Local isometric immersions and breakdown of manifolds determined by Cauchy problems of the Degasperis-Procesi equation (Q6635934)

The author discusses local and non-local formulations of the Degasperis-Procesi (DP) equation \(u_t - u_{txx} +(b+1) u u_x = b u_x u_{xx}+u u_{xxx}\), where \(b\) is a real constant. The author describes when solutions in the Sobolev class of the DP equation may describe surfaces of constant and negative Gaussian curvatures. It is shown that any non-trivial initial datum in a certain Sobolev class gives rise to a dual coframe of one-forms defined on a simply connected subset contained in a strip uniquely determined by the initial datum. Any non-trivial, compactly supported, initial datum gives rise to a surface that is only defined on the right side of a certain curve determined by the initial datum. Solutions emanating from non-trivial initial data allow for the existence of a dual coframe for which the coefficients of the corresponding connection forms are independent of a particular solution. Blow-up scenario and a global solution are investigated.