Các dòng hình học mới trên đa tạp Riemann và ứng dụng cho các dòng Schrödinger-Airy

Bejenaru I, Ionescu A D, Kenig C E, et al. Global Schrödinger maps in dimensions d ⩾ 2: Small data in the critical Sobolev spaces. Ann Math, 2011, 173: 1443–1506 Chang N H, Shatah J, Uhlenbeck K. Schrödinger maps. Comm Pure Appl Math, 2000, 53: 590–602 Colliander J, Keel M, Staffilani G, et al. Sharp global well-posedness for KdV and modified KdV on ℝ and \(\mathbb{T}\). J Amer Math Soc, 2003, 16: 705–749 Date E, Jimbo M, Kashiwara M, et al. Landau-Lifshitz equation: Solitons, quasi-periodic solutions and infinite-dimensional Lie algebras. J Phys A, 1983, 16: 221–236 Ding Q, Wang Y D. Geometric KdV flows, motions of curves and the third-order system of the AKNS hierarchy. Internat J Math, 2011, 22: 1013–1029 Ding W Y. On the Schrödinger flows. Proc ICM Beijing, 2002, 283–292 Ding W Y, Wang Y D. Schrödinger flows of maps into symplectic manifolds. Sci China Ser A, 1998, 41: 746–755 Ding W Y, Wang Y D. Local Schrödinger flow into Kähler manifolds. Sci China Ser A, 2001, 44: 1446–1464 Friedman A. Partial Differential Equations of Parabolic Type. Englewood Cliffs, NJ: Prentice-Hall Inc., 1964 Fukumoto Y, Miyazaki T. Three-dimensional distortions of a vortex filament with axial velocity. J Fluid Mech, 1991, 222: 369–416 Gerdjikov V S, Kostov N A. Reductions of multicomponent mKdV equations on symmetric spaces of DIII-type. Symmatry Integrability Geom Methods Appl, 2008, 4: 29–58 Golubchik I Z, Sokolov V V. Multicomponent generalization of the hierarchy of the Landau-Lifshitz equation. Theo Math Phys, 2000, 124: 909–917 Hasegawa A, Kodama Y. Nonlinear pulse propagation in a monomode dielectric guide. IEEE J Quantum Elec, 1987, 23: 510–524 Hasimoto H. A soliton on a vortex filament. J Fluid mech, 1972, 51: 477–485 Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces. Providence, RI: Amer Math Soc, 2001 Hirota R. Exact envelope-soliton solutions of a nonlinear wave equation. J Math Phys, 1973, 14: 805–809 Meshkov A G, Sokolov V V. Integrable evolution equations on the n-dimensional sphere. Comm Math Phys, 2002, 232: 1–18 Moffatt H K, Ricca R L. Interpretation of invariants of the Betchov-Da Rios equations and of the Euler equations. Global Geom Turbulence, 1991, 268: 257–264 Nishiyama T, Tani A. Initial and initial-boundary value problems for a vortex filament with or without axial flow. SIAM J Math Anal, 1996, 27: 1015–1023 Onodera E. A third-order dispersive flow for closed curves into Kähler manifolds. J Geom Anal, 2008, 18: 889–918. Petersen P. Riemannian Geometry. New York: Springer Verlag, 1998 Ricca R L. Rediscovery of Da Rios equations. Nature, 1991, 352: 561–562 Rodnianski I, Rubinstein Y A, Staffilani G. On the global well-posedness of the one-dimensional Schrodinger map flow. Analysis PDE, 2009, 2: 187–209 Song C. The KdV curve and Schrödinger-Airy curve. Proc Amer Math Soc, 2012, 140: 635–644 Song C, Yu J. The Cauchy problem of generalized Landau-Lifshitz equation into S n. Sci China Math, 2013, 56: 283–300 Sun X W, Wang Y D. KdV geometric flows on Kähler manifolds. Internat J Math, 2011, 22: 1–62 Sun X W, Wang Y D. Geometric Schrödinger-Airy Flows on Kähler Manifolds. Acta Math Sin (Engl Ser), 2013, 29: 209–240 Tani A, Nishiyama T. Solvability of equations for motion of a vortex filament with or without axial flow. Publ Res Inst Math Sci, 1997, 33: 509–526 Taylor M E. Partial Differential Equations III: Nonlinear Equations. New York: Springer-Verlag, 1997 Tsuchida T. New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems. J Math Phys, 2010, 51: 053511–27 Tsuchida T, Wadati M. The coupled modified Korteweg-de Vries equations. J Phys Soc Japan, 1998, 67: 1175–1187 Veselov A P. Finite-gap potentials and integrable systems on the sphere with a quadratic potential. Funct Anal Appl, 1980, 14: 48–50 Wang B, Han L, Huang C. Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data. Ann Inst H Poincaré Anal Non Linéaire, 2009, 26: 2253–2281