Publications:

• Turner, J. C. & M. I. Weinstein (2025). Resonance-induced Nonlinear Bound States. arXiv:2510.19538 [math-ph].
  
  We study a special class of bound state solutions to the nonlinear Schrödinger / Gross–Pitaevskii equation. Nonlinear bound states are known to emerge from the usual square-integrable bound states of the underlying linear Schrödinger operator.
  
  In this work, we show a broader and more surprising phenomenon. For the one-dimensional equation, nonlinear bound states can also form from resonant structures of the linear operator—specifically, from scattering resonances and transmission resonances. These resonances are not square-integrable and do not decay at infinity. They arise as imaginary poles and zeros of the reflection coefficient in the scattering theory. We prove that each resonance generates its own family of nonlinear bound states, which arise through a bifurcation mechanism at a strictly positive norm threshold determined by the location of the resonance in the complex spectral plane.
  
  
• Turner, J. C. & D. Crews (2025). Linear Stability Analysis of Ideal Magnetohydrodynamics in Cylindrical Shear Flows. Manuscript in preparation.
  
  In this work, we analyze the linear stability of the ideal magnetohydrodynamics (MHD) equations in a cylindrical geometry with shear flow and axisymmetric profiles. We identify and study a class of shear-driven instabilities known as Mack modes, which arise in the presence of flow shear even when classical MHD theory predicts stability. For example, a pressure–magnetic Kadomtsev equilibrium is known to be stable to axisymmetric disturbances, yet with shear included we show that new unstable modes appear at high wavenumbers and high frequencies, occurring at discrete resonance-like intervals. These modes are fundamentally different from the more familiar incompressible Kelvin–Helmholtz instability, despite also being driven by shear.
  
  To conduct this analysis, we develop and implement a high-precision eigenvalue solver that improves on previous approaches. Our method uses an adaptive radial grid to resolve boundary-layer behavior and employs parameter continuation techniques—motivated by the Implicit Function Theorem—to efficiently track eigenvalues across parameter regimes.
  
  
• Kao, C.-Y., Osting, B., & Turner, J. C. (2023). Flat Tori with Large Laplacian Eigenvalues in Dimensions up to Eight. SIAM Journal on Applied Algebra and Geometry, 7, 172–193. https://doi.org/10.1137/22M1478823
  
  We study an optimization problem in spectral geometry: how to choose the shape of a flat, d-dimensional torus (a multi-dimensional periodic space) so that a chosen Laplacian eigenvalue is as large as possible. For the first eigenvalue, this problem is closely related to classical sphere packing: the optimal torus corresponds to the densest lattice arrangement of spheres. For higher eigenvalues, the problem becomes significantly more complex and connects to finding lattices with unusually long shortest vectors.
  
  Through extensive computations in dimensions up to eight, we identify a family of flat tori whose k-th Laplacian eigenvalues are remarkably large. In each dimension, these optimized eigenvalues exceed the predictions of the classical asymptotic Weyl law by a factor between roughly 1.5 and 2. We derive the stationarity conditions that these optimal tori must satisfy and verify them numerically. Finally, we describe how the geometry of these tori changes as the eigenvalue index becomes large.
  
  
• Turner, J. C., Cherkaev, E., & Wang, D. (2022). A Generalized Expansion Method for Computing Laplace–Beltrami Eigenfunctions on Manifolds. arXiv:2210.10982 [math.NA].
  
  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. (preprint)
  
  
• Turner, J. C. (2021). Expansion Method for Eigenvalue Problems: Theories, Algorithms, and Applications. Honors Thesis, University of Utah. Link
  
  The Laplacian operator plays a ubiquitous role in the differential equations that describe many physical systems. These include, for example, vibrating membranes, fluid flow, heat flow, and solutions to the Schrödinger equation. In this paper, we investigate a method to find the eigenfunctions of the Laplacian operator and extend it to other surfaces: spherical and flat tori. We code this method and give various examples and applications of the computation results. We develop a novel robust and fast method to solve for the Laplacian eigenfunctions on various surfaces – called the partitioned expansion method – which we describe in full and give examples of computational results. Lastly, we utilize the expansion method to study the dependence of the spectra of domains on some physical parameter.