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

@article{Sacchetti2020StationaryST,
  title={Stationary solutions to cubic nonlinear Schr{\"o}dinger equations with quasi-periodic boundary conditions},
  author={Andrea Sacchetti},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2020}
}
  • A. Sacchetti
  • Published 3 February 2020
  • Mathematics
  • Journal of Physics A: Mathematical and Theoretical
In this paper we give the \emph {quantization rules} to determine the normalized stationary solutions to the cubic nonlinear Schr\"odinger equation with quasi-periodic conditions on a given interval. \ Similarly to what happen in the Floquet's theory for linear periodic operators, also in this case some kind of band functions there exist. 

Figures from this paper

Vacuum Energy for a Scalar Field with Self-Interaction in (1 + 1) Dimensions
We calculate the vacuum (Casimir) energy for a scalar field with ϕ4 self-interaction in (1 + 1) dimensions non perturbatively, i.e., in all orders of the self-interaction. We consider massive and
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

References

SHOWING 1-10 OF 36 REFERENCES
Localization in Periodic Potentials: From Schrodinger Operators to the Gross-Pitaevskii Equation
Preface 1. Formalism of the nonlinear Schrodinger equations 2. Justification of the nonlinear Schrodinger equations 3. Existence of localized modes in periodic potentials 4. Stability of localized
Stationary Real Solutions of the Nonlinear Schrödinger Equation on a Ring with a Defect
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
Stationary solutions of the one-dimensional nonlinear Schroedinger equation: II. Case of attractive nonlinearity
All stationary solutions to the one-dimensional nonlinear Schroedinger equation under box or periodic boundary conditions are presented in analytic form for the case of attractive nonlinearity. A
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
Spectral Properties of Nonlinear Schrödinger Equation on a Ring
The stationary states of nonlinear Schrodinger 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
Orbital stability of periodic waves for the nonlinear Schrödinger equation
The nonlinear Schrödinger equation has several families of quasi-periodic traveling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the
On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations
In this paper we generalize previous work on the spectral and orbital stability of waves for infinite-dimensional Hamiltonian systems to include those cases for which the skew-symmetric operator
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
...
...