Dispersion of long gravity-capillary surface waves and asymptotic equations for solitons

Article

Stepanyants, Yury. 2005. "Dispersion of long gravity-capillary surface waves and asymptotic equations for solitons." Proceedings of the Russian Academy of Engineering Sciences Series: Applied Mathematics and Mechanics. 14, pp. 33-40.
Article Title

Dispersion of long gravity-capillary surface waves and asymptotic equations for solitons

Article CategoryArticle
Authors
AuthorStepanyants, Yury
Journal TitleProceedings of the Russian Academy of Engineering Sciences Series: Applied Mathematics and Mechanics
Journal Citation14, pp. 33-40
Number of Pages8
Year2005
Abstract

It is shown that in the description of long surface gravity-capillary waves, accounting for air density is essential when the fluid depth is close to the critical value. At critical depth the dispersive term of the third order (KdV-dispersion) in the Taylor's series of frequency on wavenumber vanishes due to expostulate actions of gravity and capillary effects. Estimates show that in the critical case the dispersive term of the second order (BO-dispersion), rather than the fifth order, as was thought before, becomes determinative. The main dispersive term, in the corresponding evolution equation for weakly nonlinear perturbations, determines a structure of solitary waves. Solitary wave solutions numerically obtained for the combined BO-KdV equation, as well as their Fourier spectra,
are compared with the known soliton solutions of the BO and KdV equations.

Keywordssurface wave, capillary effect, air density, shallow water, critical depth, dispersion relation, evolution equation, solitary wave, numerical modelling
ANZSRC Field of Research 2020370803. Physical oceanography
490105. Dynamical systems in applications
490109. Theoretical and applied mechanics
Public Notes

(Published in Russian) ISBN: 5-93496-048-2.

