Abstract
The modeling of unidirectional propagation of long water waves in dispersive media is presented. The Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations are derived from water waves models. New traveling solutions of the KdV and BBM equations are obtained by implementing the extended direct algebraic and extended sech-tanh methods. The stability of the obtained traveling solutions is analyzed and discussed.
Highlights
Many nonlinear evolution equations are playing important role in the analysis of some phenomena and including ion acoustic waves in plasmas, dust acoustic solitary structures in magnetized dusty plasmas, and electromagnetic waves in size-quantized films [1–4]
The BBM equation has a phase velocity and a group velocity both of which are bounded for all k
The main argument that was used in [18] to derive the BBM equation is that, to the first order in ε, the scaled KdV equation is equivalent to ut + ux + uux − a2uxxx − b2uxxt = 0
Summary
Many nonlinear evolution equations are playing important role in the analysis of some phenomena and including ion acoustic waves in plasmas, dust acoustic solitary structures in magnetized dusty plasmas, and electromagnetic waves in size-quantized films [1–4]. The Benjamin-Bona-Mahony (BBM) equation is well known in physical applications [15]; it describes the model for propagation of long waves which incorporates nonlinear and dissipative effects; it is used in the analysis of the surface waves of long wavelength in liquids, hydromagnetic waves in cold plasma, acoustic-gravity waves in compressible fluids, and acoustic waves in harmonic crystals [15]. Many mathematicians paid their attention to the dynamics of the BBM equation [16].
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