An exactly solvable problem of wave fronts and applications to the asymptotic theory. (English) Zbl 1484.35105
Summary: A nontrivially solvable 4-dimensional Hamiltonian system is applied to the problem of wave fronts and to the asymptotic theory of partial differential equations. The Hamilton function we consider is \(H(\mathbf{x}, \mathbf{p}) = \sqrt{D(\mathbf{x})}|\mathbf{p}|\). Such Hamiltonians arise when describing the fronts of linear waves generated by a localized source in a basin with a variable depth. We consider two realistic types of bottom shape: 1) the depth of the basin is determined, in the polar coordinates, by the function \(D(\varrho, \varphi) = (\varrho^2 + b)/(\varrho^2 + a)\) and 2) the depth function is \(D(x, y) = (x^2 + b)/(x^2 + a)\). As an application, we construct the asymptotic solution to the wave equation with localized initial conditions and asymptotic solutions of the Helmholtz equation with a localized right-hand side.
MSC:
| 35C05 | Solutions to PDEs in closed form |
| 35L10 | Second-order hyperbolic equations |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
Keywords:
exactly solvable Hamiltonian systems; wave fronts; asymptotic theory; elliptic functions; linear waves generated by a localized source; localized initial conditions; Helmholtz equation with a localized right-hand sideReferences:
| [1] | Akhiezer, N. I., (Elements of the Theory of Elliptic Functions. Elements of the Theory of Elliptic Functions, Transl. Math. Monogr., vol. 79 (1990), AMS Providence: AMS Providence R. I) · Zbl 0694.33001 |
| [2] | Pelinovsky, E., Hydrodynamics of Tsunami Waves. Waves in Geophysical Fluids, Vol. 489 (2006), CISM International Centre for Mechanical Sciences, Springer · Zbl 1241.76092 |
| [3] | Stocker, J. J., Waves on Water. the Mathematical Theory and Applications (1957), Interscience: Interscience New York-London · Zbl 0078.40805 |
| [4] | Alber, M. S.; Luther, G. G.; Marsden, J. E., Energy dependent Schrödinger operators and complex Hamiltonian systems on Riemann surfaces, Nonlinearity, 10, 223-241 (1997) · Zbl 0906.35082 |
| [5] | Gesztesy, F.; Holden, H., Soliton Equations and their Algebro-Geometric Solutions. \( ( 1 + 1 )\)-Dimensional Continuous Models. Cambridge Studies in Advanced Mathematics 79 (2003), Cambridge University Press · Zbl 1061.37056 |
| [6] | Dickey, L. A., Soliton Equations and Hamiltonian Systems (2003), World Scientific · Zbl 1140.35012 |
| [7] | Belokolos, E. D.; Bobenko, A. I.; Enol’skii, V. Z.; Its, A. R.; Matveev, V. B., Algebro-Geometric Approach To Nonlinear Integrable Equations (1994), Springer · Zbl 0809.35001 |
| [8] | Alber, M. S.; Fedorov, Yu. N., Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians, J. Phys. A: Math. Gen., 33, 8409-8425 (2000) · Zbl 0960.37036 |
| [9] | Brezhnev, Yu. V., Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials, Contemp. Math., 563, 1-31 (2012) · Zbl 1248.81032 |
| [10] | Sherratt, J. A.; Brezhnev, Yu. V., The mean values of the weierstrass elliptic function \(\wp \): Theory and application, Physica D, 263, 86-98 (2013) · Zbl 1293.33023 |
| [11] | Babic, V. M.; Buldyrev, V. S., Asymptotic Methods in Short-Wavelength Diffraction Theory (2009), Alpha Science International · Zbl 0742.35002 |
| [12] | Maslov, V. P., Operational Methods (1976), Mir · Zbl 0449.47002 |
| [13] | Maslov, V. P.; Fedoriuk, M. V., Semi-Classical Approximation in Quantum Mechanics (2001), Reidel: Reidel Dordrecht |
| [14] | Yu. Dobrokhotov, S.; Ya. Sekerzh-Zenkovich, S.; Tirozzi, B.; Volkov, B., Explicit asymptotics for tsunami waves in framework of the piston model, Russ. J. Earth Sciences, 8, ES403, 1-12 (2006) |
| [15] | Yu. Dobrokhotov, S.; Shafarevich, A. I.; Tirozzi, B., Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations, Russ. J. Math. Phys., 15, 2, 192-221 (2008) · Zbl 1180.35336 |
| [16] | Yu. Dobrokhotov, S.; Nazaikinskii, V. E., Asymptotics of localized wave end vortex solutions of a linearized system of shallow water equations, (Actual Problems of Mechanics (2015), Nauka: Nauka Moscow), 98-139, (in Russian) |
| [17] | Yu. Anikin, A.; Yu. Dobrokhotov, S.; Nazaikinskii, V. E.; Rouleux, M., The maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solution of stationary equations with localized right-hand sides, Dokl. Math., 96, 1, 406-410 (2017) · Zbl 1377.35062 |
| [18] | Błaszak, M., Multi-Hamiltonian Theory of Dynamical Systems (1998), Springer · Zbl 0912.58017 |
| [19] | Yu. Dobrokhotov, S.; Nazaikinskii, V. E.; Tirozzi, B., Asymptotic solution of 2D wave equation with variable velocity and localized right-hand side, Russ. J. Math. Phys., 17, 1, 66-76 (2010) · Zbl 1192.35110 |
| [20] | Mishchenko, A. S.; Shatalov, V. E.; Sternin, B. Y., Lagrangian Manifolds and the Maslov Operator (1990), Springer · Zbl 0727.58001 |
| [21] | Heading, J., An Introduction To Phase-Integral Methods (1962), Wiley · Zbl 0115.07102 |
| [22] | Eidus, D. M., On the Principle of Limiting Absorption (1963), New York University · Zbl 0149.30602 |
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