Abstract.
The existence, stability and other dynamical properties of a new type of multi-dimensional (2D or 3D) solitons supported by a transverse low-dimensional (1D or 2D, respectively) periodic potential in the nonlinear Schrödinger equation with the self-defocusing cubic nonlinearity are studied. The equation describes propagation of light in a medium with normal group-velocity dispersion (GVD). Strictly speaking, solitons cannot exist in the model, as its spectrum does not support a true bandgap. Nevertheless, the variational approximation (VA) and numerical computations reveal stable solutions that seem as completely localized ones, an explanation to which is given. The solutions are of the gap-soliton type in the transverse direction(s), in which the periodic potential acts in combination with the diffraction and self-defocusing nonlinearity. Simultaneously, in the longitudinal (temporal) direction these are ordinary solitons, supported by the balance of the normal GVD and defocusing nonlinearity. Stability of the solitons is predicted by the VA, and corroborated by direct simulations.
Similar content being viewed by others
References
B.A. Malomed, D. Mihalache, F. Wise, L. Torner, J. Opt. B: Quant. Semicl. Opt. 7, R53 (2005)
B.B. Baizakov, V.V. Konotop, M. Salerno, J. Phys. B 35, 5105 (2002); E.A. Ostrovskaya, Y.S. Kivshar, Phys. Rev. Lett. 90, 160407 (2003); H. Sakaguchi, B.A. Malomed, J. Phys. B 37, 1443 (2004)
B.B. Baizakov, B.A. Malomed, M. Salerno, Europhys. Lett. 63, 642 (2003)
J. Yang, Z. Musslimani, Opt. Lett. 23, 2094 (2003); Y.V. Kartashov, A.A. Egorov, L. Torner, D.N. Christodoulides, Opt. Lett. 29, 1918 (2004)
B.B. Baizakov, M. Salerno, B.A. Malomed, in Nonlinear Waves: Classical and Quantum Aspects, edited by F.Kh. Abdullaev, V.V. Konotop (Kluwer Academic Publishers, Dordrecht, 2004), p. 61; also available at http://rsphy2.anu.edu.au/~asd124/ Baizakov_2004_61_Nonlinear Waves.pdf
B.B. Baizakov, B.A. Malomed, M. Salerno, Phys. Rev. A 70, 053613 (2004)
D. Mihalache, D. Mazilu, F. Lederer, Y.V. Kartashov, L.-C. Crasovan, L. Torner, Phys. Rev. E 70, 055603 (2004)
P. Xie, Z.-Q. Zhang, X. Zhang, Phys. Rev. E 67, 026607 (2003); A. Ferrando, M. Zacarés, P. Fernández de Córdoba, D. Binosi, J.A. Monsoriu, Opt. Exp. 11, 452 (2003); P. Fernández de Córdoba, D. Binosi, J.A. Monsoriu, Opt. Exp. 12, 817 (2004)
Y.V. Kartashov, V.A. Vysloukh, L. Torner, Phys. Rev. Lett. 93, 093904 (2004); Phys. Rev. Lett. 94, 043902 (2005)
D. Mihalache, D. Mazilu, F. Lederer, B.A. Malomed, Y.V. Kartashov, L.-C. Crasovan, L. Torner, Phys. Rev. Lett. 95, 023902 (2005)
N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, M. Segev, Phys. Rev. E 66, 046602 (2002)
J.W. Fleischer, M. Segev, N.K. Efremidis, D.N. Christodoulides, Nature 422, 147 (2003)
D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, Phys. Rev. Lett. 92, 123903 (2004); J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D.N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004); G. Bartal, O. Manela, O. Cohen, J.W. Fleischer, M. Segev, Phys. Rev. Lett. 95, 053904 (2005)
D. Cheskis, S. Bar-Ad, R. Morandotti, J.S. Aitchison, H.S. Eisenberg, Y. Silberberg, D. Ross, Phys. Rev. Lett. 91, 223901 (2003)
D. Anderson, Phys. Rev. A 27, 1393 (1983); B.A. Malomed, Progr. Opt. 43, 69 (2002)
M.G. Vakhitov, A.A. Kolokolov, Radiophys. Quant. Electron. 16, 783 (1973)
N.K. Efremidis, J. Hudock, D.N. Christodoulides, J.W. Fleischer, O. Cohen, M. Segev, Phys. Rev. Lett. 91, 213906 (2003)
L. Bergé, Phys. Rep. 303, 259 (1998)
B.B. Baizakov, M. Salerno, Phys. Rev. A 69, 013602 (2004)
M. Trippenbach, M. Matuszewski, B.A. Malomed, Europhys. Lett. 70, 8 (2005); M. Matuszewski, E. Infeld, B.A. Malomed, M. Trippenbach, Phys. Rev. Lett. 95, 050403 (2005)
M. Matuszewski, E. Infeld, B.A. Malomed, M. Trippenbach, Stabilization of three-dimensional light bullets by a transverse lattice in a Kerr medium with dispersion management, Opt. Commun. (in press, 2005)
B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, M.K. Oberthaler, Phys. Rev. Lett. 92, 230401 (2004)
O. Bang, C.B. Clausen, P.L. Christiansen, L. Torner, Opt. Lett. 24, 1413 (1999)
J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, D.N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003)
M. Matuszewski, M. Trippenbach, B.A. Malomed, E. Infeld, M. Skorupski, Phys. Rev. E 70, 016603 (2004)
I.N. Towers, B.A. Malomed, F.W. Wise, Phys. Rev. Lett. 90, 123902 (2003)
K. Beckwitt, Y.-F. Chen, F.W. Wise, B.A. Malomed, Phys. Rev. E 68, 057601 (2003)
T. Dohnal, A.B. Aceves, Stud. Appl. Math. 115, 209 (2005)
A. Gubeskys, B.A. Malomed, I.M. Merhasin, Stud. Appl. Math. 115, 255 (2005)
P.J.Y. Louis, E.A. Ostrovskaya, C.M. Savage, Y.S. Kivshar, Phys. Rev. A 67, 013602 (2003); Y.V. Kartashov, V.A. Vysloukh, L. Torner, Opt. Expr. 12, 2831 (2004)
I.M. Merhasin, B.V. Gisin, R. Driben, B.A. Malomed, Phys. Rev. E 71, 016613 (2005)
R. Grimshaw, J. He, B.A. Malomed, Phys. Scripta 53, 385 (1996)
R. Driben, B.A. Malomed, Opt. Commun. 185, 439 (2000)
E.A. Ostrovskaya, Y.S. Kivshar, Phys. Rev. Lett. 93, 160405 (2004); H. Sakaguchi, B.A. Malomed, J. Phys. B 37, 2225 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baizakov, B., Malomed, B. & Salerno, M. Multidimensional semi-gap solitons in a periodic potential. Eur. Phys. J. D 38, 367–374 (2006). https://doi.org/10.1140/epjd/e2006-00004-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjd/e2006-00004-8