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Published: 10 March 2023

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1

Henry, David, Konstantinos Kalimeris, Emilian I. Părău, Jean-Marc Vanden-Broeck, and Erik Wahlén, eds. Nonlinear Water Waves. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-33536-6.

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2

Horikawa, Kiyoshi, and Hajime Maruo, eds. Nonlinear Water Waves. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/978-3-642-83331-1.

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3

Constantin, Adrian, Joachim Escher, Robin Stanley Johnson, and Gabriele Villari. Nonlinear Water Waves. Edited by Adrian Constantin. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-31462-4.

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4

Debnath, Lokenath. Nonlinear water waves. Boston: Academic Press, 1994.

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5

Debnath, Lokenath. Nonlinear water waves. Boston: Academic Press, 1994.

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6

Oskar, Mahrenholtz, and Markiewicz M, eds. Nonlinear water wave interaction. Southampton: WIT Press, 1999.

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7

Ma, Qingwei. Advances in numerical simulation of nonlinear water waves. Hackensack, NJ: World Scientific, 2010.

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8

Kiyoshi, Horikawa, Maruo H. 1922-, Internation Union of Theoretical and Applied Mechanics., and Symposium on Non-linear Water Waves (1987 : Tokyo, Japan), eds. Nonlinear water waves: IUTAM Symposium, Tokyo/Japan, August 25-28, 1987. Berlin: Springer-Verlag, 1988.

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9

Nonlinear water waves with applications to wave-current interactions and tsunamis. Philadelphia: Society for Industrial and Applied Mathematics, 2011.

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10

John, Grue, Gjevik Bjørn, and Weber Jan Erik, eds. Waves and nonlinear processes in hydrodynamics. Dordrecht: Kluwer Academic Publishers, 1996.

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11

Grue, John. Waves and Nonlinear Processes in Hydrodynamics. Dordrecht: Springer Netherlands, 1996.

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12

Horikawa, Kiyoshi. Nonlinear Water Waves: IUTAM Symposium, Tokyo/Japan, August 25-28, 1987. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988.

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13

Sander, Johannes. Weakly nonlinear unidirectional shallow water waves generated by a moving boundary. Zurich: Ruegger, 1990.

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14

Green, A. E. A nonlinear theory of water waves for finite and infinite depths. London: RoyalSociety, 1986.

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15

1962-, Dias F., Ghidaglia J. -M, and Saut J. -C, eds. Mathematical problems in the theory of water waves: A workshop on the problems in the theory of nonlinear hydrodynamic waves, May 15-19, 1995, Luminy, France. Providence, R.I: American Mathematical Society, 1996.

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16

W, Hutchinson John. Advances in Applied Mechanics, 32. Burlington: Elsevier, 1996.

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17

Abreu, Manuel A. Nonlinear transformation of directional wave spectra in shallow water. Monterey, Calif: Naval Postgraduate School, 1991.

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18

Abbasov, Iftikhar B. 3D Modeling of Nonlinear Wave Phenomena on Shallow Water Surfaces. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2018. http://dx.doi.org/10.1002/9781119488187.

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19

Debnath, Lokenath. Nonlinear Water Waves. Elsevier Science & Technology Books, 1994.

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20

Kalimeris, Konstantinos, Emilian Părău, Erik Wahlén, Jean-Marc Vanden-Broeck, and David Henry. Nonlinear Water Waves: An Interdisciplinary Interface. Springer International Publishing AG, 2019.

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21

Kalimeris, Konstantinos, Erik Wahlén, Emilian I. Părău, Jean-Marc Vanden-Broeck, and David Henry. Nonlinear Water Waves: An Interdisciplinary Interface. Springer International Publishing AG, 2020.

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22

Escher, Joachim, Adrian Constantin, Robin Stanley Johnson, and Gabriele Villari. Nonlinear Water Waves: Cetraro, Italy 2013. Springer, 2016.

