The geometry of the two-component Camassa–Holm and Degasperis–Procesi equations

Abstract

Abstract. We use geometric methods to study two natural twocomponent generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff(S 1 ) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm equation, giving explicit examples of large subspaces of positive curvature.

Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system

“…Several types of 2-component Camassa-Holm equations have been studied in [41][42][43][44][45][46][47][48][49][50][51]7]. Several types of 2-component Camassa-Holm equations have been studied in [41][42][43][44][45][46][47][48][49][50][51]7].…”

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

“…The DP equation belongs to the class of non-metric Euler equations [17,24]. An analogous geometric framework has been established for the 2CH equation without vorticity in [12,20] where the authors showed that it can be recast as a geodesic flow on the semidirect product Diff ∞ (S) C ∞ (S) equipped with the H 1 metric for the first component plus the L 2 metric for the second component.…”

Continuity for a generalized cross-coupled Camassa–Holm system with waltzing peakons and higher-order nonlinearities

“…The 2DP was shown to have solitons, kink, and antikink solutions [57]. Escher, Kohlmann and Lenells studied the geometric properties of the 2DP and local well-posedness in various function spaces [19]. However, peakon and superposition of multi-peakons were not investigated yet.…”