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Existence of solitary waves in nonlinear equations of Schrödinger type
Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices
1. | Fakultät für Mathematik, Universität Karlsruhe, Karlsruhe 76128, Germany |
2. | Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1 |
References:
[1] |
M. Beck, J. Knobloch, D. J. B. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localised patterns, SIAM J. Math. Anal., 41 (2009), 936-972.
doi: doi:10.1137/080713306. |
[2] |
G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala and F. Sanchez, Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun., 219 (2003), 427-433.
doi: doi:10.1016/S0030-4018(03)01341-5. |
[3] |
R. Carretero-Gonzáles, J. D. Talley, C. Chong and B. A. Malomed, Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation, Physica D, 216 (2006), 77-89.
doi: doi:10.1016/j.physd.2006.01.022. |
[4] |
C. Chong, R. Carretero-González, B. A. Malomed and P. G. Kevrekidis, Multistable solitons in higher-dimensional cubic-quintic nonlinear Schrödinger lattices, Physica D, 238 (2009), 126-136.
doi: doi:10.1016/j.physd.2008.10.002. |
[5] |
D. N. Christodoulides and R. I. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides, Opt. Lett., 13 (1988), 794-796.
doi: doi:10.1364/OL.13.000794. |
[6] |
J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis and B. A. Malomed, Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D, 238 (2009), 67-76.
doi: doi:10.1016/j.physd.2008.08.013. |
[7] |
J. Ch. Eilbeck and M. Johansson, The discrete nonlinear Schrödinger equations-20 years on, in "Localization and Energy Transfer in Nonlinear Systems" (eds. L. Vazquez, R. S. MacKay and M. P. Zorzano), World Scientific, (2003), 44-67.
doi: doi:10.1142/9789812704627_0003. |
[8] |
L. Hadžievski, A. Maluckov, M. Stepić and D. Kip, Power controlled soliton stability and steering in lattices with saturable nonlinearity, Phys. Rev. Lett., 93 (2004), 033901.
doi: doi:10.1103/PhysRevLett.93.033901. |
[9] |
R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu and H. Kuroda, Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm, J. Opt. A: Pure Appl. Opt., 6 (2004), 282-287.
doi: doi:10.1088/1464-4258/6/2/021. |
[10] |
D. J. Kaup, Variational solutions for the discrete nonlinear Schrödinger equation, Math. Comput. Simulat., 69 (2005), 322-333.
doi: doi:10.1016/j.matcom.2005.01.015. |
[11] |
P. G. Kevrekidis, K. Ø. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833-2900.
doi: doi:10.1142/S0217979201007105. |
[12] |
D. J. B. Lloyd and B. Sandstede, Localized radial solutions of the Swift-Hohenberg equation, Nonlinearity, 22 (2009), 485-524.
doi: doi:10.1088/0951-7715/22/2/013. |
[13] |
B. A. Malomed, Variational methods in nonlinear fiber optics and related fields, Prog. Opt., 43 (2002), 71-193. |
[14] |
B. A. Malomed and M. I. Weinstein, Soliton dynamics in the discrete nonlinear Schrödinger equation, Phys. Lett. A, 220 (1996), 91-96.
doi: doi:10.1016/0375-9601(96)00516-6. |
[15] |
A. Maluckov, L. Hadžievski and B. A. Malomed, Dark solitons in dynamical lattices with the cubic-quintic nonlinearity, Phys. Rev. E, 76 (2007), 046605.
doi: doi:10.1103/PhysRevE.76.046605. |
[16] |
A. Maluckov, L. Hadžievski and B. A. Malomed, Staggered and moving localized modes in dynamical lattices with the cubic-quintic nonlinearity, Phys. Rev. E, 77 (2008), 036604.
doi: doi:10.1103/PhysRevE.77.036604. |
[17] |
M. Öster and M. Johansson, Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling, Physica D, 238 (2009), 88-99.
doi: doi:10.1016/j.physd.2008.08.006. |
[18] |
M. Öster, M. Johansson and A. Eriksson, Enhanced mobility of strongly localized modes in waveguide arrays by inversion of stability, Phys. Rev. E, 67 (2003), 056606.
doi: doi:10.1103/PhysRevE.67.056606. |
[19] |
I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed and D. J. Frantzeskakis, Soliton collisions in the discrete nonlinear Schrödinger equation, Phys. Rev., 68 (2003), 046604. |
[20] |
D. E. Pelinovsky, P. G. Kevrekidis and D. Frantzeskakis, Stability of discrete solitons in nonlinear Schrödinger lattices, Physica D, 212 (2005), 1-19.
doi: doi:10.1016/j.physd.2005.07.021. |
[21] |
C. Taylor and J. H. P. Dawes, Snaking and isolas of localised states in bistable discrete lattices, Phys. Lett. A, 375 (2010), 4968-4976.
doi: doi:10.1016/j.physleta.2010.10.010. |
[22] |
R. A. Vicencio and M. Johansson, Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity, Phys. Rev. E, 73 (2006), 046602.
doi: doi:10.1103/PhysRevE.73.046602. |
[23] |
C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao and Y. Nie, Third- and fifth-order optical nonlinearities in a new stilbazolium derivative, J. Opt. Soc. Am. B, 19 (2002), 369-375.
