# Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

N. H. Ibragimov; R. N. Ibragimov

Mathematical Modelling of Natural Phenomena (2012)

- Volume: 7, Issue: 2, page 52-65
- ISSN: 0973-5348

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topIbragimov, N. H., and Ibragimov, R. N.. "Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences." Mathematical Modelling of Natural Phenomena 7.2 (2012): 52-65. <http://eudml.org/doc/222193>.

@article{Ibragimov2012,

abstract = {Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.},

author = {Ibragimov, N. H., Ibragimov, R. N.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {internal gravity waves; invariant solutions},

language = {eng},

month = {2},

number = {2},

pages = {52-65},

publisher = {EDP Sciences},

title = {Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences},

url = {http://eudml.org/doc/222193},

volume = {7},

year = {2012},

}

TY - JOUR

AU - Ibragimov, N. H.

AU - Ibragimov, R. N.

TI - Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

JO - Mathematical Modelling of Natural Phenomena

DA - 2012/2//

PB - EDP Sciences

VL - 7

IS - 2

SP - 52

EP - 65

AB - Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

LA - eng

KW - internal gravity waves; invariant solutions

UR - http://eudml.org/doc/222193

ER -

## References

top- H. Cho, T. Shepherd, V. Vladimirov. Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere. J. Atmosph. Sci., (1993), 50 (6), 822-836.
- W. Craig, P. Guyenne, H. Kalisch. Hamiltonian long-wave expansions for the free surfaces and interfaces. Comm. Pure Appl. Math., (2005), 58, 1587-1641.
- E. Dewan, R. Picard, R. O’Neil, H. Gardiner, J. Gibson. MSX satellite observations of thunderstorm-generated gravity waves in mid-wave infrared images of the upper stratosphere. Geophys. Res. Lett., (1998), 25, 939-942.
- S. Dalziel, G. Hughes, B. Sutherland. Whole field density measurements by synthetic schlieren. Experiments in Fluids, (2000), 28, 322-337.
- T. Dauxois, W. Young. Near-critical reflection of internal waves. J. Fluid Mech., (1999), 390, 271-295.
- R. Fjortoft. R, Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geophys. Publ., (1950), 17(6), 1-52.
- M. Flynn, K. Onu, B. Sutherland. Internal wave excitation by a vertically oscillating sphere. J. Fluid Mech., (2003), 494, 65-93.
- C. Garrett. Internal tides and ocean mixing. Science, (2003), 301 (5641), 1858-1859, doi :. URI10.1126/science.1090002
- C. Garrett, W. Munk. Space time scale of internal waves.A progress report. J. Geophys. Res., (1975), 80, 291-297.
- A. Gill. Atmosphere-Ocean Dynamics. New York, etc., Academic Press, (1983).
- J. Hadamard. Lectures on Cauchy’s problem in linear partial differential equations. Yale University Press, New Haven, (1983).
- J. Hadamard. The problem of diffusion of waves. Annals of Mathematics, Ser. 2 43 : 510-522, (1942).
- N. Ibragimov. Elementary Lie group analysis of ordinary differential equations. John Wiley & Sons, Chichester, (1999).
- N. Ibragimov. Transformation Groups Applied to Mathematical Physics. Nauka, Moscow (1983), English. transl., Reidel, Dordrecht.
- N. Ibragimov Ed. CRC Handbook of Lie group analysis of differential equations (CRC Press, Boca Raton) ; Vol. 1 (1994, 429 p), Vol. 2 (1995, 546 p.), Vol. 3 (1996, 536 p.).
- N. Ibragimov. A new conservation theoremJ. Math. Anal. Appl., (2007), 333 : 311-328.
- N. Ibragimov. Group analysis - a microscope of mathematical modelling. I : Galilean relativity in diffusion models. Selected works (ALGA Publications, Karlskrona), (2006), 2 : 225-243.
- N. Ibragimov. Conformal invariance and Huygens’ principle. Soviet Mathematics Doklady, (1970), 11(5) : 1153-1157.
- N. Ibragimov. Application of group analysis to liquid metal systems. Archives of ALGA, (2010), 6 : 91-101.
- N. Ibragimov. Lie group analysis of Moffatt’s model in metallurgical industry. Nonlinear Math. Phys., (2011), 18, 143-162.
- N. Ibragimov, R. Ibragimov, V. Kovalev. Group analysis of nonlinear internal waves in oceans. Archives of ALGA, (2009), 6, 45-54.
