Stationary Real Solutions of the Nonlinear Schrödinger Equation on a Ring with a Defect

@article{PerezObiol2018StationaryRS,
  title={Stationary Real Solutions of the Nonlinear Schr{\"o}dinger Equation on a Ring with a Defect},
  author={Axel P'erez-Obiol and Taksu Cheon},
  journal={Journal of the Physical Society of Japan},
  year={2018}
}
We analyze the 1D cubic nonlinear stationary Schr\"odinger equation on a ring with a defect for both focusing and defocusing nonlinearity. All possible $\delta$ and $\delta'$ boundary conditions are considered at the defect, computing for each of them the real eigenfunctions, written as Jacobi elliptic functions, and eigenvalues for the ground state and first few excited energy levels. All six independent Jacobi elliptic functions are found to be solutions of some boundary condition. We also… 
4 Citations

Figures and Tables from this paper

Stationary solutions to cubic nonlinear Schrödinger equations with quasi-periodic boundary conditions

  • A. Sacchetti
  • Mathematics
    Journal of Physics A: Mathematical and Theoretical
  • 2020
In this paper we give the quantization rules to determine the normalized stationary solutions to the cubic nonlinear Schrödinger equation with quasi-periodic conditions on a given interval. Similarly

Bose-Einstein condensate confined in a one-dimensional ring stirred with a rotating delta link.

A Bose-Einstein condensate with repulsive interactions confined in a one-dimensional ring where a Dirac δ is rotating at constant speed is considered and a set of adiabatic cycles is proposed in which gray and dark solitons, and vortex states of arbitrary quantized angular momenta are obtained from the ground state by setting and unsetting a rotating δ.

Current production in ring condensates with a weak link

We consider attractive and repulsive condensates in a ring trap stirred by a weak link, and analyze the spectrum of solitonic trains dragged by the link, by means of analytical expressions for the

Physics-informed neural networks for operator equations with stochastic data

It is remarked that application of PINNs—referred to as TPINNs—allows to solve the induced tensor operator equations under minimal changes of existing PINNs code, which can overcome the curse of dimensionality and covers non-linear and time-dependent operators.

References

SHOWING 1-10 OF 18 REFERENCES

Spectral properties of nonlinear Schrödinger equation on a ring

The stationary states of nonlinear Schrödinger equation on a ring with a defect is numerically analyzed. Unconventional connection conditions are imposed on the point defect, and it is shown that the

Stationary solutions of the one-dimensional nonlinear Schrodinger equation: II. Case of attractive nonlinearity

In this second of two papers, we present all stationary so- lutions of the nonlinear Schrodinger equation with box or pe- riodic boundary conditions for the case of attractive nonlin- earity. The

Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph.

In this paper we give the complete classification of solitons for a cubic nonlinear Schrödinger equation on the simplest network with a nontrivial topology: the tadpole graph, i.e., a ring with a

FAST SOLITONS ON STAR GRAPHS

We define the Schrodinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary

An initial-boundary value problem for the nonlinear Schrödinger equation

Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media

It is demonstrated that the equation iol/J/ot + l/Jxx + K 1¢12 1/1 = 0, which describes plane self-focusing and one-dimensional self-modulation can be solved exactly by reducing it to the inverse

Symmetric and asymmetric solitons trapped in H-shaped potentials

We report results of numerical and analytical studies of the spontaneous symmetry breaking in solitons, both two- and one-dimensional, which are trapped in H-shaped potential profiles, built of two

Dynamics of one-dimensional Bose liquids: Andreev-like reflection at Y junctions and the absence of the Aharonov-Bohm effect.

In a ring-interferometer-type configuration, the transport is completely insensitive to the (effective) flux contained in the ring, in contrast with the Aharonov-Bohm effect of a single particle in the same geometry.

Symmetries of Schrödinger Operators with Point Interactions

The transformations of all the Schrodinger operators with point interactions in dimension one under space reflection P , time reversal T and (Weyl) scaling W are presented. In particular, those