WELL-POSEDNESS AND ILL-POSEDNESS OF THE FIFTH-ORDER MODIFIED KDV EQUATION

Abstract

We consider the initial value problem of the fifth-order modified KdV equation on the Sobolev spaces. partial derivative(t)u - partial derivative(5)(x)u + c(1)partial derivative(3)(x)(u(3)) + c(2)u partial derivative(x)u partial derivative(2)(x)u + c(3)uu partial derivative(3)(x)u = 0 u(x,0) = u(0)(x) where u : R x R -> R and c(j)'s are real. We show the local well-posedness in H-s(R) for s >= 3/4 via the contraction principle on X-s,X-b space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below H-3/4(R). The counter example is obtained by approximating the fifth order mKdV equation by the cubic NLS equation.