×

zbMATH — the first resource for mathematics

The local ill-posedness of the modified KdV equation. (English) Zbl 0858.35130
Summary: We find a new method for proving the local ill-posedness of the Cauchy problem for nonlinear partial differential equations. The method is used to prove that the Cauchy problem for the modified KdV equation is ill-posed in Sobolev spaces \(H^s(\mathbb{R})\), \(s<-1/2\).

MSC:
35R25 Ill-posed problems for PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Tables of Integral Transforms, Vol. I, (1954), Mc Graw-Hill New York
[2] B. Birnir, Kenig C. E., Ponce G., Svanstead N. and Vega L., On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, to appear in the Journal of the London Math. Soc., 1996.
[3] Gardner, C. S., Korteweg de Vries equation and generalizations IV. the Korteweg de Vries equation as a Hamiltonian system, J. Math. Phys., Vol. 12, 1548-1551, (1971) · Zbl 0283.35021
[4] Glassey, R. T., On the blowing-up solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., Vol. 18, 1794-1797, (1979) · Zbl 0372.35009
[5] Gardner, C. S.; Greene, H.; Kruskal, M. D.; Miura, R. M., Methods for solving the Korteweg-de Vries equation, Physical Review Letters, Vol. 19, 1095-1097, (1967) · Zbl 1103.35360
[6] Kenig, C. E.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle, Comm. Pure Appl. Math., Vol. 46, 527-620, (1993) · Zbl 0808.35128
[7] Miura, R. M., Korteweg-de Vries equation and generalizations I. A remarkable explicit nonlinear transformation, J. Math. Phys., Vol. 9, 1202-1204, (1968) · Zbl 0283.35018
[8] Rauch, J., Nonlinear superposition and absorption of delta waves in one space dimension, J. Funct. Anal., Vol. 73, 152-178, (1987) · Zbl 0661.35058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.