Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice

We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k ! 0) our solutions model a quasi-one dimensional quantum degenerate Bose- Fermi mixture trapped in optical lattice. In the limit k ! 1 the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.

[1]  V. Gerdjikov,et al.  Two-component Bose-Einstein condensates in periodic potential. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  S. V. Manakov.  On the theory of two-dimensional stationary self-focusing of electromagnetic waves , 1973 .

[3]  M. Oberthaler,et al.  Dynamics of Bose-Einstein condensates in optical lattices , 2006 .

[4]  N. A. Kostov,et al.  Quasiperiodic and periodic solutions for vector nonlinear Schrödinger equations , 2000 .

[5]  Mario Salerno.  Matter-wave quantum dots and antidots in ultracold atomic Bose-Fermi mixtures , 2005 .

[6]  M. Wadati,et al.  Dynamics of Magnetically Trapped Boson-Fermion Mixtures , 2000 .

[7]  Sarah B. McKinney.  Dynamics of Bose-Einstein condensates in optical lattices , 2004 .

[8]  Bernard Deconinck,et al.  Dynamics of periodic multi-component Bose-Einstein condensates , 2003 .

[9]  M Inguscio,et al.  Fermi-Bose quantum degenerate 40K-87Rb mixture with attractive interaction. , 2002, Physical review letters.

[10]  Víctor M. Pérez-García,et al.  Modulational instability, solitons and periodic waves in a model of quantum degenerate boson–fermion mixtures , 2005 .

[11]  J R Anglin,et al.  Dark-bright solitons in inhomogeneous Bose-Einstein condensates. , 2001, Physical review letters.

[12]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[13]  F. Dalfovo,et al.  Theory of Bose-Einstein condensation in trapped gases , 1998, cond-mat/9806038.

[14]  R. Brecha,et al.  Collapse of a Degenerate Fermi Gas , 2002, Science.

[15]  M. Mewes,et al.  Sympathetic cooling of bosonic and fermionic lithium gases towards quantum degeneracy , 2001 .

[16]  E. Belokolos,et al.  Algebro-geometric approach to nonlinear integrable equations , 1994 .

[17]  Alexey V. Porubov,et al.  Some general periodic solutions to coupled nonlinear Schrödinger equations , 1999 .

[18]  Sadhan K. Adhikari.  Fermionic bright soliton in a boson-fermion mixture (7 pages) , 2005 .

[19]  J C Bronski,et al.  Bose-Einstein condensates in standing waves: the cubic nonlinear Schrödinger equation with a periodic potential. , 2001, Physical review letters.

[20]  V. M. Kenkre,et al.  Matter waves of Bose-Fermi mixtures in one-dimensional optical lattices (14 pages) , 2006 .

[21]  J C Bronski,et al.  Stability of repulsive Bose-Einstein condensates in a periodic potential. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  L D Carr,et al.  Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose-Einstein condensate. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  N. A. Kostov,et al.  Quasi–periodic and periodic solutions for coupled nonlinear Schrödinger equations of Manakov type , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[24]  P. Maddaloni,et al.  Dynamics of two colliding Bose-Einstein condensates in an elongated magnetostatic trap , 2000 .