Abstract

We theoretically study the role of the mode structure of a multicomponent Bose–Einstein condensate (BEC) in the potential created by a nonlinear optical lattice. We describe a multisoliton complex (MSC) as a superposition of different fundamental soliton modes in the matter-wave system. Using a similarity transformation, we solve the nonlinear evolution equation of the multimode coupled matter-wave field and construct a set of analytical bright soliton solutions. A perturbation method is used to examine the linear stability of the constructed solitons. Based on these particular solutions, we numerically analyze the mode structure of a MSC. The results show that the periodicity causes a Bloch modulation in the envelopes of the density distribution. When different fundamental modes collide with each other in the nonlinear lattice, the collision-induced shifts, and the space-dependent modulation of external potentials change the density profile of the multimode soliton complex. Therefore, the mode structure, which is absent in a one-mode BEC, provides the possible multiscale modeling of the matter-wave field with extra degrees of freedom.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (2)

2011 (3)

D. Tilahun, R. A. Duine, and A. H. MacDonald, “Quantum theory of cold bosonic atoms in optical lattices,” Phys. Rev. A 84, 033622 (2011).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

J. Belmonte-Beitia, V. M. Prez-Garca, and V. Brazhnyi, “Solitary waves in coupled nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities,” Commun. Nonlinear Sci. Numer. Simul. 16, 158–172 (2011).
[CrossRef]

2010 (3)

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010).
[CrossRef]

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Rev. Mod. Phys. 82, 1225–1286 (2010).
[CrossRef]

M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef]

2009 (1)

J. Belmonte-Beitia, V. V. Konotop, V. M. Pérez-García, and V. E. Vekslerchik, “Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: linear and nonlinear lattices,” Chaos Solitons Fractals 41, 1158–1166 (2009).
[CrossRef]

2008 (4)

D. Pelinovsky and G. Schneider, “Moving gap solitons in periodic potentials,” Math. Methods Appl. Sci. 31, 1739–1760 (2008).
[CrossRef]

S. Li, L. Fu, W. Duan, and J. Liu, “Nonlinear Ramsey interferometry with Rosen–Zener pulses on a two-component Bose–Einstein condensate,” Phys. Rev. A 78, 063621 (2008).
[CrossRef]

A.-C. Ji, W. M. Liu, J. L. Song, and F. Zhou, “Dynamical creation of fractionalized vortices and vortex lattices,” Phys. Rev. Lett. 101, 010402 (2008).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).
[CrossRef]

2007 (3)

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

F. K. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A 76, 043611 (2007).
[CrossRef]

M. Lewenstein, A. Sanpera, and V. Ahufinger, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[CrossRef]

2006 (1)

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

2005 (2)

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose–Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E 72, 046610 (2005).
[CrossRef]

2004 (3)

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

Z.-W. Xie and W. M. Liu, “Superfluid-Mott-insulator transition of dipolar bosons in an optical lattice,” Phys. Rev. A 70, 045602 (2004).
[CrossRef]

W. Bao, “Ground states and dynamics of multicomponent Bose–Einstein condensates,” Multiscale Model. Simul. 2, 210–236 (2004).
[CrossRef]

2001 (1)

P. Öhberg, and L. Santos, “Dark solitons in a two-component Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2918–2921 (2001).
[CrossRef]

2000 (2)

R. Grimm and M. Weidemuller, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys. 42, 95–170 (2000).
[CrossRef]

N. Akhmediev and A. Ankiewicz, “Multi-soliton complexes,” Chaos 10, 600–612 (2000).
[CrossRef]

1999 (4)

A. Ankiewicz, W. Królikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solution,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

A. A. Sukhorukov and N. N. Akhmediev, “Coherent and incoherent contributions to multisoliton complexes,” Phys. Rev. Lett. 83, 4736–4739 (1999).
[CrossRef]

H. J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and W. Ketterle, “Observation of metastable states in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 82, 2228–2231 (1999).
[CrossRef]

W. Królikowski, N. Akhmediev, and B. Luther-Davies, “Collision-induced shape transformations of partially coherent solitons,” Phys. Rev. E 59, 4654–4658 (1999).
[CrossRef]

1998 (1)

D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Measurements of relative phase in two-component Bose–Einstein condensates,” Phys. Rev. Lett. 81, 1543–1546 (1998).
[CrossRef]

