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Discontinuity, Nonlinearity, and Complexity 13(2) (2024) 279--289 | DOI:10.5890/DNC.2024.06.006

Yu. A. Chirkunov

Department Novosibirsk State University of Architecture and Civil Engineering (Sibstrin), 113 Leningradskaya st., 630008 Novosibirsk, Russa

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Abstract

In this paper, we study a system of shallow water equations with a rectilinear sloping bottom. Two fundamentally different cases are considered. In the first case, the Jacobi matrix of dependent variables with respect to independent variables is degenerate. This case has not been studied by anyone before. We have obtained all such solutions of the system. Each solution for specific values of its parameters is illustrated by a graph of the distribution of the excess of the free surface above the bottom and a graph of the distribution of the propagation velocity of a surface wave. The physical meaning of these solutions is indicated. In the second case, the Jacobi matrix of dependent variables with respect to independent variables is nondegenerate. This allows you to linearize the original system using a special hodograph transformation. Two exact solutions of this linear system are found: an invariant solution and a partially invariant solution. With the help of these solutions, we have obtained the exact solutions of the original system. Each solution for specific values of its parameters also is illustrated by a graph of the distribution of the excess of the free surface above the bottom and a graph of the distribution of the velocity of propagation of a surface wave. The physical meaning of these solutions is indicated. Using the method of ${\bm A}$-operators, we have found all zero-order conservation laws for the original system with an inclined bottom.

References

1. [1]	Stoker, J.J. (1958), Water Waves: The Mathematical Theory with Applications, John Wiley and Sons, New York.

2. [2]	Whitham, G.B. (1974), Linear and Nonlinear Waves, John Wiley and Sons, New York.

3. [3]	Kowalik, Z. (2012), Introduction to Numerical Modeling of Tsunami Waves, Fairbank: University of Alaska, p. 167.

4. [4]	Sepic, J., Vilibic, I., and Fine, I. (2015), Northern adriatic meteorological tsunamis: Assessment of their potential through ocean modeling experiments, Journal of Geophysical Research: Oceans, 120(4), 2993-3010.

5. [5]	Carrier, G.F. and Yeh, H. (2005), Tsunami propagation from a finite source, CMES - Computer Modeling in Engineering and Sciences, 10(2), 113-122.

6. [6]	Bonacci, O. and Oskorus, D. (2010), The changes in the lower Drava river water level, discharge and suspended sediment regime, Environmental Earth Sciences, 59, 1661-1670.

7. [7]	Singh, J., Altinakar, M.S., and Ding, Y. (2015), Numerical modeling of rainfall-generated overland flow using nonlinear shallow-water equations, Journal of Hydrologic Engineering, 20(8), 04014089.

8. [8]	Hu, K., Mingham, C.G., and Causon, D.M. (2000), Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations, Coastal Engineering, 41(4), 433-465.

9. [9]	Jeong, W. (2015), A study on simulation of flood inundation in a coastal urban area using a two-dimensional well-balanced finite volume model, Natural Hazards, 77(1), 337-354.

10. [10]	Gioia, G. and Bombardelli, F.A. (2001), Scaling and similarity in rough channel flows, Physical Review Letters, 88(1), p.014501.

11. [11]	Stoker, J.J. (1948), The formation of breakers and bores the theory of nonlinear wave propagation in shallow water and open channels, Communications on Pure and Applied Mathematics, 1(1),1-87.

12. [12]	Chirkunov, Y.A., Dobrokhotov, S.Y., Medvedev, S.B., and Minenkov, D.S. (2014), Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms, Theoretical and Mathematical Physics, 178, 278-298.

13. [13]	Carrier, G.F. and Greenspan, H.P. (1958), Water waves of finite amplitude on a sloping beach, Journal of Fluid Mechanics, 4(1), 97-109.

14. [14]	Tuck, E.O. and Hwang, L.S. (1972), Long wave generation on a sloping beach, Journal of Fluid Mechanics, 51(3), 449-461.

15. [15]	Pelinovsky, E.N. and Mazova, R.K. (1992), Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles, Natural Hazards, 6, 227-249.

16. [16]	Aksenov, A.V., Dobrokhotov, S.Y., and Druzhkov, K.P. (2018), Exact step-like solutions of one-dimensional shallow-water equations over a sloping bottom, Mathematical Notes, 104, 915-921.

17. [17]	Ugboh, J.A. and Esuabana, I.M. (2019), Marching method: a new numerical method for finding roots of algebraic and transcendental equations, American Journal of Computational and Applied Mathematics, 9(1), 6-11.

18. [18]	Ovsyannikov, L.V. (1982), Group Analysis of Differential Equations, Academic Press, New York, 1-399.

19. [19]	Chirkunov, Yu.A. and Khabirov, S.V. (2012), The Elements of Symmetry Analysis of Differential Equations of Continuous Medium Me Chanics, NSTU, Novosibirsk, 21659 (in Russian).

20. [20]	Chirkunov, Yu.A. (2009), Method of a-operators and conservation laws for the equations of gas dynamics, Journal of Applied Mechanics and Technical Physics, 50, 213-219.