Curvature computations for a two-component Camassa-Holm equation with vorticity

Abstract

Abstract. In the present paper, a two-component Camassa-Holm (2CH) system with vorticity is studied as a geodesic flow on a suitable Lie group. The paper aims at presenting various details of the geometric formalism and a major result is the computation of the sectional curvature K of the underlying configuration manifold. As a further result, we show that there are directions for which K is strictly positive and bounded away from zero.

Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy

“…For instance, its relevance to shallow water theory is discussed in [10], [37], well-posedness and blow-up are studied in [10], [16], [19], [28], [41], various types of solutions (global, dissipative, conservative, etc.) are treated in [16], [24]- [27], [29], the inverse scattering transform is applied to the CH-2 system in [12], [34], N solitary waves are discussed in [10], [33], [34], [45], traveling waves are studied in [47], [49], the geometry of CH-2 is investigated in [17], [33], [34], [42], the periodic CH-2 system is discussed in [25], [36], [51]. For connections to other integrable systems see [3], [8], [18].…”