Journals Information
Universal Journal of Applied Mathematics Vol. 2(1), pp. 10 - 23
DOI: 10.13189/ujam.2014.020103
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Symmetry Properties and Solutions of Shallow Water Equations
Yu. A. Chirkunov 1,*, E. O. Pikmullina 2
1 Novosibirsk State Technical University, pr. K. Marksa 20, Novosibirsk, 630092, Russia
2 Institute of Computational Technologies, Siberian Branch, Russian Academy of Sciences,pr. Akademika Lavrentjeva 6, Novosibirsk, 630090 Russia
ABSTRACT
Within one-dimensional model of shallow water it is investigated a one-parameter family of equations describes the propagation of surface waves above a straight bottom. The parameter of the family is the slope of the bottom. This family is generated by the equations of one-dimensional model of shallow water with a horizontal bottom. By means of the method of A- operators it is found that this system has an infinite aggregate of non-trivial zero-order conservation laws generated by the system of linear differential equations. As a result of special choice of the hodograph transformation, the system of equations of one-dimensional model of shallow water with a horizontal bottom is generated by the same system of linear differential equations. The group analysis of the systems is carried out. An infinite aggregate of non-degenerate solutions of the equations of one-dimensional model of shallow water with a straight bottom is received. All of the degenerate solutions of these equations are found. Thus, the data base of exact solutions of the equations of one-dimensional model of shallow water with a straight bottom is created. The solutions obtained in this paper may be used in the study of tsunami waves and fluid distribution in channels.
KEYWORDS
Shallow Water, Conservation Laws, the Method of A -Operators, the Group Analysis of Differential Equations, Partially Invariant Solutions, Invariant Solutions, Non-Degenerate Solutions, Degenerate Solutions
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Yu. A. Chirkunov , E. O. Pikmullina , "Symmetry Properties and Solutions of Shallow Water Equations," Universal Journal of Applied Mathematics, Vol. 2, No. 1, pp. 10 - 23, 2014. DOI: 10.13189/ujam.2014.020103.
(b). APA Format:
Yu. A. Chirkunov , E. O. Pikmullina (2014). Symmetry Properties and Solutions of Shallow Water Equations. Universal Journal of Applied Mathematics, 2(1), 10 - 23. DOI: 10.13189/ujam.2014.020103.