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

TL;DR

This paper investigates the well-posedness of the fifth order modified KdV equation in Sobolev spaces, establishing local well-posedness for s ≥ 3/4 and demonstrating ill-posedness below this threshold through approximation techniques.

Contribution

It provides the first rigorous analysis of the well-posedness threshold for the fifth order mKdV, including both positive results and ill-posedness evidence.

Findings

Local well-posedness in H^s for s ≥ 3/4

Failure of uniform continuity of the solution map below H^{3/4}

Ill-posedness demonstrated via approximation by cubic NLS

Abstract

We consider the initial value problem of the fifth order modified KdV equation on the Sobolev spaces. \partial_t u - \partial_x^5u + c_1\partial_x^3(u^3) + c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0, u(x,0)= u_0(x) where $ u:R\timesR \to R $ and $c_j$'s are real. We show the local well-posedness in H^s(R) for s \geq 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 $H^{3/4}(R)$. The counter example is obtained by approximating the fifth order mKdV equation by the cubic NLS equation.