@article{3359fac3498346e2b1720964dce121fa,

title = "Simple two-layer dispersive models in the Hamiltonian reduction formalism",

abstract = "A Hamiltonian reduction approach is defined, studied, and finally used to derive asymptotic models of internal wave propagation in density stratified fluids in two-dimensional domains. Beginning with the general Hamiltonian formalism of Benjamin (1986 J. Fluid Mech. 165 445-74) for an ideal, stably stratified Euler fluid, the corresponding structure is systematically reduced to the setup of two homogeneous fluids under gravity, separated by an interface and confined between two infinite horizontal plates. A long-wave, small-amplitude asymptotics is then used to obtain a simplified model that encapsulates most of the known properties of the dynamics of such systems, such as bidirectional wave propagation and maximal amplitude travelling waves in the form of fronts. Further reductions, and in particular devising an asymptotic extension of Dirac{\textquoteright}s theory of Hamiltonian constraints, lead to the completely integrable evolution equations previously considered in the literature for limiting forms of the dynamics of stratified fluids.",

keywords = "dispersive internal wave models, Hamiltonian PDEs, Hamiltonian reductions, stratified fluids, travelling wave solutions",

author = "R. Camassa and G. Falqui and G. Ortenzi and M. Pedroni and {Vu Ho}, {T. T.}",

note = "Funding Information: We thank R Barros, P Lorenzoni, and R Vitolo for useful discussions. Thanks are also due to the anonymous referees, for providing remarks and suggestions which helped improve the presentation, and for suggesting additional references. This project has received funding from the European Union{\textquoteright}s Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie Grant No. 778010 IPaDEGAN. We also gratefully acknowledge the auspices of the GNFM section of INdAM, under which part of this work was carried out, and the financial support of the project MMNLP (Mathematical Methods in Nonlinear Physics) of the INFN. RC thanks the support by the National Science Foundation under Grants RTG DMS-0943851, CMG ARC-1025523, DMS-1009750, DMS-1517879, DMS-1910824, and by the Office of Naval Research under Grants N00014-18-1-2490, N00014-23-1-2478, and DURIP N00014-12-1-0749. R C and M P thank the Department of Mathematics and its Applications of the University of Milano-Bicocca for its hospitality. R C would also like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme HYD2 where work on this paper was completed, with support by EPSRC Grant EP/R014604/1. Publisher Copyright: {\textcopyright} 2023 IOP Publishing Ltd & London Mathematical Society.",

year = "2023",

month = sep,

day = "1",

doi = "10.1088/1361-6544/ace3a0",

language = "English",

volume = "36",

journal = "Nonlinearity",

issn = "0951-7715",

publisher = "IOP Publishing",

number = "9",

}