Analyticity in a Dispersive Camassa–Holm Equation with Cubic Nonlinearities

The Cauchy problem of a third-order dispersive Camassa–Holm equation with cubic nonlinearities having initial data \(\varphi (x)\) in analytic spaces is studied. First, local well-posedness in analytic Gevrey spaces \(G^{\delta , s}(\mathbb {R})\) , \(s>1/2\) , is established by using trilinear estimates in analytic Bourgain spaces. Then, using the fact that solutions of this equation conserve their \(H^1\) -norm, an almost conservation law in the corresponding analytic Gevrey spaces is derived. Finally, using this almost conservation law, it is shown that the solution \(u(t)\) exists for all time and a lower bound for its radius of spatial analyticity of the form \(c_0/t e^{ct}\) is established, for some positive constants \(c_0\) and c.