Soliton interaction with external forcing within the Korteweg–de Vries equation

Article


Ermakov, Andrei and Stepanyants, Yury. 2019. "Soliton interaction with external forcing within the Korteweg–de Vries equation." Chaos: an interdisciplinary journal of nonlinear science. 29 (1). https://doi.org/10.1063/1.5063561
Article Title

Soliton interaction with external forcing within the Korteweg–de Vries equation

ERA Journal ID87
Article CategoryArticle
AuthorsErmakov, Andrei (Author) and Stepanyants, Yury (Author)
Journal TitleChaos: an interdisciplinary journal of nonlinear science
Journal Citation29 (1)
Article Number013117
Number of Pages14
Year2019
PublisherAIP Publishing
Place of PublicationUnited States
ISSN1054-1500
1089-7682
Digital Object Identifier (DOI)https://doi.org/10.1063/1.5063561
Web Address (URL)https://aip.scitation.org/doi/abs/10.1063/1.5063561
Abstract

We revise the solutions of the forced Korteweg–de Vries equation describing a resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to previous work where only the limiting cases of a very narrow forcing in comparison with the initial soliton or a very narrow soliton in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of soliton and external perturbation of a relatively small amplitude. In many particular cases, exact solutions of the forced Korteweg–de Vries equation can be obtained for the specific forcings of arbitrary amplitude. We use the earlier developed asymptotic method to derive an approximate set of equations up to the second-order on a small parameter characterising the amplitude of external force. The analysis of exact solutions of the derived equations is presented and illustrated graphically. It is shown that the theoretical outcomes obtained by the asymptotic method are in a good agreement with the results of direct numerical modeling within the framework of forced Korteweg–de Vries equation.

KeywordsSurface waves; Soliton solutions; Korteweg-de Vries equation; Dynamical systems; Internal waves; Perturbation theory; Asymptotic analysis; Numerical methods
ANZSRC Field of Research 2020401207. Fundamental and theoretical fluid dynamics
401204. Computational methods in fluid flow, heat and mass transfer (incl. computational fluid dynamics)
401208. Geophysical and environmental fluid flows
490199. Applied mathematics not elsewhere classified
Public Notes

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 29, 013117 (2019)
and may be found at https://doi.org/10.1063/1.5063561

Byline AffiliationsSchool of Agricultural, Computational and Environmental Sciences
Institution of OriginUniversity of Southern Queensland
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Kurkin, A. A., Semin, S. V. and Stepanyants, Yu A.. 2015. "Transformation of surface waves over a bottom step." Russian Academy of Sciences, Izvestiya: Atmospheric and Oceanic Physics. 51 (2), pp. 214-223. https://doi.org/10.1134/S0001433815020097
Modeling of transformation of surface wave packets interacting with abrupt bottom change
Brovchenko, Igor, Maderich, Vladimir, Semin, Sergey, Stepanyants, Yury and Terletska, Kateryna. 2015. "Modeling of transformation of surface wave packets interacting with abrupt bottom change." Applied Hydromechanics. 17 (1), pp. 3-9.
Dynamics of two charged particles in a creeping flow
Hassan, H. K. and Stepanyants, Y. A.. 2015. "Dynamics of two charged particles in a creeping flow." Journal of Physical Mathematics. 6 (2). https://doi.org/10.4172/2090-0902.1000145
Beyond the KdV: post-explosion development
Ostrovsky, L., Pelinovsky, E., Shrira, V. and Stepanyants, Y.. 2015. "Beyond the KdV: post-explosion development." Chaos: an interdisciplinary journal of nonlinear science. 25 (9), pp. 1-13. https://doi.org/10.1063/1.4927448
Generalized Korteweg–de Vries equation for internal waves in two-layer fluid
Giniyatullin, A. R., Kurkin, A. A., Kurkina, O. E. and Stepanyants, Y. A.. 2014. "Generalized Korteweg–de Vries equation for internal waves in two-layer fluid." Fundamental and Applied Hydrophysics (in Russian). 7 (4), pp. 16-28.
Transformation of internal waves passing over a bottom step
Churaev, Egor N., Semin, Sergey V. and Stepanyants, Yuri A.. 2015. "Transformation of internal waves passing over a bottom step." Journal of Fluid Mechanics. 768, pp. R3-1. https://doi.org/10.1017/jfm.2015.92
Structure of internal solitary waves in two-layer fluid at near-critical situation
Kurkina, O., Singh, N. and Stepanyants, Y.. 2015. "Structure of internal solitary waves in two-layer fluid at near-critical situation." Communications in Nonlinear Science and Numerical Simulation. 22 (1-3), pp. 1235-1242. https://doi.org/10.1016/j.cnsns.2014.09.018
Dispersion properties of waves on a surface of viscous fluid covered by an elastic film
Averbukh, E. L., Kurkin, A. A., Stepanyants, Y. A. and Talipova, T. G.. 2014. "Dispersion properties of waves on a surface of viscous fluid covered by an elastic film." Fluid Dynamics. 49 (6), pp. 796-803. https://doi.org/10.1134/S0015462814060118
Transformation of narrowband wavetrains of surface gravity waves passing over a bottom step
Giniyatullin, A. R., Kurkin, A. A., Semin, S. V. and Stepanyants, Y. A.. 2014. "Transformation of narrowband wavetrains of surface gravity waves passing over a bottom step." Mathematical Modelling of Natural Phenomena (MMNP). 9 (5), pp. 73-82. https://doi.org/10.1051/mmnp/20149505
On stationary solutions of the reduced Gardner–Ostrovsky equation
Obregon, Maria and Stepanyants, Yury. 2014. "On stationary solutions of the reduced Gardner–Ostrovsky equation." Discontinuity, Nonlinearity and Complexity. 3 (4), pp. 445-456. https://doi.org/10.5890/DNC.2014.12.007
Nonlinear vector waves of a flexural mode in a chain model of atomic particles
Nikitenkova, S. P., Raj, N. and Stepanyants, Y. A.. 2015. "Nonlinear vector waves of a flexural mode in a chain model of atomic particles." Communications in Nonlinear Science and Numerical Simulation. 20 (3), pp. 731-742. https://doi.org/10.1016/j.cnsns.2014.05.031
Interaction of gaseous bubbles under the action of radiation modified Bjerknes force
Stepanyants, Yury and Yeoh, Guan. 2008. "Interaction of gaseous bubbles under the action of radiation modified Bjerknes force." 22nd International Congress of Theoretical and Applied Mechanics (IC-TAM2008). Adelaide, Australia 24 - 29 Aug 2008
Recent results on the problem of wave-current interaction including water depth, surface tension/amplitude and vorticity effects
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.
Dispersion of long gravity-capillary surface waves and asymptotic equations for solitons
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.
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