• Corpus ID: 211082855

Threshold for Blowup and Stability for Nonlinear Schrödinger Equation with Rotation

@article{Basharat2020ThresholdFB,
  title={Threshold for Blowup and Stability for Nonlinear Schr{\"o}dinger Equation with Rotation},
  author={Nyla Basharat and Hichem Hajaiej and Y. Hu and Shijun Zheng},
  journal={arXiv: Analysis of PDEs},
  year={2020}
}
We consider the focusing NLS with an angular momentum and a harmonic potential, which models Bose-Einstein condensate under a rotating magnetic trap. We give a sharp condition on the global existence and blowup in the mass-critical case. We further consider the stability of such systems via variational method. We determine that at the critical exponent $p=1+4/n$, the mass of $Q$, the ground state for the NLS with zero potential, is the threshold for both finite time blowup and orbital… 
View PDF on arXiv
8 Citations
Background Citations
2

8 Citations

Remarks on nonlinear Schrödinger equations arising on rotational Bose–Einstein condensates
Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed
We study the existence and stability of standing waves associated to the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic…
On stability of rotational 2D binary Bose-Einstein condensates
We consider a two-dimensional nonlinear Schrodinger equation proposed in Physics to model rotational binary Bose-Einstein condensates. The nonlinearity is a logarithmic modification of the usual…
Blowup rate for mass critical rotational nonlinear Schrödinger equations
We consider the blowup rate for blowup solutions to $L^2$-critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the…
Multiplicity, asymptotics and stability of standing waves for nonlinear Schrödinger equation with rotation
The Nonexistence of Vortices for Rotating Bose-Einstein Condensates in Non-Radially Symmetric Traps
We consider ground states of rotating Bose-Einstein condensates with attractive interactions in a homogeneous trap $V(x)$ of degree $2$ in $R^2$, which can be non-radially symmetric. For any fixed…
Universal Upper Bound on the Blowup Rate of Nonlinear Schrödinger Equation with Rotation
In this paper, we prove a universal upper bound on the blowup rate of a focusing nonlinear Schrödinger equation with an angular momentum under a trapping harmonic potential, assuming that the initial…
Global Well-Posedness, Blow-Up and Stability of Standing Waves for Supercritical NLS with Rotation
We consider the focusing mass supercritical nonlinear Schrodinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}\Delta u+\frac{1}{2}V(x)u-|u|^{p-1}u+L_{\Omega}u,\quad (x,t)\in…

References

SHOWING 1-10 OF 56 REFERENCES
Vortex Collapse for the L2-Critical Nonlinear Schr\
The focusing cubic nonlinear Schrodinger equation in two dimensions admits vortex solitons, standing wave solutions with spatial structure, Q(m)(r, θ) = eimθR(m)(r). In the case of spin m = 1, we…
Blow-up profile of rotating 2D focusing Bose gases
We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation…
ON THE CAUCHY PROBLEM FOR NONLINEAR SCHR ODINGER EQUATIONS WITH ROTATION
We consider the Cauchy problem for (energy-subcritical) nonlinear Schrodinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a…
Blowup rate for mass critical rotational nonlinear Schrödinger equations
We consider the blowup rate for blowup solutions to $L^2$-critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the…
ON THE STABILITY OF STATIONARY STATES FOR NONLINEAR SCHRODINGER EQUATIONS WITH AN EXTERNAL MAGNETIC FIELD
We study the stability properties of the standing waves for nonlinear Schr6dinger equations in R3. in presence of an external, constant magnetic field. For that purpose, we first establish…
Gross-Pitaevskii Theory of the Rotating Bose Gas
Abstract: We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state; more precisely,…
The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases
We study the ground state of a trapped Bose gas, starting from the full many-body Schrodinger Hamiltonian, and derive the nonlinear Schrodinger energy functional in the limit of large particle…
Blow-up dynamics and spectral property in the L2-critical nonlinear Schrödinger equation in high dimensions
We study stable blow-up dynamics in the L2-critical nonlinear Schrödinger (NLS) equation in high dimensions. First, we show that in dimensions d  =  4 to d  =  12 generic blow-up behavior confirms…
Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases
We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body…
Stability of Attractive Bose–Einstein Condensates
We propose the critical nonlinear Schrödinger equation with a harmonic potential as a model of attractive Bose–Einstein condensates. By an elaborate mathematical analysis we show that a sharp…
...
...