Advertisement

A Model Description of Rolling Waves on Water

  • Oleg Novik
  • Feodor Smirnov
  • Maxim Volgin
Chapter

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 U0(t, y) is approximated with the smooth rw-solutions Uε(t, y): Uε(t, y) → U0(t, y), ε → 0, according to the rule Ф(Uε(t, y)) = Ф (U0(t, y)), ε > 0, where Ф is the functional determined on Uε(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; Uε(t, y) is determined uniquely by ε > 0.

Keywords

Modified Burgers equation Vanishing viscosity Self-similar solutions 

References

  1. Bec, J., & Khanin, K. (2007). Burgers turbulence. Physics Reports, 447((1–2), 1–66.CrossRefGoogle Scholar
  2. Bianchini, S., & Bressan, A. (2005). Vanishing viscosity solution of non-linear hyperbolic systems. Annals of Mathematics, 161, 223–342.CrossRefGoogle Scholar
  3. 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
  4. 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
  5. Calogero, F. (2008). Isochronous systems. Oxford: Oxford University Press.CrossRefGoogle Scholar
  6. 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
  7. 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
  8. Danford, N., & Schwartz, J. (1988). Linear operators, part I: General theory (Vol. I, p. 872). Hoboken: Wiley.Google Scholar
  9. 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
  10. 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
  11. 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
  12. 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
  13. Henkin, G. M., & Shananin, A. A. (2014). Cauhy–Gelfand problem for quasilinear conservation law. Bulletin des Sciences Mathématiques, 138, 783–804.CrossRefGoogle Scholar
  14. 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
  15. Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics (Vol. 6, 2nd ed.). Oxford: Butterworth–Heinemann.Google Scholar
  16. LeVeque, R. (2002). Finite volume methods for hyperbolic problems. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  17. 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
  18. Oleinik, O. A. (1963). Discontinuous solutions of nonlinear differential equations. American Mathematical Society Translations: Series, 2(26), 95–172.Google Scholar
  19. Rossinger, E. E. (1987). Generalized solutions of nonlinear differential equations. Amsterdam: Elsevier.Google Scholar
  20. Saks, P. (2017). Techniques of functional analysis for differential and integral equations (p. 320). Amsterdam: Elsevier.Google Scholar
  21. 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
  22. Sternberg, S. (2013). Dynamical systems. New York: Dover.Google Scholar
  23. 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
  24. Wesseling, P. (2001). Principles of computational fluid dynamics. Heidelberg: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Oleg Novik
    • 1
  • Feodor Smirnov
    • 1
  • Maxim Volgin
    • 1
  1. 1.IZMIRAN of the Russian Academy of SciencesMoscowRussia

Personalised recommendations