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In the work of Colliander et al. (2020) a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of ‘‘antidark’’ type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in 1 + 1 dimensions, and pave the way for exploration of corresponding configurations in higher dimensions.
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Standing and traveling waves in a model of periodically modulated one-dimensional waveguide arrays
In the present work, we study coherent structures in a one-dimensional discrete nonlinear Schrodinger lattice in which the coupling between waveguides is periodically modulated. Numerical experiments with single-site initial conditions show that, depending on the power, the system exhibits two fundamentally different behaviors. At low power, initial conditions with intensity concentrated in a single site give rise to transport, with the energy moving unidirectionally along the lattice, whereas high power initial conditions yield stationary solutions. We explain these two behaviors, as well as the nature of the transition between the two regimes, by analyzing a simpler model where the couplings between waveguides are given by step functions. For the original model, we numerically construct both stationary and moving coherent structures, which are solutions reproducing themselves exactly after an integer multiple of the coupling period. For the stationary solutions, which are true periodic orbits, we use Floquet analysis to determine the parameter regime for which they are spectrally stable. Typically, the traveling solutions are characterized by having small-amplitude, oscillatory tails, although we identify a set of parameters for which these tails disappear. These parameters turn out to be independent of the lattice size, and our simulations suggest that for these parameters, numerically exact traveling solutions are stable.
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- Award ID(s):
- 1840260
- PAR ID:
- 10534524
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical review E
- ISSN:
- 2470-0053
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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In the work of Colliander et al. (2020) a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of ‘‘antidark’’ type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in 1 + 1 dimensions, and pave the way for exploration of corresponding configurations in higher dimensions.more » « less
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