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Using the Abstract Cauchy-Kowalewski Theorem we prove that the $b$-familyequation admits, locally in time, a unique analytic solution. Moreover, if theinitial data is real analytic and it belongs to $H^s$ with $s > 3/2$, and themomentum density $u_0 - u_{0,{xx}}$ does not change sign, we prove that thesolution stays analytic globally in time, for $b\geq 1$. Using pseudospectralnumerical methods, we study, also, the singularity formation for the $b$-familyequations with the singularity tracking method. This method allows us to followthe process of the singularity formation in the complex plane as thesingularity approaches the real axis, estimating the rate of decay of theFourier spectrum.

Analytic solutions and Singularity formation for the Peakon b--Family equations