1997 (1)

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

1996 (1)

M. Mitchell, Z. Chen, M.-F. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

1995 (3)

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef]

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef]

1974 (1)

C. B. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

1973 (1)

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–789 (1973).
[CrossRef]

1961 (2)

E. P. Gross, “Structure of a quantized vortex in boson systems,” Nuovo Cimento Soc. Ital. Fis. B 20, 454–477 (1961).
[CrossRef]

L. P. Pitaevsk, “Vortex lines in an imperfect Bose gas,” Sov. Phys. JETP 13, 451–454 (1961).

Abdullaev, F. K.

F. K. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A 76, 043611 (2007).
[CrossRef]

Ahufinger, V.

M. Lewenstein, A. Sanpera, and V. Ahufinger, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[CrossRef]

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, “Multi-soliton complexes,” Chaos 10, 600–612 (2000).
[CrossRef]

W. Królikowski, N. Akhmediev, and B. Luther-Davies, “Collision-induced shape transformations of partially coherent solitons,” Phys. Rev. E 59, 4654–4658 (1999).
[CrossRef]

Akhmediev, N. N.

A. Ankiewicz, W. Królikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solution,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

A. A. Sukhorukov and N. N. Akhmediev, “Coherent and incoherent contributions to multisoliton complexes,” Phys. Rev. Lett. 83, 4736–4739 (1999).
[CrossRef]

Anderson, M. H.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef]

Andrews, M. R.

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, “Multi-soliton complexes,” Chaos 10, 600–612 (2000).
[CrossRef]

A. Ankiewicz, W. Królikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solution,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

Bao, W.

W. Bao, “Ground states and dynamics of multicomponent Bose–Einstein condensates,” Multiscale Model. Simul. 2, 210–236 (2004).
[CrossRef]

Belmonte-Beitia, J.

J. Belmonte-Beitia, V. M. Prez-Garca, and V. Brazhnyi, “Solitary waves in coupled nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities,” Commun. Nonlinear Sci. Numer. Simul. 16, 158–172 (2011).
[CrossRef]

J. Belmonte-Beitia, V. V. Konotop, V. M. Pérez-García, and V. E. Vekslerchik, “Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: linear and nonlinear lattices,” Chaos Solitons Fractals 41, 1158–1166 (2009).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

Böhi, P.

M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef]

Bradley, C. C.

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef]

Brazhnyi, V.

J. Belmonte-Beitia, V. M. Prez-Garca, and V. Brazhnyi, “Solitary waves in coupled nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities,” Commun. Nonlinear Sci. Numer. Simul. 16, 158–172 (2011).
[CrossRef]

Byrnes, T.

T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose–Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

Chen, T.

Chen, X.

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

Chen, Z.

M. Mitchell, Z. Chen, M.-F. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Chikkatur, A. P.

H. J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and W. Ketterle, “Observation of metastable states in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 82, 2228–2231 (1999).
[CrossRef]

Chin, C.

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Rev. Mod. Phys. 82, 1225–1286 (2010).
[CrossRef]

Christodoulides, D. N.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

Cornell, E. A.

D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Measurements of relative phase in two-component Bose–Einstein condensates,” Phys. Rev. Lett. 81, 1543–1546 (1998).
[CrossRef]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef]

Coskun, T. H.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

da Luz, H. L. F.

F. K. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A 76, 043611 (2007).
[CrossRef]

Davis, K. B.

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

Denschlag, J. H.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

Duan, W.

S. Li, L. Fu, W. Duan, and J. Liu, “Nonlinear Ramsey interferometry with Rosen–Zener pulses on a two-component Bose–Einstein condensate,” Phys. Rev. A 78, 063621 (2008).
[CrossRef]

Duine, R. A.

D. Tilahun, R. A. Duine, and A. H. MacDonald, “Quantum theory of cold bosonic atoms in optical lattices,” Phys. Rev. A 84, 033622 (2011).
[CrossRef]

Durfee, D. S.

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

Ensher, J. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef]

Fang, Y.

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

Fu, L.

S. Li, L. Fu, W. Duan, and J. Liu, “Nonlinear Ramsey interferometry with Rosen–Zener pulses on a two-component Bose–Einstein condensate,” Phys. Rev. A 78, 063621 (2008).
[CrossRef]

Gammal, A.

