The present study theoretically carries out a derivation of the Korteweg–de Vries–Burgers (KdVB) equation and the nonlinear Schrödinger (NLS) equation for weakly nonlinear propagation of plane (i.e., one-dimensional) progressive waves in water flows containing many spherical gas bubbles that oscillate due to the pressure wave approaching the bubble. Main assumptions are as follows: (i) bubbly liquids are not at rest initially; (ii) the bubble does not coalesce, break up, extinct, and appear; (iii) the viscosity of the liquid phase is taken into account only at the bubble–liquid interface, although that of the gas phase is omitted; (iv) the thermal conductivities of the gas and liquid phases are dismissed. The basic equations for bubbly flows are composed of conservation equations for mass and momentum for the gas and liquid phases in a two-fluid model, the Keller-Miksis equation (i.e., the equation for radial oscillations as the expansion and contraction), and so on. By using the method of multiple scales and the determination of size of three nondimensional ratios that are wavelength, propagation speed and incident wave frequency, we can derive two types of nonlinear wave equations describing long range propagation of plane waves. One is the KdVB equation for a low frequency long wave, and the other is the NLS equation for an envelope wave for a moderately high frequency short carrier wave.

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