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On the ill-posedness of some canonical dispersive equations. (English) Zbl 1034.35145
This paper is concerned with the local well-posedness of the initial value problem for some classical nonlinear dispersive equations on the real line, namely \[ i\partial_tu+\partial_x^2 u\pm| u|^2 u= 0 \] (cubic nonlinear Schrödinger), \[ \partial_tu+ \partial^3_x u+ | u|^2\partial_xu=0 \] (complex modified Korteweg-de Vries), \[ \partial_t u+\partial_x^3 u+u^k\partial_xu=0,\;k\in \mathbb{Z}^+ \] (\(k\)-generalized Korteweg-de Vries). The authors establish the minimum regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces they show ill-posedness results for Sobolev index above the value suggested by the scaling argument.

MSC:
35R25 Ill-posed problems for PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
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