Electron. J. Diff. Eqns., Vol. 2008(2008), No. 01, pp. 1-15.

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

Soonsik Kwon

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

Submitted November 7, 2007. Published January 2, 2008.
Math Subject Classifications: 35Q53.
Key Words: Local well-posedness; ill-posedness; mKdV hierarchy.

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Soonsik Kwon
Department of Mathematics, University of California
Los Angeles, CA 90095-1555, USA
email: rhex2@math.ucla.edu

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