doi: doi:10.1364/JOSAB.19.000369. |
show all references
References:
[1] |
M. Beck, J. Knobloch, D. J. B. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localised patterns, SIAM J. Math. Anal., 41 (2009), 936-972.
doi: doi:10.1137/080713306. |
[2] |
G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala and F. Sanchez, Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun., 219 (2003), 427-433.
doi: doi:10.1016/S0030-4018(03)01341-5. |
[3] |
R. Carretero-Gonzáles, J. D. Talley, C. Chong and B. A. Malomed, Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation, Physica D, 216 (2006), 77-89.
doi: doi:10.1016/j.physd.2006.01.022. |
[4] |
C. Chong, R. Carretero-González, B. A. Malomed and P. G. Kevrekidis, Multistable solitons in higher-dimensional cubic-quintic nonlinear Schrödinger lattices, Physica D, 238 (2009), 126-136.
doi: doi:10.1016/j.physd.2008.10.002. |
[5] |
D. N. Christodoulides and R. I. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides, Opt. Lett., 13 (1988), 794-796.
doi: doi:10.1364/OL.13.000794. |
[6] |
J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis and B. A. Malomed, Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D, 238 (2009), 67-76.
doi: doi:10.1016/j.physd.2008.08.013. |
[7] |
J. Ch. Eilbeck and M. Johansson, The discrete nonlinear Schrödinger equations-20 years on, in "Localization and Energy Transfer in Nonlinear Systems" (eds. L. Vazquez, R. S. MacKay and M. P. Zorzano), World Scientific, (2003), 44-67.
doi: doi:10.1142/9789812704627_0003. |
[8] |
L. Hadžievski, A. Maluckov, M. Stepić and D. Kip, Power controlled soliton stability and steering in lattices with saturable nonlinearity, Phys. Rev. Lett., 93 (2004), 033901.
doi: doi:10.1103/PhysRevLett.93.033901. |
[9] |
R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu and H. Kuroda, Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm, J. Opt. A: Pure Appl. Opt., 6 (2004), 282-287.
doi: doi:10.1088/1464-4258/6/2/021. |
[10] |
D. J. Kaup, Variational solutions for the discrete nonlinear Schrödinger equation, Math. Comput. Simulat., 69 (2005), 322-333.
doi: doi:10.1016/j.matcom.2005.01.015. |
[11] |
P. G. Kevrekidis, K. Ø. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833-2900.
doi: doi:10.1142/S0217979201007105. |
[12] |
D. J. B. Lloyd and B. Sandstede, Localized radial solutions of the Swift-Hohenberg equation, Nonlinearity, 22 (2009), 485-524.
doi: doi:10.1088/0951-7715/22/2/013. |
[13] |
B. A. Malomed, Variational methods in nonlinear fiber optics and related fields, Prog. Opt., 43 (2002), 71-193. |
[14] |
B. A. Malomed and M. I. Weinstein, Soliton dynamics in the discrete nonlinear Schrödinger equation, Phys. Lett. A, 220 (1996), 91-96.
doi: doi:10.1016/0375-9601(96)00516-6. |
[15] |
A. Maluckov, L. Hadžievski and B. A. Malomed, Dark solitons in dynamical lattices with the cubic-quintic nonlinearity, Phys. Rev. E, 76 (2007), 046605.
doi: doi:10.1103/PhysRevE.76.046605. |
[16] |
A. Maluckov, L. Hadžievski and B. A. Malomed, Staggered and moving localized modes in dynamical lattices with the cubic-quintic nonlinearity, Phys. Rev. E, 77 (2008), 036604.
doi: doi:10.1103/PhysRevE.77.036604. |
[17] |
M. Öster and M. Johansson, Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling, Physica D, 238 (2009), 88-99.
doi: doi:10.1016/j.physd.2008.08.006. |
[18] |
M. Öster, M. Johansson and A. Eriksson, Enhanced mobility of strongly localized modes in waveguide arrays by inversion of stability, Phys. Rev. E, 67 (2003), 056606.
doi: doi:10.1103/PhysRevE.67.056606. |
[19] |
I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed and D. J. Frantzeskakis, Soliton collisions in the discrete nonlinear Schrödinger equation, Phys. Rev., 68 (2003), 046604. |
[20] |
D. E. Pelinovsky, P. G. Kevrekidis and D. Frantzeskakis, Stability of discrete solitons in nonlinear Schrödinger lattices, Physica D, 212 (2005), 1-19.
doi: doi:10.1016/j.physd.2005.07.021. |
[21] |
C. Taylor and J. H. P. Dawes, Snaking and isolas of localised states in bistable discrete lattices, Phys. Lett. A, 375 (2010), 4968-4976.
doi: doi:10.1016/j.physleta.2010.10.010. |
[22] |
R. A. Vicencio and M. Johansson, Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity, Phys. Rev. E, 73 (2006), 046602.
doi: doi:10.1103/PhysRevE.73.046602. |
[23] |
C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao and Y. Nie, Third- and fifth-order optical nonlinearities in a new stilbazolium derivative, J. Opt. Soc. Am. B, 19 (2002), 369-375.
doi: doi:10.1364/JOSAB.19.000369. |
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