American Institute of Mathematical Sciences

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October  2011, 4(5): 1019-1031. doi: 10.3934/dcdss.2011.4.1019

Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices

Christopher Chong 1, and  Dmitry Pelinovsky 2,

1. 

Fakultät für Mathematik, Universität Karlsruhe, Karlsruhe 76128, Germany
2. 

Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1

Received  April 2009 Revised  September 2009 Published  December 2010

Using a variational approximation we study discrete solitons of a nonlinear Schrödinger lattice with a cubic-quintic nonlinearity. Using an ansatz with six parameters we are able to approximate bifurcations of asymmetric solutions connecting site-centered and bond-centered solutions and resulting in the exchange of their stability. We show that the numerical and variational approximations are quite close for solitons of small powers.
Keywords: Variational approximations., Discrete nonlinear Schrödinger equations, Bifurcations of discrete solitons.
Mathematics Subject Classification: Primary: 35Q55, 37K60, 35B32; Secondary: 58E3.
Citation: Christopher Chong, Dmitry Pelinovsky. Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1019-1031. doi: 10.3934/dcdss.2011.4.1019
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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