Abstract
This chapter builds on the material of Chap. 14 and reviews other standard nonlinear models that can be described by partial differential equations. We begin with the model equations of gas dynamics, expand the Burgers–KPZ model to the multidimensional case and the related concentration fields, study in detail the Korteweg–de Vries (KdV) equations, in which one can observe the creation of solitary waves (solitons), and finally, discuss nonlinear flows in porous media.
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Notes
- 1.
For more and in-depth information, see, S. Gurbatov, A. Malakhov and A. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles, Manchester University Press, Manchester–New York, 1991, and W.A. Woyczyński, Burgers–KPZ Turbulence: Göttingen Lectures, Springer-Verlag, Berlin–Heidelberg, 1998.
- 2.
Note that the form of the first two terms in (4) forces v(x, t) to have the dimensionality of velocity.
- 3.
For clarity’s sake, note that the first equality in (12) defines a dimensionless dispersive factor, while the right-hand side of the second equation contains its dimensional counterpart.
- 4.
R. Hirota. Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. (1971) 27, 1192–1194.
- 5.
Fascinating videos of KdV solitons, both simulated and photographed at ocean beaches, can be found at www.youtube.com/watch?v=ZsTe2N5_eZE.
- 6.
See, e.g., F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Communications in Partial Differential Equations 26 (2001), 101–174.
- 7.
A comprehensive calculation showing how Darcy’s law can be obtained from the general equations of flow in porous media can be found, e.g., in W.G. Grey and K. O’Neil, On the general equations of flow in porous media and their reduction to Darcy’s law, Water Resources Research 12 (1976), 148–154, or U. Hornung, Homogenization and Porous Media, Springer-Verlag, New York 1997, pp. 16ff.
- 8.
See, e.g., G.I. Barenblatt, On some unsteady motions of a liquid and a gas in a porous medium, Prikladnaya Matematika i Mekhanika 16 (1952), 67–78; D.G. Aronson, The Porous Medium Equation, Lecture Notes in Mathematics 1224, Springer-Verlag 1224, 1986; M.E. Gurtin and R.C. MacCamy, On the diffusions of biological populations, Mathematical Biosciences 33 (1977), 35–49.
- 9.
See, e.g., J.L. Vazquez, Asymptotic behavior for the porous medium equation posed in the whole space, Journal of Evolution Equations 2 (2002), 1–52.
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Saichev, A.I., Woyczyński, W.A. (2013). Other Standard Nonlinear Models of Higher Order. In: Distributions in the Physical and Engineering Sciences, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4652-3_7
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DOI: https://doi.org/10.1007/978-0-8176-4652-3_7
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