| Abstract: |
| We investigate the Cauchy problem for the Camassa-Holm (CH) equation with a small dispersion parameter $\epsilon>0$.
Under a negative analytic initial data, we prove that before a certain time, the solution could be well approximated
by the solution to Hopf equation which corresponds to $\epsilon=0$. Near the point of gradient catastrophe to the Hopf equation,
we show that the solution of CH equation is approximated by a particular Painlev\`e transcendent in the double scaling limit.
This proves the validity of the Dubrovin conjecture concerning the critical asymptotics for a broad class of Hamiltonian perturbative
hyperbolic equations. We established our results by performing the steepest descent analysis to an associated Riemann-Hilbert problem. |
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