On the davey-stewartson and ishimori systems

Mathematical Physics, Analysis and Geometry

Volumn 2, Issue 1, 1999, Pages 53-81

  Hayashi, Nakao ^a  

^a TOKYO UNIVERSITY OF SCIENCE   (Japan)

Author keywords

Davey stewartson system; Elliptic hyperbolic case; Ishimori system; Nonlocal nonlinear schr dinger equation

Indexed keywords

EID: 0002424103     PISSN: 13850172     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1009875019021     Document Type: Article

References (34)

1. Ablowitz, J. M. and Haberman, R.: Nonlinear evolution equations in two and three dimensions, Phys. Rev. Lett. 35 (1975), 1185-1188.
2. Ablowitz, J. M. and Fokas, A. S.: On the inverse scattering transform of multidimensional nonlinear equations, J. Math. Phys. 25 (1984), 2494-2505.
3. Ablowitz, J. M. and Segur, A.: Solitons and Inverse Scattering Transform, PA, SLAM, 1981.
4. Anker, D. and Freeman, N. C.: On the soliton solutions of the Davey-Stewartson equation for long waves, Proc. R. Soc. A 360 (1978), 529-540.
5. Bardos, C., Sulem, C. and Sulem, P. L.: On the continuous limit for a system of classical spins, Comm. Math. Phys. 107 (1986), 431-454.
6. Benney, D. J. and Roskes, G. L.: Wave instabilities, Stud. Appl. Math. 48 (1969), 377-385.
7. Beals, R. and Coifman, R. R.: The spectral problem for the Davey-Stewartson and Ishimori hierarchies, Proc. Conf. on Nonlinear Evolution Equations: Integrability and Spectral Methods, Manchester, U.K., 1988.
8. Chihara, H.: The initial value problem for the elliptic-hyperbolic Davey-Stewartson equation, Preprint, 1995.
9. Constantin, P.: Decay estimates of Schrödinger equations, Comm. Math. Phys. 127 (1990), 101-108.
10. Cornille, H.: Solutions of the generalized nonlinear Schrödinger equation in two spatial dimensions, J. Math. Phys. 20 (1979), 199-209.
11. Davey, A. and Stewartson, K.: On three-dimensional packets of surface waves, Proc. R. Soc. A 338(1974), 101-110.
12. Djordjevic, V. D. and Redekopp, L. G.: On two-dimensional packets of capillary-gravity waves, J. Fluid Mech. 79 (1977), 703-714.
13. Doi, S.: On the Cauchy problem Schrödinger type equations and regularity of solutions, J. Math. Kyoto Univ. 34 (1994), 319-328.
14. Fokas, A. S. and Santini, P. M.: Recursion operators and bi-Hamiltonian structures in multidimensions. I, II, Comm. Math. Phys. 115 (1988), 375-419, 449-474.
15. Fokas, A. S. and Sung, L. Y.: On the solvability of the AT-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations, Inverse Problems 8 (1992), 673-708.
16. Ghidaglia, J. M. and Saut, J. C.: On the initial value problem for the Davey-Stewartson systems, Nonlinearity 3 (1990), 475-506.
17. Hayashi, N.: Local existence in time of small solutions to the Davey-Stewartson system, Annales de l'I.H.P. Physique Théorique 65 (1996), 313-366.
18. Hayashi, N.: On the initial value problem for the Davey-Stewartson and the Ishimori systems, in The Proceedings of the 4th MSJ International Reserch Institute on "Nonlinear Waves", Sapporo, July 10-21, 1995, GAKUTO International Series, Mathematical Sciences and Applications, Gakkotosho, vol. 10, 1997, pp. 145-154.
19. Hayashi, N.: Local existence in time of solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data, J. Analysé Mathématique 73 (1997), 133-164.
20. Hayashi, N.: Local existence in time of small solutions to the Ishimori system, Diff. Integral Eqs., to appear.
21. Hayashi, N. and Hirata, H.: Global existence and asymptotic behavior in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity 9 (1996), 1387-1409.
22. Hayashi, N. and Hirata, H.: Local existence in time of small solutions to the elliptichyperbolic Davey-Stewartson system in the usual Sobolev space, Proceedings of the Edinburgh Mathematical Society 40 (1997), 563-581.
23. Hayashi, N. and Naumkin, P. I.: Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited, Discrete and Continuous Dynamical Systems 3 (1997), 383-400.
24. Hayashi, N. and Saut, J. C.: Global existence of small solutions to the Davey-Stewartson and the Ishimori systems, Diff. Integral Eqs. 8 (1995), 1657-1675.
25. Hayashi, N. and Saut, J. C.: Global existence of small solutions to the Ishimori system without exponential decay of the data, Diff. Integral Eqs. 9 (1996), 1183-1195.
26. Ishimori, Y.: Multi vortex solutions of a two dimensional nonlinear wave equation, Progr. Theor. Phys. 72 (1984), 33-37.
27. Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Appl. Math. 8 (1983), 93-128.
28. Kenig, C. E., Ponce, G. and Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle, Comm. Pure Appl. Math. 46 (1993), 527-620.
29. Konopelchenko, B. G. and Matkarimov, B. T.: Inverse spectral transform for the nonlinear evolution equation generating the Davey-Stewartson and Ishimori equations, Stud. Appl. Math. 82(1990), 319-359.
30. Linares, F. and Ponce, G.: On the Davey-Stewartson systems, Ann. Inst. Henri Poincaré, Anal, non linéaire 10 (1993), 523-548.
31. Soyeur, A.: The Cauchy problem for the Ishimori equations, J. Funct. Anal. 105 (1992), 233-255.
32. Stein, E. M.: Singular Integral and Differentiability Properties of Functions, Princeton Univ. Press, Princeton Math. Series 30,1970.
33. Sung, L. Y.: An inverse-scattering transform for the Davey-Stewartson II equations, Part III, J. Math. Anal. Appl. 183 (1994), 477-494.
34. Sung, L. Y.: The Cauchy problem for the Ishimori equation, Preprint, 1993.