F. K. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A 76, 043611 (2007).
[CrossRef]

Grimm, R.

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Rev. Mod. Phys. 82, 1225–1286 (2010).
[CrossRef]

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

R. Grimm and M. Weidemuller, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys. 42, 95–170 (2000).
[CrossRef]

Gross, E. P.

E. P. Gross, “Structure of a quantized vortex in boson systems,” Nuovo Cimento Soc. Ital. Fis. B 20, 454–477 (1961).
[CrossRef]

Gu, Z.

Guo, H.

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

Hall, D. S.

D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Measurements of relative phase in two-component Bose–Einstein condensates,” Phys. Rev. Lett. 81, 1543–1546 (1998).
[CrossRef]

Hänsch, T. W.

M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef]

Hellwig, M.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

Hulet, R. G.

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef]

Inouye, S.

H. J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and W. Ketterle, “Observation of metastable states in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 82, 2228–2231 (1999).
[CrossRef]

Ji, A.-C.

A.-C. Ji, W. M. Liu, J. L. Song, and F. Zhou, “Dynamical creation of fractionalized vortices and vortex lattices,” Phys. Rev. Lett. 101, 010402 (2008).
[CrossRef]

Julienne, P.

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Rev. Mod. Phys. 82, 1225–1286 (2010).
[CrossRef]

Kartashov, Y. V.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

Ketterle, W.

H. J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and W. Ketterle, “Observation of metastable states in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 82, 2228–2231 (1999).
[CrossRef]

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

Kolokolov, A. A.

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–789 (1973).
[CrossRef]

Konotop, V. V.

J. Belmonte-Beitia, V. V. Konotop, V. M. Pérez-García, and V. E. Vekslerchik, “Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: linear and nonlinear lattices,” Chaos Solitons Fractals 41, 1158–1166 (2009).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).
[CrossRef]

Królikowski, W.

W. Królikowski, N. Akhmediev, and B. Luther-Davies, “Collision-induced shape transformations of partially coherent solitons,” Phys. Rev. E 59, 4654–4658 (1999).
[CrossRef]

A. Ankiewicz, W. Królikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solution,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

Kurn, D. M.

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

Lewenstein, M.

M. Lewenstein, A. Sanpera, and V. Ahufinger, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[CrossRef]

Li, S.

S. Li, L. Fu, W. Duan, and J. Liu, “Nonlinear Ramsey interferometry with Rosen–Zener pulses on a two-component Bose–Einstein condensate,” Phys. Rev. A 78, 063621 (2008).
[CrossRef]

Li, Y.

M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef]

Liang, Z. X.

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose–Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

Liu, J.

S. Li, L. Fu, W. Duan, and J. Liu, “Nonlinear Ramsey interferometry with Rosen–Zener pulses on a two-component Bose–Einstein condensate,” Phys. Rev. A 78, 063621 (2008).
[CrossRef]

Liu, W. M.

A.-C. Ji, W. M. Liu, J. L. Song, and F. Zhou, “Dynamical creation of fractionalized vortices and vortex lattices,” Phys. Rev. Lett. 101, 010402 (2008).
[CrossRef]

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose–Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

Z.-W. Xie and W. M. Liu, “Superfluid-Mott-insulator transition of dipolar bosons in an optical lattice,” Phys. Rev. A 70, 045602 (2004).
[CrossRef]

Luo, S.

Luther-Davies, B.

W. Królikowski, N. Akhmediev, and B. Luther-Davies, “Collision-induced shape transformations of partially coherent solitons,” Phys. Rev. E 59, 4654–4658 (1999).
[CrossRef]

Ma, X.

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

MacDonald, A. H.

D. Tilahun, R. A. Duine, and A. H. MacDonald, “Quantum theory of cold bosonic atoms in optical lattices,” Phys. Rev. A 84, 033622 (2011).
[CrossRef]

Malomed, B. A.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E 72, 046610 (2005).
[CrossRef]

Manakov, C. B.

C. B. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Matthews, M. R.

D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Measurements of relative phase in two-component Bose–Einstein condensates,” Phys. Rev. Lett. 81, 1543–1546 (1998).
[CrossRef]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef]

Mewes, M.-O.

