximated theory of Korteweg de Vries equation and a quantity called Maximal Temporal Amplitude (MTA) for seeking nonlinear evolution of wave group with three frequencies is discussed in Ramli [22].…”