Abstract
The present chapter studies behavior of two standard 1-D nonlinear dynamics models described by partial differential equations of order two and higher: the Burgers equation and the related KPZ model. We shall concentrate our attention on the theory of nonlinear fields of hydrodynamic type, where the basic features of the temporal evolution of nonlinear waves can be studied in the context of competition between the strengths of nonlinear and dissipative and/or dispersive effects. Apart from being model equations for specific physical phenomena, Burgers–KPZ equations are generic nonlinear equations that often serve as a testing ground for ideas for analysis of other nonlinear equations. They also produce a striking typically nonlinear phenomenon: shock formation.
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Notes
- 1.
Edwin Hubble was an astronomer who in the 1920s, discovered that the universe is continuously expanding, with galaxies moving away from each other.
- 2.
Note that the Reynolds number (53) is four times the size of the R previously introduced in (39). There is no inconsistency here, since the Reynolds number has a semiqualitative character and can be enlarged or reduced a little without changing the essence of its meaning. The definition of R in (39) was selected in a way that was convenient for the subsequent formulas. In what follows, we will stick to the latter definition.
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See also E. Zuazua, Weakly nonlinear large time behavior in scalar convection–diffusion equation, Differential and Integral Equations 6 (1993), 1481–1491.
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Saichev, A.I., Woyczyński, W.A. (2013). Nonlinear Waves and Growing Interfaces: 1-D Burgers–KPZ Models. 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_6
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DOI: https://doi.org/10.1007/978-0-8176-4652-3_6
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