Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis On the ill-posedness of some canonical dispersive equations. (English) Zbl 1034.35145 Duke Math. J. 106, No. 3, 617-633 (2001). 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. Reviewer: Viorel Iftimie (Bucureşti) Cited in 4 ReviewsCited in 174 Documents 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) Keywords:generalized Korteweg-de Vries; local well-posedness; nonlinear Schrödinger; modified Korteweg-de Vries; Sobolev spaces; ill-posedness; Sobolev index PDF BibTeX XML Cite \textit{C. E. Kenig} et al., Duke Math. J. 106, No. 3, 617--633 (2001; Zbl 1034.35145) Full Text: DOI References: [1] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems , Stud. Appl. Math. 53 (1974), 249–315. · Zbl 0408.35068 [2] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I, II , Arch. Rational Mech. Anal. 82 (1983), 313–345., 347–375. · Zbl 0533.35029 [3] B. Birnir, C. E. Kenig, G. Ponce, N. Svanstedt, and L. 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