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

Miesner, H. J.

H. J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and W. Ketterle, “Observation of metastable states in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 82, 2228–2231 (1999).
[CrossRef]

Mitchell, M.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M.-F. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Öhberg, P.

P. Öhberg, and L. Santos, “Dark solitons in a two-component Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2918–2921 (2001).
[CrossRef]

Pelinovsky, D.

D. Pelinovsky and G. Schneider, “Moving gap solitons in periodic potentials,” Math. Methods Appl. Sci. 31, 1739–1760 (2008).
[CrossRef]

Peng, J.

Pérez-García, V. M.

J. Belmonte-Beitia, V. V. Konotop, V. M. Pérez-García, and V. E. Vekslerchik, “Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: linear and nonlinear lattices,” Chaos Solitons Fractals 41, 1158–1166 (2009).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

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L. P. Pitaevsk, “Vortex lines in an imperfect Bose gas,” Sov. Phys. JETP 13, 451–454 (1961).

Prez-Garca, V. M.

J. Belmonte-Beitia, V. M. Prez-Garca, and V. Brazhnyi, “Solitary waves in coupled nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities,” Commun. Nonlinear Sci. Numer. Simul. 16, 158–172 (2011).
[CrossRef]

Qian, K.

Riedel, M. F.

M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef]

Ruff, G.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

Sackett, C. A.

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef]

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E 72, 046610 (2005).
[CrossRef]

Sanpera, A.

M. Lewenstein, A. Sanpera, and V. Ahufinger, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[CrossRef]

Santos, L.

P. Öhberg, and L. Santos, “Dark solitons in a two-component Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2918–2921 (2001).
[CrossRef]

Schneider, G.

D. Pelinovsky and G. Schneider, “Moving gap solitons in periodic potentials,” Math. Methods Appl. Sci. 31, 1739–1760 (2008).
[CrossRef]

Segev, M.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M.-F. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Shen, Q.

Shih, M.-F.

M. Mitchell, Z. Chen, M.-F. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Sinatra, A.

M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef]

Song, J. L.

A.-C. Ji, W. M. Liu, J. L. Song, and F. Zhou, “Dynamical creation of fractionalized vortices and vortex lattices,” Phys. Rev. Lett. 101, 010402 (2008).
[CrossRef]

Stamper-Kurn, D. M.

H. J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and W. Ketterle, “Observation of metastable states in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 82, 2228–2231 (1999).
[CrossRef]

Stenger, J.

H. J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and W. Ketterle, “Observation of metastable states in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 82, 2228–2231 (1999).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov and N. N. Akhmediev, “Coherent and incoherent contributions to multisoliton complexes,” Phys. Rev. Lett. 83, 4736–4739 (1999).
[CrossRef]

Thalhammer, G.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

Theis, M.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

Tiesinga, E.

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Rev. Mod. Phys. 82, 1225–1286 (2010).
[CrossRef]

Tilahun, D.

D. Tilahun, R. A. Duine, and A. H. MacDonald, “Quantum theory of cold bosonic atoms in optical lattices,” Phys. Rev. A 84, 033622 (2011).
[CrossRef]

Tollett, J. J.

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef]

Tomio, L.

F. K. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A 76, 043611 (2007).
[CrossRef]

Torner, L.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

Torres, P. J.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

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L. N. Trefethen, Spectral Methods in MATLAB (SIAM, 2000).

Treutlein, P.

M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef]

Vakhitov, N. G.

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–789 (1973).
[CrossRef]

van Druten, N. J.

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

Vekslerchik, V.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

Vekslerchik, V. E.

J. Belmonte-Beitia, V. V. Konotop, V. M. Pérez-García, and V. E. Vekslerchik, “Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: linear and nonlinear lattices,” Chaos Solitons Fractals 41, 1158–1166 (2009).
[CrossRef]

Wang, Y.

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

Weidemuller, M.

R. Grimm and M. Weidemuller, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys. 42, 95–170 (2000).
[CrossRef]

Wen, K.

T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose–Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

Wieman, C. E.

D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Measurements of relative phase in two-component Bose–Einstein condensates,” Phys. Rev. Lett. 81, 1543–1546 (1998).
[CrossRef]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef]

Winkler, K.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

Xia, L.

