Multistable solitons in the cubic quintic discrete nonlinear Schrödinger equation

We analyze the existence and stability of localized solutions in the one-dimensional discrete nonlinear Schrodinger (DNLS) equation with a combination of competing self-focusing cubic and defocusing quintic onsite nonlinearities. We produce a stability diagram for different families of soliton solutions that suggests the (co)existence of infinitely many branches of stable localized solutions. Bifurcations that occur with an increase in the coupling constant are studied in a numerical form. A variational approximation is developed for accurate prediction of the most fundamental and next-order solitons, together with their bifurcations. Salient properties of the model, which distinguish it from the well-known cubic DNLS equation, are the existence of two different types of symmetric solitons and stable asymmetric soliton solutions that are found in narrow regions of the parameter space. The asymmetric solutions appear from and disappear back into the symmetric ones via loops of forward and backward pitchfork bifurcations.

[1]  Johann Troles,et al.  Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses , 2003 .

[2]  Kim Ø. Rasmussen,et al.  THE DISCRETE NONLINEAR SCHRÖDINGER EQUATION: A SURVEY OF RECENT RESULTS , 2001 .

[3]  Richard H. Enns,et al.  Quasi-soliton and other behaviour of the nonlinear cubic-quintic Schrodinger equation , 1986 .

[4]  B. Hubbard,et al.  Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array. , 2003, Physical review letters.

[5]  M. Gell-Mann,et al.  Physics Today. , 1966, Applied optics.

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  Keith Promislow,et al.  Localized breathing oscillations of Bose-Einstein condensates in periodic traps , 2001, cond-mat/0105600.

[8]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[9]  Y Silberberg,et al.  Strong spatiotemporal localization in a silica nonlinear waveguide array. , 2003, Physical review letters.

[10]  Robert S. MacKay,et al.  Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators , 1994 .

[11]  A. Scott,et al.  Quantum signatures of breather-breather interactions. , 2003, Physical review letters.

[12]  Mordechai Segev,et al.  Spatial photonics in nonlinear waveguide arrays. , 2005, Optics express.

[13]  B. Malomed,et al.  Three-dimensional nonlinear lattices: from oblique vortices and octupoles to discrete diamonds and vortex cubes. , 2005, Physical review letters.

[14]  T. Hänsch,et al.  Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms , 2002, Nature.

[15]  Fangwei Ye,et al.  Lattice solitons supported by competing cubic–quintic nonlinearity , 2005 .

[16]  I. V. Tomov,et al.  Self-action of light beams in nonlinear media: soliton solutions , 1979 .

[17]  A Smerzi,et al.  Discrete solitons and breathers with dilute Bose-Einstein condensates. , 2001, Physical review letters.

[18]  Exact solutions of the saturable discrete nonlinear Schrödinger equation , 2004, nlin/0409057.

[19]  Yuri S. Kivshar,et al.  Localizing Energy Through Nonlinearity and Discreteness , 2004 .

[20]  L. Hadzievski,et al.  One-dimensional bright discrete solitons in media with saturable nonlinearity. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  J. S. Aitchison,et al.  Discrete Spatial Optical Solitons in Waveguide Arrays , 1998 .

[22]  J. C. Ross,et al.  Multibreathers and homoclinic orbits in 1-dimensional nonlinear lattices , 2000 .

[23]  F Minardi,et al.  Josephson Junction Arrays with Bose-Einstein Condensates , 2001, Science.

[24]  M. Segev,et al.  Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays. , 2005, Optics express.

[25]  C. R. Willis,et al.  Discrete Breathers , 1997 .

[26]  Daoben Zhu,et al.  Third- and fifth-order optical nonlinearities in a new stilbazolium derivative , 2002 .

[27]  姜祈傑.  「Science」與「Nature」之科學計量分析 , 2008 .

[28]  B A Malomed,et al.  Stability of multiple pulses in discrete systems. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Andreas Tünnermann,et al.  Discrete diffraction in two-dimensional arrays of coupled waveguides in silica. , 2004, Optics letters.

[30]  D. Christodoulides,et al.  Discrete self-focusing in nonlinear arrays of coupled waveguides. , 1988, Optics letters.

[31]  Turitsyn,et al.  Stability of discrete solitons and quasicollapse to intrinsically localized modes. , 1994, Physical review letters.

[32]  A. J. Sievers,et al.  Direct observation of the discrete character of intrinsic localized modes in an antiferromagnet , 2004, Nature.

[33]  A. A. Kolokolov,et al.  Stationary solutions of the wave equation in a medium with nonlinearity saturation , 1973 .

[34]  Vincent Couderc,et al.  Non-linear optical properties of chalcogenide glasses measured by Z-scan , 2000 .

[35]  B. Malomed,et al.  Bistable guided solitons in the cubic-quintic medium , 2004 .

[36]  Boris A. Malomed,et al.  Soliton dynamics in the discrete nonlinear Schrödinger equation , 1996 .

[37]  P. Kevrekidis,et al.  Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Detlef Kip,et al.  Power controlled soliton stability and steering in lattices with saturable nonlinearity. , 2004, Physical review letters.

[39]  Michel Peyrard,et al.  Physics of Solitons , 2006 .

[40]  Anders Eriksson,et al.  Enhanced mobility of strongly localized modes in waveguide arrays by inversion of stability. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  B. Malomed,et al.  Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice. , 2004, Physical review letters.

[42]  G. Tsironis.  If "discrete breathers" is the answer, what is the question? , 2003, Chaos.

[43]  Rodislav Driben,et al.  Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Boris A. Malomed,et al.  Chapter 2 - Variational methods in nonlinear fiber optics and related fields , 2002 .

[45]  Mason A Porter,et al.  Nonlinear lattice dynamics of Bose-Einstein condensates. , 2005, Chaos.

[46]  K. Kladko,et al.  Energy thresholds for discrete breathers in one-, two- and three-dimensional lattices , 1997, patt-sol/9701003.

[47]  Yaron Silberberg,et al.  Discretizing light behaviour in linear and nonlinear waveguide lattices , 2003, Nature.

[48]  V. Koukouloyannis,et al.  Existence of multibreathers in chains of coupled one-dimensional Hamiltonian oscillators. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.