Global smooth solution to a coupled Schrödinger system in atomic Bose-Einstein condensates with two-dimensional spaces

Abstract

We obtain the global smooth solution of a nonlinear Schrödinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global existence and uniqueness of the smooth solution.

Keywords

Schrödinger equation / Galerkin method / a priori estimate / global smooth solution

References

[1] Adhikari S K. Numerical study of the spherically symmetric Gross-Pitaevskii equation in two space dimensions. Phys Rev E, 2000, 62(2): 2937–2944

[2] Adhikari S K, Muruganandam P. Bose-Einstein condensation dynamics from the numerical solution of the Gross-Pitaevskii equation. J Phys B, 2002, 35: 2831–2847

[3] Bao W Z, Jaksch D, Markowich P A. Numerical solution of the Gross-Pitaevskii equation for Boes-Einstein condensation. J Comput Phys, 2003, 187: 318–342

[4] Bao W Z, Tang W J. Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional. J Comput Phys, 2003, 187: 230–254

[5] Gross E P. Structure of a quantized vortex in boson systems. Nuovo Cimento, 1961, 20: 454–477

[6] Guo B L. The global solution for some systems of nonlinear Schrödinger equations. In: Chern S S, Wu W, eds. Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol 3. Beijing: Science Press, 1982, 1227–1246

[7] Guo B L. The initial and periodic value problem of one class nonlinear Schrödinger equations describing excitons in molecular crystals. Acta Math Sci Ser B Engl Ed, 1982, 2(3): 269–276

[8] Guo B L. The initial value problems and periodic boundary value problem of one class of higher order multi-dimensional nonlinear Schrödinger equations. Chin Sci Bull, 1982, 6: 324–327

[9] Guo B L. Nonlinear Evolution Equations.Shanghai: Shanghai Scientific and Technological Education Publishing House, 1985 (in Chinese)

[10] Lee M D, Morgan S A, Davis M J, Burnett K. Energy-dependent scattering and the Gross-Pitaevskii equation in two-dimensional Bose-Einstein condensates. J Phys Rev A, 2002, 65: 043617–043638

[11] Pérez-Garca V M, Michinel, H J, Cirac I, Lewenstein M, Zoller P. Dynamics of Bose-Einstein condensates: Variational solutions of the Gross-Pitaevskii equations. Phys Rev A, 1997, 56(2): 1424–1532

[12] Sadhan K, Adhikari. Numerical solution of the two-dimensional Gross-Pitaevskii equation for trapped interacting atoms. Phys Lett A, 2000, 265: 91–96

[13] Timmermans E, Tommasini P, Hussein M, Kerman A. Feshbach resonances in atomic Bose-Einstein condensates. Phys Rep, 1999, 315: 199–230

[14] Wang T C, Zhao X F. Optimal l^∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions. Sci China Math, 2014, 57(10): 2189–2214

[15] Zhou Y, Guo B L. Periodic boundary problem and initial value problem for the generalized Korteweg-de Vries systems of higher order. Acta Math Sinica (Chin Ser), 1984, 27: 154–176 (in Chinese)

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Higher Education Press and Springer-Verlag Berlin Heidelberg