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23

Escher, Joachim, Adrian Constantin, Robin Stanley Johnson, and Gabriele Villari. Nonlinear Water Waves: Cetraro, Italy 2013. Springer London, Limited, 2016.

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24

(Editor), O. Mahrenholtz, and M. Markiewicz (Editor), eds. Nonlinear Water Wave Interaction (Advances in Fluid Mechanics Volume 24). Computational Mechanics, Inc., 1999.

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25

Dinamika voln na poverkhnosti zhidkosti. Moskva: Nauka Fizmatlit, 1999.

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26

Horikawa, K. Nonlinear Water Waves Iutam Symposium, Tokyo/Japan, August 25-28, 1987. Springer, 1988.

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27

Horikawa, Kiyoshi, and Hajime Maruo. Nonlinear Water Waves: IUTAM Symposium Tokyo/Japan 1987. Springer London, Limited, 2011.

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28

Ma, Qingwei. Advances in Numerical Simulation of Nonlinear Water Waves. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/7087.

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29

Ahn, Kyungmo. Nonlinear analysis of waves in finite water depth. 1993.

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30

Abbasov, Iftikhar B. 3D Modeling of Nonlinear Wave Phenomena on Shallow Water Surfaces. Wiley & Sons, Incorporated, John, 2018.

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31

Abbasov, Iftikhar B. 3D Modeling of Nonlinear Wave Phenomena on Shallow Water Surfaces. Wiley & Sons, Incorporated, John, 2018.

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32

Abbasov, Iftikhar B. 3D Modeling of Nonlinear Wave Phenomena on Shallow Water Surfaces. Wiley & Sons, Limited, John, 2018.

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33

Abbasov, Iftikhar B. 3D Modeling of Nonlinear Wave Phenomena on Shallow Water Surface. Wiley & Sons, Limited, John, 2018.

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34

West, B. J. On the Simpler Aspect of Nonlinear Fluctuating Deep Water Gravity Waves. Springer, 2014.

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35

Zeitlin, Vladimir. Getting Rid of Fast Waves: Slow Dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.003.0005.

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After analysis of general properties of horizontal motion in primitive equations and introduction of principal parameters, the key notion of geostrophic equilibrium is introduced. Quasi-geostrophic reductions of one- and two-layer rotating shallow-water models are obtained by a direct filtering of fast inertia–gravity waves through a choice of the time scale of motions of interest, and by asymptotic expansions in Rossby number. Properties of quasi-geostrophic models are established. It is shown that in the beta-plane approximations the models describe Rossby waves. The first idea of the classical baroclinic instability is given, and its relation to Rossby waves is explained. Modifications of quasi-geostrophic dynamics in the presence of coastal, topographic, and equatorial wave-guides are analysed. Emission of mountain Rossby waves by a flow over topography is demonstrated. The phenomena of Kelvin wave breaking, and of soliton formation by long equatorial and topographic Rossby waves due to nonlinear effects are explained.

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36

Zeitlin, Vladimir. Rotating Shallow-Water Models as Quasilinear Hyperbolic Systems, and Related Numerical Methods. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.003.0007.

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The chapter contains the mathematical background necessary to understand the properties of RSW models and numerical methods for their simulations. Mathematics of RSW model is presented by using their one-dimensional reductions, which are necessarily’one-and-a-half’ dimensional, due to rotation and include velocity in the second direction. Basic notions of quasi-linear hyperbolic systems are recalled. The notions of weak solutions, wave breaking, and shock formation are introduced and explained on the example of simple-wave equation. Lagrangian description of RSW is used to demonstrate that rotation does not prevent wave-breaking. Hydraulic theory and Rankine–Hugoniot jump conditions are formulated for RSW models. In the two-layer case it is shown that the system loses hyperbolicity in the presence of shear instability. Ideas of construction of well-balanced (i.e. maintaining equilibria) shock-resolving finite-volume numerical methods are explained and these methods are briefly presented, with illustrations on nonlinear evolution of equatorial waves.