- N. Ibragimov, R. Ibragimov. Applications of Lie Group Analysis in Geophysical Fluid Dynamics. Series on Complexity, Nonlinearity and Chaos, (2011), Vol 2, World Scientific Publishers, ISBN : 978-981-4340-46-5.
- N. Ibragimov, R. Ibragimov. Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics. Comm. Nonlinear Sci. Num. Simulat., (2010), 15, 1989-2002.
- R. Ibragimov. Oscillatory nature and dissipation of the internal wave energy spectrum in the deep ocean. Eur. Phys. J. Appl. Phys., (2007), 40, 315-334.
- R. Ibragimov. Generation of internal tides by an oscillating background flow along a corrugated slope. Phys. Scr., (2008), 78, 065801.
- R. Ibragimov, D. Pelinovsky. Incompressible viscous fluid flows in a thin spherical shell. J. Math. Fluid. Mech., (2009), 11, 60-90.
- R. Ibragimov, N. Ibragimov. Effects of rotation on self-resonant internal gravity waves. Ocean Modelling, (2010), 31, 80-87.
- R. Ibragimov, M. Dameron. Spinning phenomena and energetics of spherically pulsating patterns in stratified fluids. Phys. Scr., (2011), 84, 015402.
- A. Javam, J. Imberger, S. Armfield. Numerical study of internal wave-wave interactions in a stratified fluid. J. Fluid Mech., (2000), 415, 65-87.
- H. Kalisch, N. Nguyen. On the stability of internal waves. J. Phys. A, (2010), 43, 495205.
- H. Kalisch, J. Bona. Modes for internal waves in deep water. Disc. Cont. Dyn. Sys. (2000), 6, 1-19.
- A. Kistovich, Y. Chashechkin. Nonlinear interactions of two dimensional packets of monochromatic internal waves. Izv. Atmos. Ocean. Phys., (1991), 27 (12) 946-951.
- F. Lam, L. Mass, T. Gerkema. Spatial structure of tidal and residual currents as observed over the shelf break in the Bay of Biscay. Deep-See Res., (2004), I 51, 10751096.
- P. Lombard, J. Riley. On the breakdown into turbulence of propagating internal waves. Dyn. Atmos. Oceans, (1996), 23, 345-355.
- H. Moffatt. High frequency excitation of liquid metal systems. Metallurgical Applications of Magnetohydrodynamics, (1984), (Metals Society, London) 180-189.
- P. Müller, G. Holloway, F. Henyey, N. Pomphrey. Nonlinear interactions among internal gravity waves. Rev. Geophys., (1986), 24, 3, 493-536.
- J. Nash, E. Kunze, C. Lee, T. Sanford. Structure of the baroclinic tide generated at Keana Ridge, Hawaii. J. Phys. Oceanogr., (2006), 36, 1123-1135.
- P. Olver. Applications of Lie groups to differential equations. Springer-Verlag, New York, 2nd ed. 1993.
- L. Ovsyannikov. Group Analysis of Differential Equations. Nauka, Moscow, (1978), English transl., ed. W.F. Ames, Academic Press, New York (1982).
- D. Ramsden, G. Holloway. Energy transfers across an internal wave-vortical mode spectrum. J. Geophys. Res., (1992), 97, 3659-3668.
- J. Riley, R. Metcalfe, M. Weissman. Direct numerical simulations of homogeneous turbulence in density-stratified fluids. Nonlinear properties of internal waves, (1981), 76, edited by B.J. West, pp. 79-112, Americal Institute of Physics, New York.
- T. Shepherd. Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Advances in Geophysics, (1990), 32, 287-338
- C. Staquet, J. Sommeria. Internal Gravity Waves : From instabilities to turbulence. Annu. Rev. Fluid Mech., (2002), 34, 559-593.
- A. Tabaei, T. Akylas, K. Lamb. Nonlinear effects in reflecting and colliding internal wave beams. J. Fluid Mech., (2005), 526, 217-243.
- A. Tabaei, T. Akylas. Nonlinear internal gravity wave beams. J. Fluid Mech., (2003), 482, 141-161.
- S. Teoh, J. Imberger, G. Ivey. Laboratory study of the interactions between two internal wave rays. J. Fluid Mech., (1997), 336:91.
- K. Winters, E. D’Asaro. Direct simulation of internal wave energy transfer. J. Phys. Oceanogr., (1997), 9, 235-243.
- C. Wunsch, R. Ferrari. Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech., (2004), 36, 281-314.
- H. Zhang, B. King, H. Swinney. Resonant generation of internal waves on a model continental slope. Phys. Rev. Let., (2008), 100, 244504.

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