On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations

Roberto Camassa ; Gregorio Falqui ; Giovanni Ortenzi ; Marco Pedroni

Symmetry, integrability and geometry: methods and applications, Tome 15 (2019)

Résumé

Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrödinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.

Détail

Keywords: bi-Hamiltonian geometry, self-similar solutions, shallow water models.
Mots-clés : Poisson reductions

Bibliographie

[1] Ablowitz M. J., Segur H., Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981 | MR | Zbl

[2] Airy G. B., “Tides and waves”, Encyclopaedia Metropolitana, 5 (1845), 241–396

[3] Alber M. S., Camassa R., Fedorov Yu.N., Holm D. D., Marsden J. E., “The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type”, Comm. Math. Phys., 221 (2001), 197–227 | DOI | MR | Zbl

[4] Barenblatt G. I., Zel'dovich Ya. B., “Self-similar solutions as intermediate asymptotics”, Ann. Rev. Fluid Mech., 4 (1972), 285–312 | DOI | MR | Zbl

[5] Barré de Saint-Venant A. J. C., “Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et à l'introduction de marées dans leurs lits”, C. R. Acad. Sci. Paris, 73 (1871), 147–154; 237–240

[6] Bogojavlenskii O. I., Novikov S. P., “The relationship between Hamiltonian formalisms of stationary and nonstationary problems”, Funct. Anal. Appl., 10 (1976), 8–11 | DOI | MR | Zbl

[7] Brazhnyi V. A., Kamchatnov A. M., Konotop V. V., “Hydrodynamic flow of expanding Bose–Einstein condensates”, Phys. Rev. A, 68 (2003), 035603, 4 pp., arXiv: cond-mat/0307191 | DOI

[8] Calogero F., Degasperis A., Spectral transform and solitons, v. I, Studies in Mathematics and its Applications, 13, Tools to solve and investigate nonlinear evolution equations, North-Holland Publishing Co., Amsterdam–New York, 1982 | MR | Zbl

[9] Camassa R., Falqui G., Ortenzi G., “Two-layer interfacial flows beyond the Boussinesq approximation: a Hamiltonian approach”, Nonlinearity, 30 (2017), 466–491, arXiv: 1512.07498 | DOI | MR | Zbl

[10] Camassa R., Falqui G., Ortenzi G., Pedroni M., Pitton G., “Singularity formation as a wetting mechanism in a dispersionless water wave model”, Nonlinearity, 32 (2019), 4079–4116, arXiv: 1906.11637 | DOI | MR | Zbl

[11] Camassa R., Falqui G., Ortenzi G., Pedroni M., Thomson C., “Hydrodynamic models and confinement effects by horizontal boundaries”, J. Nonlinear Sci., 29 (2019), 1445–1498, arXiv: 1812.02963 | DOI | MR | Zbl

[12] Cordani B., “Conformal regularization of the Kepler problem”, Comm. Math. Phys., 103 (1986), 403–413 | DOI | MR | Zbl

[13] Evans L. C., Partial differential equations, Graduate Studies in Mathematics, 19, Amer. Math. Soc., Providence, RI, 1998 | MR | Zbl

[14] Falqui G., Lorenzoni P., “Exact Poisson pencils, $\tau$-structures and topological hierarchies”, Phys. D, 241 (2012), 2178–2187, arXiv: 1106.1546 | DOI | MR | Zbl

[15] Gardner C. S., Greene J. M., Kruskal M. D., Miura R. M., “Korteweg–de Vries equation and generalization. VI Methods for exact solution”, Comm. Pure Appl. Math., 27 (1974), 97–133 | DOI | MR | Zbl

[16] Gelfand I. M., Zakharevich I., “On the local geometry of a bi-Hamiltonian structure”, The Gelfand Mathematical Seminars (1990–1992), Birkhäuser Boston, Boston, MA, 1993, 51–112 | DOI | MR | Zbl

[17] Gurevich A. V., Krylov A. L., “Dissipationless shock waves in media with positive dispersion”, Sov. Phys. JETP, 65 (1987), 944–953

[18] Lamb H., Hydrodynamics, Cambridge Mathematical Library, 6th ed., Cambridge University Press, Cambridge, 1993 | MR | Zbl

[19] Liu T. P., Smoller J. A., “On the vacuum state for the isentropic gas dynamics equations”, Adv. in Appl. Math., 1 (1980), 345–359 | DOI | MR | Zbl

[20] Magri F., “A simple model of the integrable Hamiltonian equation”, J. Math. Phys., 19 (1978), 1156–1162 | DOI | MR | Zbl

[21] Magri F., Casati P., Falqui G., Pedroni M., “Eight lectures on integrable systems”, Integrability of Nonlinear Systems, Lecture Notes in Phys., 638, eds. Y. Kosmann-Schwarzbach, K.M. Tamizhmani, B. Grammaticos, Springer, Berlin, 2004, 209–250 | DOI | MR

[22] Maltsev A. Ya., Novikov S. P., “On the local systems Hamiltonian in the weakly non-local Poisson brackets”, Phys. D, 156 (2001), 53–80, arXiv: nlin.SI/0006030 | DOI | MR | Zbl

[23] Nutku Y., Pavlov M. V., “Multi-Lagrangians for integrable systems”, J. Math. Phys., 43 (2002), 1441–1459, arXiv: hep-th/0108214 | DOI | MR | Zbl

[24] Oevel W., Fuchssteiner B., Zhang H., Ragnisco O., “Mastersymmetries, angle variables, and recursion operator of the relativistic Toda lattice”, J. Math. Phys., 30 (1989), 2664–2670 | DOI | MR | Zbl

[25] Ovsyannikov L. V., “Two-layer “shallow water” model”, J. Appl. Mech. Tech. Phys., 20 (1979), 127–135 | DOI | MR

[26] Praught J., Smirnov R. G., “Andrew Lenard: a mystery unraveled”, SIGMA, 1 (2005), 005, 7 pp., arXiv: nlin.SI/0510055 | DOI | MR | Zbl

[27] Rauch-Wojciechowski S., “Newton representation for stationary flows of the KdV hierarchy”, Phys. Lett. A, 170 (1992), 91–94 | DOI | MR

[28] Sedov L. I., Similarity and dimensional methods in mechanics, Academic Press, New York–London, 1959 | MR | Zbl

[29] Stoker J. J., Water waves: The mathematical theory with applications, Pure and Applied Mathematics, 4, Interscience Publishers, Inc., New York, 1957 | MR | Zbl

[30] Stokes G. G., “On the theory of oscillatory waves”, Trans. Camb. Phil. Soc., 8 (1847), 411–455

[31] Talanov V. I., “Self-focusing of wave beams in nonlinear media”, JETP Lett., 2 (1965), 138–141

[32] Vaisman I., Lectures on the geometry of Poisson manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994 | DOI | MR | Zbl

[33] Whitham G. B., Linear and nonlinear waves, Pure and Applied Mathematics, John Wiley Sons, Inc., New York, 1999 | DOI | MR

[34] Zakharov V. E., “Benney equations and quasiclassical approximation in the method of the inverse problem”, Funct. Anal. Appl., 14 (1980), 89–98 | DOI | MR | Zbl

[35] Zubelli J. P., Magri F., “Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV”, Comm. Math. Phys., 141 (1991), 329–351 | DOI | MR | Zbl