Byline AffiliationsAustralian Nuclear Science and Technology Organisation
Institution of OriginUniversity of Southern Queensland
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Maissa, Philippe, Rousseaux, Germain and Stepanyants, Yury. 2013. "Recent results on the problem of wave-current interaction including water depth, surface tension/amplitude and vorticity effects." 7th International Conference on Coastal Dynamics. Arcachon, France 24 - 28 Jun 2013 Arcachon, France.
Nonlinear spectra of shallow water waves
Giovanangeli, J. -P., Kharif, C., Raj, N. and Stepanyants, Y.. 2013. "Nonlinear spectra of shallow water waves." Oceans - San Diego, 2013. San Diego, United States 23 - 26 Sep 2013 United States. IEEE (Institute of Electrical and Electronics Engineers). https://doi.org/10.23919/OCEANS.2013.6741132
Numerical study of nonlinear wave processes by means of discrete chain models
Obregon, M., Raj, N. and Stepanyants, Y.. 2012. "Numerical study of nonlinear wave processes by means of discrete chain models." Gu, Y. T. and Saha, Suvash C. (ed.) 4th International Conference on Computational Methods (ICCM 2012). Gold Coast, Australia 25 - 28 Nov 2012 Brisbane, Australia.
On numerical solution of the Gardner Ostrovsky equation
Obregon, M. A. and Stepanyants, Y. A.. 2012. "On numerical solution of the Gardner Ostrovsky equation." Mathematical Modelling of Natural Phenomena (MMNP). 7 (2), pp. 113-130. https://doi.org/10.1051/mmnp/20127210
Stationary gravity waves with the zero mean vorticity in stratified fluid
Clamond, D. and Stepanyants, Y.. 2012. "Stationary gravity waves with the zero mean vorticity in stratified fluid." Studies in Applied Mathematics. 128 (1), pp. 59-85. https://doi.org/10.1111/j.1467-9590.2011.00530.x
Measuring rate of change of oxygen concentration and intrinsic oxidation rate in a pile of material
Bennett, John, Stepanyants, Yury and Ritchie, Ian. 2004. Measuring rate of change of oxygen concentration and intrinsic oxidation rate in a pile of material. WO 2004/072636 A1
Internal solitons in laboratory experiments: Comparison with theoretical models
Ostrovsky, L A. and Stepanyants, Y. A.. 2005. "Internal solitons in laboratory experiments: Comparison with theoretical models." Chaos: an interdisciplinary journal of nonlinear science. 15. https://doi.org/10.1063/1.2107087
Measurement of hydraulic conductivity using a radioactive or activatable tracer
Waring, Christopher Leslie, Airey, Peter Lewis and Stepanyants, Yury A.. 2007. Measurement of hydraulic conductivity using a radioactive or activatable tracer. AU2007231556
Measurement of hydraulic conductivity, porosity and lithology by neutron activation borehole logging at high spatial resolution increments
Waring, C. L., Stepanyants, Y. A., Hankin, S. I., Airey, P. L. and Peterson, M. A.. 2009. "Measurement of hydraulic conductivity, porosity and lithology by neutron activation borehole logging at high spatial resolution increments." Milne-Home, William A. (ed.) IAH Groundwater in The Sydney Basin Symposium (2009). Sydney, Australia 04 - 05 Aug 2009 Sydney, NSW, Australia.
Internal solitons in the ocean and their effect on underwater sound
Apel, John R., Ostrovsky, Lev A., Stepanyants, Yury A. and Lynch, James F.. 2007. "Internal solitons in the ocean and their effect on underwater sound." The Journal of the Acoustical Society of America. 121 (2), pp. 695-722. https://doi.org/10.1121/1.2395914
Influence of bottom boundary conditions for temperature on the heat distribution within oxidising heaps
Crawford, Jagoda and Stepanyants, Yury. 2005. "Influence of bottom boundary conditions for temperature on the heat distribution within oxidising heaps." Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal. 46, pp. C1104-C1125.
Propagation of waves in shear flows
Fabrikant, Anatoly and Stepanyants, Yury. 1998. Propagation of waves in shear flows. Singapore. World Scientific Publishing.
Scalar description of three-dimensional vortex flows of incompressible fluid
Stepanyants, Yu. A. and Yakubovich, Evsey. 2011. "Scalar description of three-dimensional vortex flows of incompressible fluid." Doklady Physics. 56 (2), pp. 130-133. https://doi.org/10.1134/S1028335811020169
The inverse problem for the Gross-Pitaevskii equation
Malomed, Boris A. and Stepanyants, Yury A.. 2010. "The inverse problem for the Gross-Pitaevskii equation." Chaos: an interdisciplinary journal of nonlinear science. 20 (1). https://doi.org/10.1063/1.3367776
