Well-posedness and ill-posedness of the fifth-order modified KdV equation

We consider the initial value problem of the fifth-order modified KdV equation on the Sobolev spaces. $$displaylines{ partial_t u - partial_x^5u + c_1partial_x^3(u^3) + c_2upartial_x upartial_x^2 u + c_3uupartial_x^3 u =0cr u(x,0)= u_0(x) }$$ where $u:mathbb{R} imesmathbb{R} o mathbb{R} $ and $c_j$'s are real. We show the local well-posedness in $H^s(mathbb{R})$ for $sgeq 3/4$ via the contraction principle on $X^{s,b}$ space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below $H^{3/4}(mathbb{R})$. The counter example is obtained by approximating the fifth order mKdV equation by the cubic NLS equation.