arXiv subjects

Jackson C. Turner

Publications and source records attributed to Jackson C. Turner.

4 recordsLinked to original sources

arXiv2026

On the ideal stability of the sheared-flow Z pinch

Sheared-flow Z-pinch stability has been studied within ideal MHD primarily through growth rate calculations, which find that even trans-Alfv\'{e}nic sheared flows apparently fail to suppress the kink instability. Trans-Alfv\'{e}nic sheared flow does stabilize the MHD kink but also excites shear-driven instabilities characteristic of high-Reynolds-number supersonic flow. This distinction is evident from the dispersion relations underlying the growth rates, computed here as the analytic dispersion function in the complex-frequency plane. Regularization splits this function into adiabatic and resonant parts describing how discrete modes emerge from and interact with the continuous spectrum. The Doppler-shifted flow continuum interacts with the interchange and kink instabilities in distinct ways. For interchange, the continuum overlaps the instability branch at all wavenumbers, so even sub-Alfv\'{e}nic sheared flow stabilizes profiles modestly beyond the interchange threshold. The kink, by contrast, is shielded from the continuum by a frequency gap, and trans-Alfv\'{e}nic flow is required to Doppler-shift the continuum into resonance with it, giving a geometric picture of the stabilization threshold. But shear-driven instabilities arise at this same threshold, including reflection modes and an acoustic kink. It is these shear-driven modes, not the original MHD instabilities, that dominate the ideal-MHD spectrum in trans-Alfv\'{e}nic conditions. The ideal analysis thus describes the stabilization mechanism while showing that the stability of the sheared-flow Z pinch ultimately rests on non-ideal physics, including finite orbit width and dissipation.

physics.plasm-ph↗

arXiv2025

Resonance-induced nonlinear bound states

We study nonlinear bound states -- time-harmonic and spatially decaying ($L^2$) solutions -- of the nonlinear Schrödinger / Gross--Pitaevskii equations (NLS/GP) with a compactly supported linear potential. Such solutions are known to bifurcate from the $L^2$ bound states of an underlying Schrödinger operator $H_V=-\partial_x^2+V$. In this article we prove an extension of this result: for the 1D NLS/GP, nonlinear bound states also arise via bifurcation from the scattering resonance states and transmission resonance states of $H_V$, associated with the poles and zeros, respectively, of the reflection coefficients, $r_\pm(k)$, of $H_V$. The corresponding resonance states are non-decaying and only $L^2_{\rm loc}$. In contrast to nonlinear states arising from $L^2$ bound states of $H_V$, these resonance bifurcations initiate at a strictly positive $L^2$ threshold which is determined by the position of the complex scattering resonance pole or transmission resonance zero.

math-ph↗

arXiv2022

A generalized expansion method for computing Laplace-Beltrami eigenfunctions on manifolds

Eigendecomposition of the Laplace-Beltrami operator is instrumental for a variety of applications from physics to data science. We develop a numerical method of computation of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a smooth bounded domain based on the relaxation to the Schrödinger operator with finite potential on a Riemannian manifold and projection in a special basis. We prove spectral exactness of the method and provide examples of calculated results and applications, particularly, in quantum billiards on manifolds.

math.NA↗

arXiv2022

Flat tori with large Laplacian eigenvalues in dimensions up to eight

We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $λ_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For larger $k$, this is equivalent to the NP-hard problem of finding the $d$-dimensional (dual) lattice with longest $k$-th shortest lattice vector. As a result of extensive computations, for $d \leq 8$, we obtain a sequence of flat tori, $T_{k,d}$, each of volume one, such that the $k$-th Laplacian eigenvalue of $T_{k,d}$ is very large; for each (finite) $k$ the $k$-th eigenvalue exceeds the value in (the $k\to \infty$ asymptotic) Weyl's law by a factor between 1.54 and 2.01, depending on the dimension. Stationarity conditions are derived and numerically verified for $T_{k,d}$ and we describe the degeneration of the tori as $k \to \infty$.

math.SP↗