Electromagnetic Geophysical Fields pp 195-214 | Cite as

# A Model Description of Rolling Waves on Water

## Abstract

In this chapter we investigate, from the mathematical viewpoint, the problem of vertical hydrodynamic wave fronts (shock waves) arising as a result of nonlinear tsunami process development. As the model PDE (partial differential equation) we apply the Burgers equation of turbulence, modified by us with a term without differentiation. For this equation, we consider the vanishing coefficient of viscosity *ε* *→* 0 in the class of self-similar solutions, depending on the linear combination *x* = *y − ct* of the spatial (*y*) and temporal (*t*) arguments, where *c* = const is the coefficient; similar solutions are referred to by us as rolling wave solutions (rw-solutions). These solutions are smooth by *ε* > 0. The shock rw-solutions of the model PDE with *ε* = 0 are considered in the same class of the self-similar solutions depending on *x* but with an analog of the hydrodynamic conditions in the breakpoints, i.e., vertical fronts (known as the conditions on a jump). An arbitrary shock rw-solution *U*_{0}(*t*, *y*) is approximated with the smooth rw-solutions *U*_{ε}(*t*, *y*): *U*_{ε}(*t*, *y*) → *U*_{0}(*t*, *y*), *ε* → 0, according to the rule *Ф*(*U*_{ε}(*t*, *y*)) = *Ф* (*U*_{0}(*t*, *y*)), *ε >* 0, where *Ф* is the functional determined on *U*_{ε}(*t*, *y*)), *ε ≥* 0.

It was shown for the modified Burgers equation that the class of functionals allowing for an arbitrary fixed shock rw-solution to construct the approximating sequence of the smooth rw-solutions obeying this rule is not empty; *U*_{ε}(*t*, *y*) is determined uniquely by *ε >* 0.

## Keywords

Modified Burgers equation Vanishing viscosity Self-similar solutions## References

- Bec, J., & Khanin, K. (2007). Burgers turbulence.
*Physics Reports, 447(*(1–2), 1–66.CrossRefGoogle Scholar - Bianchini, S., & Bressan, A. (2005). Vanishing viscosity solution of non-linear hyperbolic systems.
*Annals of Mathematics, 161*, 223–342.CrossRefGoogle Scholar - Brezis, H. (2010).
*Functional analysis, Sobolev spaces and partial differential equations*(p. 599). Heidelberg: Springer. https://doi.org/10.1007/978-0-387-70914-7.CrossRefGoogle Scholar - Byhovsky, E. B. (1966).
*About the self-similar solutions of the type of a propagating wave of some quasi-linear differential equation and the system of equations describing the flow of water in a sloping channel*. (In Russian, Прикладная Математика и Механика, Prikladnaya Matematika i Mechanika, V. 30, Issue 2).Google Scholar - Calogero, F. (2008).
*Isochronous systems*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Chouikha, A. R. (2005). Monotonicity of the period function for some planar differential systems. Part I: Conservative and quadratic systems.
*Applicationes Mathematicae, 32*(3), 305–325.CrossRefGoogle Scholar - Colombeau, J. F., & Le Roux, A. Y. (1988). Multiplication of distributions in elasticity and hydrodynamics.
*Journal of Mathematical Physics, 29(*(2), 315–319.CrossRefGoogle Scholar - Danford, N., & Schwartz, J. (1988).
*Linear operators, part I: General theory*(Vol. I, p. 872). Hoboken: Wiley.Google Scholar - Dinjun, L., & Libang, T. (2013).
*Qualitative theory of dynamical systems*,*Advanced series in dynamical systems*. Singapore: World Scientific. https://doi.org/10.1142/1914. p. 272.CrossRefGoogle Scholar - Dressler, R. F. (1949). Mathematical solution of the problem of roll-waves in inclined open channels.
*Communications on Pure and Applied Mathematics, 2*(2–3), 149–194.CrossRefGoogle Scholar - Ginoux, J.-M., & Litellier, C. (2012). Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept.
*Chaos: An Interdisciplinary Journal of Nonlinear Science, 22*(2), 023120.CrossRefGoogle Scholar - Golitsyn, G. S. et al. (eds). (2014).
*Turbulence, atmosphere and climate dynamics*. In Collected papers of the international conference dedicated to the memory of academician A.M. Obukhov (p. 696), Moscow, GEOS.Google Scholar - Henkin, G. M., & Shananin, A. A. (2014). Cauhy–Gelfand problem for quasilinear conservation law.
*Bulletin des Sciences Mathématiques, 138*, 783–804.CrossRefGoogle Scholar - Ladyjenskaya, O. A. (1969).
*The mathematical theory of viscous incompressible flow*. New York/London: Gordon and Breach, Science Publications. (Translated from Russian, Moscow: Gos. Izdat. Fiz-Mat Lit. 1961).Google Scholar - Landau, L. D., & Lifshitz, E. M. (1987).
*Fluid mechanics*(Vol. 6, 2nd ed.). Oxford: Butterworth–Heinemann.Google Scholar - LeVeque, R. (2002).
*Finite volume methods for hyperbolic problems*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Novik, O. B. (1971).
*Model description of rolling waves*. (In Russian: Прикладная математика и механика, Vol.35, Issue 6.*Journal of RAS*, Moscow. English Translation of the Journal:*Applied Mathematics and Mechanics*. Elsevier).Google Scholar - Oleinik, O. A. (1963). Discontinuous solutions of nonlinear differential equations.
*American Mathematical Society Translations: Series, 2*(26), 95–172.Google Scholar - Rossinger, E. E. (1987).
*Generalized solutions of nonlinear differential equations*. Amsterdam: Elsevier.Google Scholar - Saks, P. (2017).
*Techniques of functional analysis for differential and integral equations*(p. 320). Amsterdam: Elsevier.Google Scholar - Simon, P. L. (2012).
*Differential equations and dynamical systems*. Budapest: Department of Applied Analysis and Computational Mathematics, Institute of Mathematics, Eötvös Lorand University.Google Scholar - Sternberg, S. (2013).
*Dynamical systems*. New York: Dover.Google Scholar - Vorobyev, A. P. (1962). About periods of solutions in the case of a center (In Russian).
*Doklady Academy of Science of Belorusskoy SSR 6*(5).Google Scholar - Wesseling, P. (2001).
*Principles of computational fluid dynamics*. Heidelberg: Springer.CrossRefGoogle Scholar