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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 $$i\partial_tu+\partial_x^2 u\pm\vert u\vert^2 u= 0$$ (cubic nonlinear Schrödinger), $$\partial_tu+ \partial^3_x u+ \vert u\vert^2\partial_xu=0$$ (complex modified Korteweg-de Vries), $$\partial_t u+\partial_x^3 u+u^k\partial_xu=0,\ k\in \bbfZ^+$$ ($k$-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.

35R25Improperly posed problems for PDE
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35Q51Soliton-like equations
35Q53KdV-like (Korteweg-de Vries) equations
Full Text: DOI
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