Dynamics of two charged particles in viscous fluid at small Reynolds numbers
Stepanyants, Y. A. and Strunin, D. V.. 2011. "Dynamics of two charged particles in viscous fluid at small Reynolds numbers." de Oliveira, Cassiano (ed.) 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Enginnering (MC 2011). Rio de Janeiro, Brazil 08 - 12 May 2011 Rio de Janeiro.
Nanoparticle dynamics in a viscous fluid at small Reynolds numbers
Stepanyants, Yury A. and Yeoh, Guan H.. 2010. "Nanoparticle dynamics in a viscous fluid at small Reynolds numbers." Teh, Kian, Davies, Ian and Howard, Ian (ed.) 6th Australasian Congress on Applied Mechanics (ACAM 6). Perth, Western Australia 12 - 15 Dec 2010 Perth, Australia.
Localised nonlinear optical modes and the corresponding support structures: exact solutions to the nonlinear Schrodinger equation with external potentials
Malomed, B. A. and Stepanyants, Y. A.. 2010. "Localised nonlinear optical modes and the corresponding support structures: exact solutions to the nonlinear Schrodinger equation with external potentials." ICEAA 2010: International Conference on Electromagnetics in Advanced Applications. Sydney, Australia 20 - 24 Sep 2010 Sydney, Australia. https://doi.org/10.1109/ICEAA.2010.5651681
Lump solutions of 2D generalized Gardner equation
Stepanyants, Y. A., Ten, I. K. and Tomita, H.. 2006. "Lump solutions of 2D generalized Gardner equation." Luo, Albert C. J., Dai, Liming and Hamidzadeh, Hamid R. (ed.) Conference on Nonlinear Science and Complexity (2006). Beijing, China 07 - 12 Aug 2006 Singapore.
Internal solitons in the ocean
Apel, John R., Ostrovsky, Lev A., Stepanyants, Yury A. and Lynch, James F.. 2006. Internal solitons in the ocean. Woods Hole, MA. United States. Woods Hole Oceanographic Institution.
Tsunami surge in a river: a hydraulic jump in an inhomogeneous channel
Caputo, Jean-Guy and Stepanyants, Y. A.. 2007. "Tsunami surge in a river: a hydraulic jump in an inhomogeneous channel." Kundu, Anjan (ed.) Tsunami and nonlinear waves. Berlin, Germany. Springer. pp. 97-112
Solutions classification to the extended reduced Ostrovsky equation
Stepanyants, Yury A.. 2008. "Solutions classification to the extended reduced Ostrovsky equation." Symmetry, Integrability and Geometry: Methods and Applications. 4. https://doi.org/10.3842/SIGMA.2008.073
Mechanisms of particle transport acceleration in porous media
Panfilov, M., Panfilova, I. and Stepanyants, Y.. 2008. "Mechanisms of particle transport acceleration in porous media." Transport in Porous Media. 74 (1), pp. 49-71. https://doi.org/10.1007/s11242-007-9201-9
Particle and bubble dynamics in a creeping flow
Stepanyants, Yury and Yeoh, Guan. 2009. "Particle and bubble dynamics in a creeping flow." European Journal of Mechanics B: Fluids. 28 (5), pp. 619-629. https://doi.org/10.1016/j.euromechflu.2009.04.004
On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons
Stepanyants, Yury. 2006. "On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons." Chaos, Solitons and Fractals. 28 (1), pp. 193-204. https://doi.org/10.1016/j.chaos.2005.05.020
The motion of a cylindrical body in a stratified fluid under the action of a radiation force
Kozyrev, O. R., Reznik, S. N. and Stepanyants, Yu. A.. 2009. "The motion of a cylindrical body in a stratified fluid under the action of a radiation force." Journal of Applied Mathematics and Mechanics. 73 (2), pp. 188-195. https://doi.org/10.1016/j.jappmathmech.2009.04.008
Front solutions of Richards' equation
Caputo, Jean-Guy and Stepanyants, Yury. 2008. "Front solutions of Richards' equation." Transport in Porous Media. 74 (1), pp. 1-20. https://doi.org/10.1007/s11242-007-9180-x
Burgers-Rott vortices with surface tension
Stepanyants, Yury A. and Yeoh, Guan H.. 2008. "Burgers-Rott vortices with surface tension." Zeitschrift fuer Angewandte Mathematik und Physik. 59 (6), pp. 1057-1068. https://doi.org/10.1007/s00033-008-7071-9
Stationary bathtub vortices and a critical regime of liquid discharge
Stepanyants, Yury and Yeoh, Guan. 2008. "Stationary bathtub vortices and a critical regime of liquid discharge." Journal of Fluid Mechanics. 604 (1), pp. 77-98. https://doi.org/10.1017/S0022112008001080
Modelling internal solitary waves on the Australian North West Shelf
Grimshaw, Roger, Pelinovsky, Efim, Stepanyants, Yury and Talipova, Tatiana. 2006. "Modelling internal solitary waves on the Australian North West Shelf." Marine and Freshwater Research. 57 (3), pp. 265-272. https://doi.org/10.1071/MF05016