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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 7

### The symmetric structure of Green–Naghdi type equations

Pages: 1925 – 1946

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n7.a7

#### Author

#### Abstract

The notion of symmetry classically defined for hyperbolic systems of conservation laws is extended to the case of evolution equations of conservative form for which the flux function can be an operator. We explain how such a symmetrization can work from a general point of view using an extension of the classical Godunov structure. We then apply it to the Green–Naghdi type equations which are a dispersive extension of the hyperbolic shallow-water equations. In fact, in the case of these equations, the general Godunov structure of the system is obtained from its Hamiltonian structure.

#### Keywords

symmetric systems, conservation law, strict convexity, variational derivative, Green–Naghdi equations

#### 2010 Mathematics Subject Classification

35L65, 35Q35, 37L50, 70S10