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

Xie, Z.-W.

Z.-W. Xie and W. M. Liu, “Superfluid-Mott-insulator transition of dipolar bosons in an optical lattice,” Phys. Rev. A 70, 045602 (2004).
[CrossRef]

Yamamoto, Y.

T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose–Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

Zhan, L.

Zhang, Z. D.

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose–Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

Zhou, F.

A.-C. Ji, W. M. Liu, J. L. Song, and F. Zhou, “Dynamical creation of fractionalized vortices and vortex lattices,” Phys. Rev. Lett. 101, 010402 (2008).
[CrossRef]

Zhou, X.

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

Adv. At. Mol. Opt. Phys. (1)

R. Grimm and M. Weidemuller, “Optical dipole traps for neutral atoms,” Adv. At. Mol. Opt. Phys. 42, 95–170 (2000).
[CrossRef]

Adv. Phys. (1)

M. Lewenstein, A. Sanpera, and V. Ahufinger, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56, 243–379 (2007).
[CrossRef]

Chaos (1)

N. Akhmediev and A. Ankiewicz, “Multi-soliton complexes,” Chaos 10, 600–612 (2000).
[CrossRef]

Chaos Solitons Fractals (1)

J. Belmonte-Beitia, V. V. Konotop, V. M. Pérez-García, and V. E. Vekslerchik, “Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: linear and nonlinear lattices,” Chaos Solitons Fractals 41, 1158–1166 (2009).
[CrossRef]

Commun. Nonlinear Sci. Numer. Simul. (1)

J. Belmonte-Beitia, V. M. Prez-Garca, and V. Brazhnyi, “Solitary waves in coupled nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities,” Commun. Nonlinear Sci. Numer. Simul. 16, 158–172 (2011).
[CrossRef]

J. Opt. Soc. Am. B (1)

Math. Methods Appl. Sci. (1)

D. Pelinovsky and G. Schneider, “Moving gap solitons in periodic potentials,” Math. Methods Appl. Sci. 31, 1739–1760 (2008).
[CrossRef]

Multiscale Model. Simul. (1)

W. Bao, “Ground states and dynamics of multicomponent Bose–Einstein condensates,” Multiscale Model. Simul. 2, 210–236 (2004).
[CrossRef]

Nature (1)

M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010).
[CrossRef]

Nuovo Cimento Soc. Ital. Fis. B (1)

E. P. Gross, “Structure of a quantized vortex in boson systems,” Nuovo Cimento Soc. Ital. Fis. B 20, 454–477 (1961).
[CrossRef]

Phys. Rev. A (7)

X. Ma, L. Xia, Y. Fang, X. Zhou, Y. Wang, H. Guo, and X. Chen, “Population oscillation of the multicomponent spinor Bose–Einstein condensate induced by nonadiabatic transitions,” Phys. Rev. A 73, 013624 (2006).
[CrossRef]

S. Li, L. Fu, W. Duan, and J. Liu, “Nonlinear Ramsey interferometry with Rosen–Zener pulses on a two-component Bose–Einstein condensate,” Phys. Rev. A 78, 063621 (2008).
[CrossRef]

T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose–Einstein condensates,” Phys. Rev. A 85, 040306 (2012).
[CrossRef]

F. K. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A 76, 043611 (2007).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010).
[CrossRef]

Z.-W. Xie and W. M. Liu, “Superfluid-Mott-insulator transition of dipolar bosons in an optical lattice,” Phys. Rev. A 70, 045602 (2004).
[CrossRef]

D. Tilahun, R. A. Duine, and A. H. MacDonald, “Quantum theory of cold bosonic atoms in optical lattices,” Phys. Rev. A 84, 033622 (2011).
[CrossRef]

Phys. Rev. E (3)

A. Ankiewicz, W. Królikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solution,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E 72, 046610 (2005).
[CrossRef]

W. Królikowski, N. Akhmediev, and B. Luther-Davies, “Collision-induced shape transformations of partially coherent solitons,” Phys. Rev. E 59, 4654–4658 (1999).
[CrossRef]

Phys. Rev. Lett. (13)

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).
[CrossRef]

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose–Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

A. A. Sukhorukov and N. N. Akhmediev, “Coherent and incoherent contributions to multisoliton complexes,” Phys. Rev. Lett. 83, 4736–4739 (1999).
[CrossRef]

