Shermenev A.

The three-dimensional nonlinear potential wave equation describes, in particular, the waves in an isentropic gas flow for the non-dissipative case. In two-dimensional case, this is the shallow water equation. Separation of variables method for the nonlinear equation in polar (and cylinder) coordinates developed in [5-6] allows to construct nonlinear corrections (see [1-4]) to a series of classic linear solutions used and studied in classic books of Rayleigh and Lamb. The aim this talk is to extend the method to vector versions of the nonlinear equation and to apply it to the nonlinear Navier-Stokes equation. Possibility to apply this approach to the cases of coordinates of elliptic and parabolic cylinder, where Mathieu functions and Weber functions are involved, will be also considered. REFERENCES [1] Shermenev, A. & Shermeneva, M. 2000 Physical Review E, No. 5, 6000–6002. [2] Shermenev, A. 2001 Geophysical and Astrophysical Fluid Dynamics, No. 1–2, 1–14. [3] Shermenev, A. 2003 Acta Acustica, vol. 89, 426–429. [4] Shermenev, A. 2003 LNCS 2630, Springer-Verlag, 375–386. [5] Shermenev A. 2004 Journal of Physics, A, 37, 1-9 [6] Shermenev A. 2005 Physica D: Nonlinear Phenomena, 212:3-4, pp 205-215