Projectors in nonlinear evolution problem: Acoustic solitons of bubbly liquid. (English) Zbl 0983.76082
Summary: We study compressible bubby fluids by using the method of splitting of one-dimensional disturbances into rightward propagating components, leftward propagating components, and stationary components by projection technique. The fundamental system of equations is transformed to three nonlinear equations for interacting components. We introduce a small parameter which determines input of nonlinear and dispersive terms, and reduce the system to one of Korteweg-de Vries type. It is shown that three-mode evolution equations can be approximately reduced to an integrable modified KdV equation, which, under special initial conditions, possesses a soliton solution.
MSC:
| 76Q05 | Hydro- and aero-acoustics |
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
| 35Q51 | Soliton equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Keywords:
splitting method; acoustic solitons; compressible bubby fluid; projection technique; small parameter; integrable modified KdV equationReferences:
| [1] | Monin, A. S.; Obukhov, A. M., The small oscillations of atmosphere and adaptation of meteorilogical fields, Izv. AN SSSR. Geophysics, 41, 1360-1373 (1958) |
| [2] | Brekhovskich, L. M.; Godin, A. O., Acoustics of Layered Media (1990), Springer-Verlag: Springer-Verlag Berlin |
| [3] | Novikov, A. A., On application of coupled waves method to non-resonance interactions analysis, Izv. Vuzov, Radiofizika, 19, 2, 321-328 (1976) |
| [4] | Pelinovsky, E. N.; Engelbrecht, Y. K.; Fridman, V. E., Nonlinear Evolution Equations (1981), Longman Sci. Tech: Longman Sci. Tech London |
| [5] | Perelomova, A. A., Nonlinear dynamics of vertically propagating acoustic waves in a stratified atmosphere, Acta Acustica, 84, 6, 1002-1006 (1998) |
| [6] | Wijngaarden, L. V., Evolving solitons in bubbly liquid, Acta Applicandae Mathematicae, 39, 507-516 (1995) · Zbl 0835.58021 |
| [7] | Prosperetti, A.; Lezzi, A., Bubbly dynamics in a compressible liquid. Part 1. First-order theory, J. Fluid Mech., 457-478 (1986) · Zbl 0597.76106 |
| [8] | Keller, J. B.; Miksis, M., Bubble oscillations of large amplitude, J. Acoust. Soc. Am., 68, 2, 628-633 (1980) · Zbl 0456.76087 |
| [9] | Leble, S. B., Nonlinear Waves in Waveguides With Stratification (1990), Springer-Verlag: Springer-Verlag Berlin |
| [10] | Kshevetsky, S. P.; Leble, S. B., Nonlinear dispersion of long internal waves, Izv. AN. SSSR. Mechanics of Liquids and Gases, 3, 1169-1174 (1988) |
| [11] | Novikov, S. P.; Manakov, S. V.; Pitaevski, L. P.; Zakharov, V. E., Theory of Solitons. The Inverse Scattering Method (1984), Plenum: Plenum New York · Zbl 0598.35002 |
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