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

P. Öhberg, and L. Santos, “Dark solitons in a two-component Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2918–2921 (2001).
[CrossRef]

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms,” Phys. Rev. Lett. 75, 3969–3973 (1995).
[CrossRef]

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Density profile of a two-mode matter-wave soliton with the initial mode amplitudes as a1=a2=3 (a) in a free space (β=0) with the mode eigenvalues as k1=0.3, k2=0.15, (b) in a lattice potential of β=0.3 with k1=0.3, k2=0.15, (c) in the same lattice potential, i.e., β=0.3, while the magnitudes of the eigenvalues of each component are increased to k1=0.5 and k2=0.25, and (d) in a decreased lattice potential with β=0.1, while the soliton parameters are k1=0.3, k2=0.15. The inset graphs are the amplitude profile of each component.

Fig. 2.
Fig. 2.

Collisions between components of a multimode soliton in the free space (β=0). Density profile of a two-mode soliton (N=2), with k1=0.25, k2=0.5, and v1=0.3, v2=0.3, (a) before the collision and (c) after the collision. Density profile of a three-mode soliton (N=3), whose parameters are k1=0.5, k2=0.4, k3=0.2 and v1=v2=0.3, v3=0.3, (b) before the collision and (d) after the collision.

Fig. 3.
Fig. 3.

Collisions between components of a multimode soliton in a nonlinear lattice (β=0.3). Density profile of a two-mode soliton (N=2), with k1=0.25, k2=0.5 and v1=0.3, v2=0.3 (a) before the collision and (c) after the collision. Density profile of a three-mode soliton (N=3), whose parameters are k1=0.5, k2=0.4, k3=0.2 and v1=v2=0.3, v3=0.3 (b) before the collision and (d) after the collision.

Equations (83)

