2D solutions of the hyperbolic discrete nonlinear Schrödinger equation

@article{DAmbroise20182DSO,
  title={2D solutions of the hyperbolic discrete nonlinear Schr{\"o}dinger equation},
  author={J. D’Ambroise and P. Kevrekidis},
  journal={Physics},
  year={2018},
  volume={94},
  pages={115203}
}
We derive stationary solutions to the two-dimensional hyperbolic discrete nonlinear Schr\"odinger (HDNLS) equation by starting from the anti-continuum limit and extending solutions to include nearest-neighbor interactions in the coupling parameter. We use pseudo-arclength continuation to capture the relevant branches of solutions and explore their corresponding stability and dynamical properties (i.e., their fate when unstable). We focus on nine primary types of solutions: single site, double… Expand
2 Citations

Figures and Tables from this paper

Strongly localized dark modes in binary discrete media with cubic-quintic nonlinearity within the anti-continuum limit
The existence of dark strongly localized modes of binary discrete media with cubic-quintic nonlinearity is numerically demonstrated by solving the relevant discrete nonlinear Schrodinger equations.Expand

References

SHOWING 1-10 OF 47 REFERENCES
A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation
TLDR
An approximate self-similar solution is found for a wide range of initial conditions -- essentially for initial lumps of small to moderate energy, which has aspects that suggest it is a universal attractor emanating from wide ranges of initial data. Expand
The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives
I Dimensions and Components.- General Introduction and Derivation of the DNLS Equation.- The One-Dimensional Case.- The Two-Dimensional Case.- The Three-Dimensional Case.- The Defocusing Case.-Expand
Stability of discrete solitons in nonlinear Schrödinger lattices
We consider the discrete solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schrodinger (NLS) lattice. The discrete soliton in the anti-continuum limit represents anExpand
Dynamics and stabilization of bright soliton stripes in the hyperbolic-dispersion nonlinear Schrödinger equation
TLDR
This work focuses on a variational approximation used to reduce the dynamics of the bright-soliton stripe to effective equations of motion for its transverse shift, and shows that the instabilities can be attenuated, up to the point of complete stabilization of the soliton stripe. Expand
Persistence and stability of discrete vortices in nonlinear Schrödinger lattices
Abstract We study discrete vortices in the two-dimensional nonlinear Schrodinger lattice with small coupling between lattice nodes. The discrete vortices in the anti-continuum limit of zero couplingExpand
Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation
We construct two families of non-localized standing waves for the hyperbolic cubic nonlinear Schrodinger equation \[iu_t+u_{xx}-u_{yy}+|u|^2u=0.\] The first family of standing waves consists ofExpand
Solitons and collapses: two evolution scenarios of nonlinear wave systems
Two alternative scenarios pertaining to the evolution of nonlinear wave systems are considered: solitons and wave collapses. For the former, it suffices that the Hamiltonian beExpand
On the evolution of packets of water waves
We consider the evolution of packets of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions. Two separateExpand
Gaussian variational ansatz in the problem of anomalous sea waves: Comparison with direct numerical simulation
The nonlinear dynamics of an obliquely oriented wave packet on a sea surface is analyzed analytically and numerically for various initial parameters of the packet in relation to the problem of theExpand
The profile decomposition for the hyperbolic Schr\"odinger equation.
In this note, we prove the profile decomposition for hyperbolic Schr\"odinger (or mixed signature) equations on $\mathbb{R}^2$ in two cases, one mass-supercritical and one mass-critical. First, as aExpand
...
1
2
3
4
5
...