The Cauchy problem for the nonlinear Schrödinger equations involving derivative terms in one spatial dimension

• Autorzy: Baoxiang W.
• Języki publikacji: EN

Abstrakty

EN

This paper is devoted to the study of the Cauchy problem for the nonlinear Schrödinger equations involving derivative terms. By introducing a generalized gauge transformation, we give some sufficient conditions for the global well-posedness of solutions in the energy space.

Słowa kluczowe

EN

Schrödinger equation; Cauchy problem

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
• Rocznik: 2003
• Tom: [Z] 43(2)
• Strony: 275--300
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Baoxiang W.

• Department of Mathematics, Peking University, Beijing 100871 People's Republic of China

Bibliografia

• [1] H. H. Chen, Y. C. Lee and C. S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scripta 20 (1979), 490-492.
• [2] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, J. of Funct. Anal. 32 (1979), 1-71.
• [3] B. L. Guo and S. B. Tan, On smooth solution to the initial value problem for the mixed nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh 119 (1991), 31-45.
• [4] N. Hayashi, Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J. 60 (1990), 717-727.
• [5] N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equations, Nonlinear Anal. Th. Meth. Appl. 20 (1993), 823-833.
• [6] N. Hayashi and T. Ozawa On the derivative nonlinear Schrödinger equations, Phys. D. 55 (1992), 14-36.
• [7] N. Hayashi and T. Ozawa Modified wave operators for the derivative nonlinear Schrödinger eguations, Math. Ann. 298 (1994), 557-576.
• [8] N. Hayashi and T. Ozawa Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal. 25 (1994), 1488-1503.
• [9] T. Kato On nonlinear Schrödinger equations, Ann. Inst. H. Poincare Phys. Theor. 46 (1986), 113-129.
• [10] D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys. 19 (1978), 798-801.
• [11] C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincare Anal. Nonlineaire 10 (1993), 255-288.
• [12] A. Kundu, Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations, J. Math. Phys. 25 (1984), 3433-3438.
• [13] A. Kundu, Exact solutions to higher-order nonlinear eguations through gauge transformation, Phys. D. 25 (1987), 399-406.
• [14] J. H. Lee, Global solvability of the derivative nonlinear Schrödinger equation, Trans. Math. Soc. 314 (1989), 107-118.
• [15] E. Mjolhus, Nonlinear Alfven waves and the DNLS equation: Oblique aspects, Phys. Scripta 40 (1989), 227-237.
• [16] T. Ozawa, Long range scattering for the nonlinear nonlinear Schrödinger equation in one space dimension, Comm. Math. Phys. 139 (1991), 479-493.
• [17] T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indian. Univ. Math. J. 45 (1996), 137-163.
• [18] S. B. Tan and L. H. Zhang, On a weak solution of the mixed nonlinear Schrödinger equations, J. Math. Anal. Appl. 182 (1994), 409-421.
• [19] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equations, Funkcial. Ekvac. 23 (1980), 259-277 and 24 (1981) 85-94.
• [20] Wang Baoxiang, On the Cauchy problem for the critical and subcritical nonlinear Schrödinger equations in Hs, Chinese Advan. in Math. 25 (1996), 471-472.
• [21] Wang Baoxiang, On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in Hs, Nonlinear Anal. Th. Meth. Appl. 31 (1998), 573-587.
• [22] Wang Baoxiang, Classical global solutions for nonlinear Klein-Gordon-Schrödinger equations, Math. Meth. in the Appl. Sci. 20 (1997), 599-616.