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Abstract In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponentκin the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the ‘mass’ (i.e., theL2norm) and the nonlinearity parameterκis explored. We observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for allκas long as their mass is less than a critical valueM*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick’s theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curveM*(κ)versus κlies below the two curves found by Derrick’s theorem and the 4CC approximation. In the absence of the external potential,κ= 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, forκ< 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the ‘frustrated’ blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimensiond.
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Stability and instability of Langmuir waves via active subspace decompositions
We study the stability and instability of Langmuir waves propagating in a hot, unmagnetized plasma modeled by the Vlasov–Poisson system and encompassing a variety of velocity distributions, including perturbations of Lorentzian, Kappa, and incomplete Maxwellian steady states. The influence of both high-frequency spatial perturbations and physical parameters on the rate of growth or decay of the plasma response to the initial perturbation is elucidated. Our methods do not rely upon analytic approximation, but instead feature a numerical approximation of the roots of the associated dielectric function that can be accurately quantified without the need for prior assumptions on the parameter regimes under consideration. In this way, the computational discovery of so-called “active” subspaces in the parameter space allows one to identify and quantify the uncertainty generated by physical parameters on the stability properties of wave-like perturbations in a collisionless plasma.
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- Award ID(s):
- 2107938
- PAR ID:
- 10591994
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Physics of Plasmas
- Volume:
- 32
- Issue:
- 2
- ISSN:
- 1070-664X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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