Nonlinear, nondispersive wave equations: Lagrangian and Hamiltonian functions in the hodograph transformation

Year: 2020

Authors: Pegoraro F., Bulanov S.

Autors Affiliation: Univ Pisa, Enrico Fermi Dept Phys, Pisa, Italy; Natl Inst Opt, Natl Res Council, Via G Moruzzi 1, Pisa, Italy; Inst Phys ASCR, ELI Beamlines Project, Slovance 2, Prague 18221, Czech Republic; Kansai Photon Sci Inst, Natl Inst Quantum & Radiol Sci & Technol QST, 8-1-7 Umemidai, Kyoto, Japan; RAS, Prokhorov Gen Phys Inst, Vavilov Str 38, Moscow 119991, Russia

Abstract: The hodograph transformation is generally used in order to associate a system of linear partial differential equations to a system of nonlinear (quasilinear) differential equations by interchanging dependent and independent variables. Here we consider the case when the nonlinear differential system can be derived from a Lagrangian density and revisit the hodograph transformation within the formalism of the Lagrangian-Hamiltonian continuous dynamical systems. Restricting to the case of nondissipative, nondispersive one-dimensional waves, we show that the hodograph transformation leads to a linear partial differential equation for an unknown function that plays the role of the Lagrangian in the hodograph variables. We then define the corresponding hodograph Hamiltonian and show that it turns out to coincide with the wave amplitude. i.e., with the unknown function of the independent variables to be solved for in the initial nonlinear wave equation.


Volume: 384 (2)      Pages from: 126064-1  to: 126064-6

More Information: Supported by the project High Field Initiative (CZ.02.1.01 0.0/0.0/15_003/0000449) from European Regional Development Fund. F.P. thanks the ELI-Beamlines project for its hospitality in September 2018.
KeyWords: Nonlinear wave propagation; Hodograph transformation; Lagrangian-Hamiltonian formalism
DOI: 10.1016/j.physleta.2019.126064

Citations: 5
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