On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation (original) (raw)
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Physica D: Nonlinear Phenomena, 1998
We review some recent results concerning the existence and stability of spatially localized and temporally quasiperiodic (non-stationary) excitations in discrete nonlinear Schrödinger (DNLS) models. In two dimensions, we show the existence of linearly stable, stationary and non-stationary localized vortex-like solutions. We also show that stationary on-site localized excitations can have internal 'breathing' modes which are spatially localized and symmetric. The excitation of these modes leads to slowly decaying, quasiperiodic oscillations. Finally, we show that for some generalizations of the DNLS equation where bistability occurs, a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value.
Localized solutions of extended discrete nonlinear Schrödinger equation
Journal of Physics: Conference Series, 2017
We consider the extended discrete nonlinear Schrödinger (EDNLSE) equation which includes the nearest neighbor nonlinear interaction in addition to the on-site cubic and quintic nonlinearities. This equation describes nonlinear excitations in dipolar Bose-Einstein condensate in a periodic optical lattice. We are particularly interested with the existence and stability conditions of localized nonlinear excitations of different types. The problem is tackled numerically, by application of Newton methods and by solving the eigenvalue problem for linearized system near the exact solution. Also the modulational instability of plane wave solution is discussed.
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Physics Letters A, 1996
Using a variational technique, based on an effective Lagrangian, we analyze static and dynamical properties of solitons in the one-dimensional discrete nonlinear Schriidinger equation with a homogeneous power nonlinearity of degree 2 CT + 1. We obtain the following results. (i) For (T < 2 there is no threshold for the excitation of a soliton; solitons of arbitrary positive energies, W = X 1 u, 1 ', exist. (ii) Range of multistability: there is a critical value of cr, uCr = 1.32, such that for oC, < (T < 2, there exist three soliton-like states in a certain finite intermediate range of energies, two stable and one unstable (while there is no multistable regime in the continuum NLS equation). For energies below and above this range, there is a unique soliton state which is stable. (iii) For (T > 2, there exists an energy threshold for formation of the soliton. For all (T > 2 there exist two soliton states, one narrow and one broad. The narrow soliton is stable, while the broad one is not. (iv) We find an energy criterion for the excitation of solitons by initial configurations which are narrowly concentrated in few lattice sites.
Stability of discrete solitons in nonlinear Schrödinger lattices
We consider the discrete solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schrödinger (NLS) lattice. The discrete soliton in the anti-continuum limit represents an arbitrary finite superposition of in-phase or anti-phase excited nodes, separated by an arbitrary sequence of empty nodes. By using stability analysis, we prove that the discrete solitons are all unstable near the anti-continuum limit, except for the solitons, which consist of alternating anti-phase excited nodes. We classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons.
Physical Review E, 2004
We present what we believe to be the first known example of an exact quasiperiodic localized stable solution with spatially symmetric large-amplitude oscillations in a non-integrable Hamiltonian lattice model. The model is a one-dimensional discrete nonlinear Schrödinger equation with alternating on-site energies, modelling e.g. an array of optical waveguides with alternating widths. The solution bifurcates from a stationary discrete gap soliton, and in a regime of large oscillations its intensity oscillates periodically between having one peak at the central site, and two symmetric peaks at the neighboring sites with a dip in the middle. Such solutions, termed 'pulsons', are found to exist in continuous families ranging arbitrarily close both to the anticontinuous and continuous limits. Furthermore, it is shown that they may be linearly stable also in a regime of large oscillations.
Journal of Physics A: Mathematical and Theoretical, 2014
We generalize a finite parity-time (PT-) symmetric network of the discrete nonlinear Schrödinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear dynamics of the dimer model, as well as on the existence and stability of large-amplitude stationary nonlinear modes. A result of particular importance and novelty is the classification of all possible stationary modes in the limit of large amplitudes. We also discover a new integrable configuration of a PTsymmetric dimer.
Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation
Nonlinearity, 1997
We show that the discrete nonlinear Schrödinger (DNLS) equation exhibits exact solutions which are quasiperiodic in time and localized in space if the ratio between the nonlinearity and the linear hopping constant is large enough. These quasiperiodic breather solutions, which also exist for a generalized DNLS equation with on-site nonlinearities of arbitrary positive power, can be constructed by continuation from the anticontinuous limit (i.e. the limit of zero hopping) of solutions where two (or more) sites are oscillating with two incommensurate frequencies. By numerical continuation from the anticontinuous limit, some quasiperiodic breathers are explicitly calculated, and their domain of existence is determined. Using Floquet analysis, we also show that the simplest quasiperiodic breathers are linearly stable close to the anticontinuous limit, and we determine numerically the stability boundaries. The nature of the bifurcations occurring at the boundaries of the stability and existence regions, respectively, is investigated by analysing the band structure of the corresponding Newton operator. We find that the way in which the breather stability and existence is lost depends qualitatively on the ratio between its frequencies. In some cases the two-site breather becomes unstable with respect to a pinning mode, so that applying a small perturbation results in a splitting of the breather into one pinned and one moving part. In other cases, the breather develops an extended tail as some harmonic of its frequencies enters the linear phonon band and becomes a 'phonobreather', which was found to be linearly stable in some domain of parameters.