The Initial Value Problem for Two Nonlinear Evolution Equations

We consider the initial value problem for two nonlinear evolution equations, first, the hyperelastic rod equation, which, under a certain choice of parameter, coincides with the Camassa-Holm equation and second, a higher-order modification of the Camassa-Holm equation. For the hyperelastic rod equation, we show that solutions to the periodic initial value problem do not depend uniformly continuously on initial data in Sobolev spaces of index s equal to 1 or s greater than or equal to 2. For the higher-order modification of the Camassa-Holm equation under consideration, we show that the non-periodic initial value problem is locally well posed for initial data in Sobolev spaces of index s greater than s' where s' is greater than or equal to 1/4 and less than 1/2 and the value of s' depends on the order of equation.