zbMATH — the first resource for mathematics

On eigenfunctions of the structures described by the “shallow-water” model on the plane. (English) Zbl 1151.76396
J. Math. Sci., New York 151, No. 1, 2639-2650 (2008); translation from Fundam. Prikl. Mat. 12, No. 6, 17-32 (2006).
Summary: We propose a new method for solving the “shallow-water” equations. We show that from the equations of “shallow water” one obtains nonlinear Liouville-type equations, Helmholtz equations, etc. This allows one to construct eigenfunctions of various structures that appear in the flow in the two-dimensional case. We obtain exact and asymptotic solutions in an elliptic domain with singularities.
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] L. D. Akulenko and S. V. Nesterov, ”Eigenvalues of an elliptic membrane,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 191–202 (2000).
[2] S. N. Aristov and V. V. Pukhnachev, ”On the equations of rotationally symmetric motion of a viscous incompressible fluid,” Dokl. Ross. Akad. Nauk, 394, No. 5, 611–614 (2004).
[3] V. V. Bulatov, Yu. V. Vladimirov, V. G. Danilov, and S. Yu. Dobrokhotov, ”An example of calculating the ’eye’ of a typhoon on the basis of the V. P. Maslov conjecture,” Dokl. Ross. Akad. Nauk, 338, 102–105 (1994). · Zbl 0872.35086
[4] F. J. Chatelon and P. Orenga, ”On a non-homogeneous shallow-water problem,” RAIRO, Modélisation Math. Anal. Numér., 31, No. 1, 27–55 (1997). · Zbl 0871.76009
[5] V. G. Danilov, V. P. Maslov, and V. M. Shelkovich, ”Algebras of singularities of generalized solutions of strictly hyperbolic systems of first-order quasilinear equations,” Teor. Mat. Fiz., 114, 3–55 (1998). · Zbl 0946.35049
[6] V. G. Danilov, V. P. Maslov, and K. A. Volosov, Mathematical Modelling of Heat and Mass Transfer Processes, Kluwer Academic, Dordrecht (1995). · Zbl 0839.35001
[7] V. G. Danilov, G. A. Omel’yanov, and D. L. Rozenkop, ”Dynamics of a point weak singularity for shallow-water equations on the sphere,” private communication.
[8] S. Yu. Dobrokhotov, ”Hugoniót-Maslov chains for solitary vortices of the shallow water equations. I, II,” Russ. J. Math. Phys., 6, No. 2, 137–173; 6, No. 3, 282–313 (1999). · Zbl 1059.76506
[9] V. P. Maslov, ”Three algebras corresponding to nonsmooth solutions of quasilinear hyperbolic equations,” Usp. Mat. Nauk, 35, No. 2, 252–253 (1980).
[10] L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1989). · Zbl 0741.20046
[11] V. V. Pukhnachev, ”Equivalence transformations and hidden symmetry of evolution equations,” Sov. Math. Dokl., 35, No. 3, 555–558 (1987). · Zbl 0681.35004
[12] K. A. Volosov, Mathematical Problems of the Theory of Transport Processes [in Russian], Candidate’s Dissertation in Physics and Mathematics, Moscow (1982).
[13] K. A. Volosov, Invariant Properties of the Ansatz of the Hirota Method for Quasilinear Parabolic Equations, arXiv:math.PH/0103014 (2001).
[14] K. A. Volosov, in: Tools for Mathematical Modeling. Third Int. Conf., St. Petersburg, June 18–23, pp. 169–171 (2001).
[15] K. A. Volosov, A property of the ansatz of the Hirota method for quasilinear parabolic equations,” Mat. Zametki, 71 (2002), No. 3, 373–389 (2002). · Zbl 1031.35055
[16] V. F. Zaitsev and A. D. Polyanin, Handbook on Partial Differential Equations [in Russian], Mezhdunarodnaya Programma Obrazovaniya, Moscow (1996). · Zbl 0855.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.