Smoothing effects for some derivative nonlinear schrödinger equations

Discrete and Continuous Dynamical Systems

Volumn 5, Issue 3, 1999, Pages 685-695

  Hayashi, Nakao ^a   Naumkin, Pavel I ^b   Pipolo, Patrick Nicolas ^a,c  

^a TOKYO UNIVERSITY OF SCIENCE   (Japan)

^b UNIVERSIDAD MICHOACANA DE SAN NICOLÁS DE HIDALGO   (Mexico)

^c UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Derivative type; Nonlinear Schr dinger; Smoothing effects

Indexed keywords

EID: 0033420172     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.1999.5.685     Document Type: Article

References (22)

1. H. Chihara, Local existence for the semilinear Schrödinger equations in one space dimension, J. Math. Kyoto Univ. 34 (1994), 353-367.
2. H. Chihara, The initial value problem for the elliptic-hyperbolic Davey-Stewartson equation, preprint.
3. P. Constantin and J.C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413-439.
4. W. Craig, T. Kappeler and W. Strauss, Gain of regularity for equations of KdV type, Ann. Inst. Henri Poincaré, Analyse non linéaire 9 (1992), 147-186.
5. S. Doi, On the Cauchy problem for Schrödinger type equations and regularity of solutions, J. Math. Kyoto Univ. 34 (1994), 319-328.
6. A. Friedman, Partial Differential Equations, Krieger, 1983.
7. N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Analysis TMA 20 (1993), 823-833.
8. N. Hayashi and H. Hirata, Global existence and asymptotic behavior in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity 9 (1996), 1387-1409.
9. N. Hayashi, K. Nakamitsu and M. Tsutsumi, On solutions of the initial value problem for non-linear Schrödinger equations in one space dimension, Math. Z. 192 (1986), 637-650.
10. N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equations, Physica D 55 (1992), 14-36.
11. N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension, Diff. Integral Eqs. 7 (1994), 453-461.
12. S. Katayama and Y. Tsutsumi, Global existence of solutions for nonlinear Schrödinger equations in one space dimension, Commun, in P.D.E. 19 (1994), 1971-1997.
13. T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Stud. Appl. Math. Adv. in Math. Supplementary Studies 18 (1983), 93-128.
14. C.E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dipersive equations, Indiana Univ. Math. J. 40 (1991), 33-69.
15. C.E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré Analyse non linéare 10 (1993), 255-288.
16. V.G. Makhankov and O.K. Pashaev, Integrable pseudospin models in condensed matter, Sov. Sci. Rev. C. Math. Phys. 9 (1992), 1-152.
17. T. Ozawa, Smoothing effects and dispersion of singularities for the Schrödinger evolution group, Arch. Rat. Mech. Anal. 110 (1990), 165-186.
18. T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), 137-163.
19. P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987), 699-715.
20. A. Soyeur, The Cauchy problem for the Ishimori equations, J. Funct. Anal. 105 (1992), 233-255.
21. E.M. Stein, Singular Integral and Differentiability Properties of Functions, Princeton Univ. Press, Princeton Math. Series 30, 1970.
22. L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878.