Symmetries of shallow water equations on a rotating plane

Article
  • 30 Downloads

Abstract

We consider a system of nonlinear differential equations which describes the spatial motion of an ideal incompressible fluid on a rotating plane in the shallow water approximation and a more general system of the theory of long waves which takes into account the specifics of shear flows. Using the group analysis methods, we calculate the 9-dimensional Lie algebras of infinitesimal operators admissible by the models. We establish an isomorphism of these Lie algebras with a known Lie algebra of operators admissible by the system of equations for the two-dimensional isentropic motions of a polytropic gas with the adiabatic exponent γ = 2. The nontrivial symmetries of the models under consideration enable us to carry out the group generation of the solutions. The class of stationary solutions to the equations of rotating shallow water transforms into a new class of periodic solutions.

Key words

equations of rotating shallow water symmetry Lie algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Pedlosky, Geophysical Fluid Dynamics (Springer, New York, 1979).MATHGoogle Scholar
  2. 2.
    A. Majda, Lecture Notes, Vol. 9: Introduction to PDEs and Waves for the Atmosphere and Ocean (Courant Inst.Math. Sci., New York, 2003).Google Scholar
  3. 3.
    L. V. Ovsyannikov, Group Analysis of Differential Equations (Nauka, Moscow, 1978) [in Russian].MATHGoogle Scholar
  4. 4.
    P. J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993).MATHGoogle Scholar
  5. 5.
    N. Kh. Ibragimov, Transformation Groups in Mathematical Physics (Nauka, Moscow, 1983) [in Russian].MATHGoogle Scholar
  6. 6.
    CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 2: Applications in Engineering and Physical Sciences, Ed. by N. H. Ibragimov (CRC Press, Boca Raton, 1995).Google Scholar
  7. 7.
    N. Bila, E. Mansfield, and P. Clarkson, “Symmetry Group Analysis of the Shallow Water and Semi-Geostrophic Equations,“ Quart. J. Mech. Appl. Math. 59(1), 95–123 (2006).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    V. E. Zakharov, “The Benney Equations and Quasiclassical Approximation in the Method of the Inverse Problem,” Funktsional. Anal. i Prilozhen. 14(2), 15–24 (1980) [Functional Anal. Appl. 14, 89–98 (1980)].MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    L. V. Ovsyannikov, “On Optimal Systems of Subalgebras,” Dokl. Akad. Nauk, Ross. Akad. Nauk 333(6), 702–704 (1994) [Russ. Acad. Sci., Dokl., Math. 48 (3), 645–649 (1994)].MathSciNetGoogle Scholar
  10. 10.
    J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, “Continuous Subgroups of the Fundamental Groups of Physics. III. The De Sitter Groups,” J. Math. Phys. 18, 2259–2288 (1977).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. S. Pavlenko, “Symmetries and Solutions to Equations of Two-Dimensional Motions of a Polytropic Gas,” Siberian Electronic Math. Rep. 2, 291–307 (2005) [see also http://semr.math.nsc.ru].MATHMathSciNetGoogle Scholar
  12. 12.
    A. A. Nikol’skii, “Invariant Transformation of the Equations of Motion of an Ideal Monatomic Gas and New Classes of Their Exact Solutions,” Prikl. Mat. Mekh. 27, 496–508 (1963) [J. Appl. Math. Mech. 27, 740–756 (1963)].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Lavrent’ev Institute of HydrodynamicsNovosibirskRussia

Personalised recommendations