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Ψ(x,t)=[ψ1(x,t),,ψN(x,t)]T,
itΨ(x,t)=2x2Ψ(x,t)+V(x)·Ψ(x,t)+U(Ψ)·Ψ(x,t),
Uj(Ψ)Uj(x,t)=αj1(x)|ψ1(x,t)|2++αjN(x)|ψN(x,t)|2.
itψj(x,t)=2x2ψj(x,t)+Vj(x)ψj(x,t)+α(x)iN|ψi(x,t)|2ψj(x,t).
ψj(x,t)=ϕj(x)eiEjt,
Ejϕj(x)=(2x2+Vj(x)+α(x)iN|ϕi(x)|2)ϕj(x).
ϕj(x)=ρ(x)Φj(X),(j=1,2N),
EjΦj(X)=(2X2+AiN|Φi(X)|2)Φj(X)
EjVj(x)+ρ(x)/ρ(x)EjX2=0,
2ρ(x)X+ρ(x)X=0,
AX2+α(x)|ρ(x)|2=0,
Ej=(EjVj(x))ρ4(x)+ρ(x)ρ3(x),
X(x)=0x1ρ2(s)ds.
α(x)=A/ρ6(x).
ρ2(x)Vj(x)+12(ρ2(x))2(ρ2(x))(Vj(x)Ej)=0,
Vj(x)=V1(x)+bj/ρ4(x)+EjE1,(j>1),
Vj(x)=bj/ρ4(x)+EjE1.
(ρ2(x))+4(ρ2(x))E1=0.
ρ(x)=(C1+C2cosωx+C3sinωx)1/2,
α(x)=A/(1+βcosωx)3,
Vj(x)=bj/(1+βcosωx)2+EjE1,
ρ(x)=(1+βcosωx)1/2,
Ej=bj+E1(1β2),
X(x)=0x11+βcosωsds.
α(x)A(13βcosωx),
Vj(x)bj(12βcosωx)+EjE1.
m=1NDj,m|A|Φm(X)2km=ej(X),
Djm=δjm+ej(X)em(X)kj+km
ej(X)=2kjajexp[kj(XXj)],
kj=Ej.
Φm(X)=2km|A|jNDjm1ej(X),
ϕm(x)=ρ(x)Φm(X)=2ρ(x)km|A|jNDjm1ej(X).
ψj(x,t)=[ϕj(x)+Aj(x,t)]eiEjt,
EjAj+iA˙j=α(x)(iN|Ai|2+iN(ϕi*Ai+ϕiAi*))(ϕj+Aj)+α(x)iN|ϕi|2Aj2x2Aj+VjAj,
iA˙j=α(x)(iNϕi(Ai+Ai*)ϕj+iN|ϕi|2Aj)2x2Aj+(VjEj)Aj.
Aj(x,t)=[pj(x)+iqj(x)]eΩt,
Ωpj=L^j()qj,
Ωqj=L^j(+)pj+ijNS^ijpi,
L^j()=VjEj2x2+α(x)iN|ϕi|2,
L^j(+)=VjEj2x2+α(x)iN|ϕi|2+2α(x)|ϕj|2,
S^ij=2α(x)ϕiϕj.
Ω2pj=L^j()L^j(+)pj+L^j()ijNS^ijpi,
λP=L^P,
P=[p1,p2,p3,,pN]T,
L^=(L^1()L^1(+)L^1()S^21L^1()S^31L^1()S^N1L^2()S^12L^2()L^2(+)L^2()S^32L^2()S^N2L^3()S^13L^3()S^23L^3()L^3(+)L^3()S^N3L^N()S^1NL^N()S^2NL^N()S^3NL^N()L^N(+)).
ψjL(x,t)=φj(ξj)exp[i(vjxϵjt)],
ξj(x,t)=xxj2vjt,
(ϵm2ivmξm)φm(ξm)=(2x22ivmx+vm2+Vm(x)+α(x)nN|φn(ξn)|2)φm(ξm).
Emφm(ξm)=(2ξm2+Vm(x)+α(x)nN|φn(ξn)|2)φm(ξm).
EmΦm(χm)=(2χm2+AnN|Φn(χn)|2)Φm(χm),
χm(ξm)=0ξm1ρ2(s)ds,
φm(ξm)=ρ(ξm)Φm(χm).
M=ζ(x,{ϕi})x+jNηj(x,{ϕi})ϕj,(i=1,2,,N)
M(2)Nj(x,ϕj,ϕj,ϕj)=(ζx+jNηjϕj+jNηj(1)ϕj+jNηj(2)ϕ)Nj(x,ϕj,ϕj,ϕj)=0,
ηj(k)(x,{ϕi},{ϕi},{ϕi},,{ϕi(k)})=Dηj(k1)Dxϕj(k)Dζ(x,{ϕi})Dx,k=1,2,
DDx=x+jϕjϕj+jϕjϕj++jϕj(n+1)ϕj(n)+.
ηj(1)=ηjx+iϕiηjϕiϕj(ζx+iϕiζϕi),
ηj(2)=(x+iϕiϕi+iϕiϕi)[ηjx+iϕiηjϕiϕj(ζx+iϕiζϕi)]ϕj(ζx+iϕiζϕi),
Nj(x,ϕj,ϕj,ϕj)=ϕjfj(x,{ϕi}),
fj(x,{ϕi})=(Vj(x)Ej+α(x)i|ϕi|2)ϕj.
M(2)Nj=ζfjxiηifjϕi+ηj(2)=0.
ζϕiϕj=0,
ηjϕjϕj=2ζϕjx,
ηjϕjϕi=ζϕix(ij),
ηjϕiϕk=0(i,kj),
ηjϕix=fjζϕi(ij),
2ηjϕjxζxx3fjζϕjijfiζϕi=0,
ηjxxiηifjϕi+ifiηjϕiζfjx2fjζx=0.
ζ(x,{ϕi})=a(x),
ηj(x,{ϕi})=c(x)ϕj,
2c(x)=a(x),
α(x)=A/a3(x),
12a(x)2a(x)(Vj(x)Ej)a(x)Vj(x)=0,
Vj(x)=V1(x)+bj/a2(x)+EjE1,
ϕj(x)=ρ(x)Φj(X),
M=X.
x=X(x)Xρ(x)ρ(x)jΦjΦj,
ϕj=1ρ(x)Φj.
a(x)X(x)=1,
a(x)ρ(x)+c(x)ρ(x)=0,
a(x)=ρ2(x).
12(ρ2(x))2(ρ2(x))(Vj(x)Ej)(ρ2(x))Vj(x)=0,
Vj(x)=V1(x)+bj/ρ4(x)+EjE1.

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