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37

Zeitlin, Vladimir. RSW Modons and their Surprising Properties: RSW Turbulence. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.003.0009.

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By using quasi-geostrophic modons constructed in Chapter 6 as initial conditions, rotating-shallow-water modons are obtained through the process of ageostrophic adjustment, both in one- and in two-layer configurations. Scatter plots show that they are solutions of the rotating shallow-water equations. A special class of modons with an internal front (shock) is shown to exist. A panorama of collision processes of the modons, leading to formation of tripoles, nonlinear modons, or elastic scattering is presented. The modon solutions are then used for initialisations of numerical simulations of decaying rotating shallow-water turbulence. The results are analysed and compared to those obtained with standard in 2D turbulence initializations, and differences are detected, showing non-universality of decaying 2D turbulence. The obtained energy spectra are steeper than theoretical predictions for ‘pure’ 2D turbulence, and pronounced cyclone–anticyclone asymmetry and dynamical separation of waves and vortices are observed.

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38

Rajeev, S. G. Integrable Models. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0009.

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Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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39

(Editor), Walter A. Strauss, and Yan Guo (Editor), eds. Nonlinear Wave Equations: A Conference in Honor of Walter A. Strauss on the Occasion of His Sixtieth Birthday, May 2-3, 1998, Brown University (Contemporary Mathematics). American Mathematical Society, 2000.

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40

Zeitlin, Vladimir. Rotating Shallow-Water model with Horizontal Density and/or Temperature Gradients. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.003.0014.

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The derivation of rotating shallow-water equations by vertical averaging and columnar motion hypothesis is repeated without supposing horizontal homogeneity of density/potential temperature. The so-called thermal rotating shallow-water model arises as the result. The model turns to be equivalent to gas dynamics with a specific equation of state. It is shown that it possesses Hamiltonian structure and can be derived from a variational principle. Its solution at low Rossby numbers should obey the thermo-geostrophic equilibrium, replacing the standard geostrophic equilibrium. The wave spectrum of the model is analysed, and the appearance of a whole new class of vortex instabilities of convective type, resembling asymmetric centrifugal instability and leading to a strong mixing at nonlinear stage, is demonstrated.

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41

Crow, John A. A nonlinear shallow water wave equation and its classical solutions of the cauchy problem. 1991.

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42

Zeitlin, Vladimir. Geophysical Fluid Dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.001.0001.

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The book explains the key notions and fundamental processes in the dynamics of the fluid envelopes of the Earth (transposable to other planets), and methods of their analysis, from the unifying viewpoint of rotating shallow-water model (RSW). The model, in its one- or two-layer versions, plays a distinguished role in geophysical fluid dynamics, having been used for around a century for conceptual understanding of various phenomena, for elaboration of approaches and methods, to be applied later in more complete models, for development and testing of numerical codes and schemes of data assimilations, and many other purposes. Principles of modelling of large-scale atmospheric and oceanic flows, and corresponding approximations, are explained and it is shown how single- and multi-layer versions of RSW arise from the primitive equations by vertical averaging, and how further time-averaging produces celebrated quasi-geostrophic reductions of the model. Key concepts of geophysical fluid dynamics are exposed and interpreted in RSW terms, and fundamentals of vortex and wave dynamics are explained in Part 1 of the book, which is supplied with exercises and can be used as a textbook. Solutions of the problems are available at Editorial Office by request. In-depth treatment of dynamical processes, with special accent on the primordial process of geostrophic adjustment, on instabilities in geophysical flows, vortex and wave turbulence and on nonlinear wave interactions follows in Part 2. Recently arisen new approaches in, and applications of RSW, including moist-convective processes constitute Part 3.

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43

Zagone, Robin L. Linear and nonlinear optical investigation of films: I. Formalism for time resolved multiphoton processes. II. Detection of solid water phase transitions on Si-SiO₂ III. Wave guided CARS spectroscopy. 1995.

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