An official website of the United States government
Abstract Galaxies and their dark-matter halos are commonly presupposed to spin. But it is an open question how this spin manifests in halos and soliton cores made of scalar dark matter (SDM, including fuzzy/wave/ultralight-axion dark matter). One way spin could manifest in a necessarily irrotational SDM velocity field is with a vortex. But recent results have cast doubt on this scenario, finding that vortices are generally unstable except with substantial repulsive self-interaction. In this paper, we introduce an alternative route to stability: in both (non-relativistic) analytic calculations and simulations, a black hole or other central mass at least as massive as a soliton can stabilize a vortex within it. This conclusion may also apply to AU-scale halos bound to the sun and stellar-mass-scale Bose stars.
more »
« less
The application of the “inverse problem” method for constructing confining potentials that make N -soliton waveforms exact solutions in the Gross–Pitaevskii equation
In this work, we discuss an application of the “inverse problem” method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross–Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose–Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov–de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.
more »
« less
- Award ID(s):
- 2204782
- PAR ID:
- 10534741
- Publisher / Repository:
- AIP Publishing
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 34
- Issue:
- 4
- ISSN:
- 1054-1500
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We prove existence, uniqueness and non-negativity of solutions of certain integral equations describing the density of states u ( z ) in the spectral theory of soliton gases for the one dimensional integrable focusing nonlinear Schrödinger equation (fNLS) and for the Korteweg–de Vries (KdV) equation. Our proofs are based on ideas and methods of potential theory. In particular, we show that the minimising (positive) measure for a certain energy functional is absolutely continuous and its density u ( z ) ⩾ 0 solves the required integral equation. In a similar fashion we show that v ( z ), the temporal analog of u ( z ), is the difference of densities of two absolutely continuous measures. Together, the integral equations for u , v represent nonlinear dispersion relation for the fNLS soliton gas. We also discuss smoothness and other properties of the obtained solutions. Finally, we obtain exact solutions of the above integral equations in the case of a KdV condensate and a bound state fNLS condensate. Our results is a step towards a mathematical foundation for the spectral theory of soliton and breather gases, which appeared in work of El and Tovbis (2020 Phys. Rev. E 101 052207). It is expected that the presented ideas and methods will be useful for studying similar classes of integral equation describing, for example, breather gases for the fNLS, as well as soliton gases of various integrable systems.more » « less
-
We consider variational principles related to V. I. Arnold’s stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the Navier–Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen’s vortex, providing a new stability proof with a stronger geometric flavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the influence of the viscous term in the particular example of the Gaussian vortex.more » « less
-
Abstract The mathematical description of localized solitons in the presence of large‐scale waves is a fundamental problem in nonlinear science, with applications in fluid dynamics, nonlinear optics, and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg–de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A topical review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton–rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite‐gap description are used to describe soliton–rarefaction wave and soliton–dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the inverse scattering transform. For transmitted solitons, far‐field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave–mean field interaction.more » « less
-
Abstract We write down and characterize a large class of nonsingular multi-soliton solutions of the defocusing Davey–Stewartson II equation. In particular we study their asymptotics at space infinities as well as their interaction patterns in the xy -plane, and we identify several subclasses of solutions. Many of these solutions describe phenomena of soliton resonance and web structure. We identify a subclass of solutions that is the analogue of the soliton solutions of the Kadomtsev–Petviashvili II equation. In addition to this subclass, however, we show that more general solutions exist, describing phenomena that have no counterpart in the Kadomtsev–Petviashvili equation, including V-shape solutions and soliton reconnection.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0183167
