Engineering topics

Cooper, Fred

Publications and source records attributed to Cooper, Fred.

10 records

DOE OSTI2024

Geometric Interpretation of a Non-Linear Extension of Quantum Mechanics

We recently introduced a particular non-linear generalization of quantum mechanics that has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics problem. In this paper, we suggest that the two components of the wave function represent the system described by the Hamiltonian H in two different asymptotic regions of spacetime and we show that the non-linear terms can be viewed as giving rise to gravitational effects.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Uniform-density Bose-Einstein condensates of the Gross-Pitaevskii equation found by solving the inverse problem for the confining potential

Here, in this work, we consider a “reverse-engineering” approach to construct confining potentials that support exact, constant density kovaton solutions to the classical Gross-Pitaevskii equation (GPE) also known as the nonlinear Schr¨odinger equation (NLSE). In the one-dimensional case, the exact solution is the sum of stationary kink and anti-kink solutions, i.e. a kovaton, and in the overlapping region, the density is constant. In higher dimensions, the exact solutions are generalizations of this wave function. In the absence of self-interactions, the confining potential is similar to a smoothed out finite square well with minima also at the edges. When self-interactions are added, a term proportional to ±gψ*ψ gets added to the confining potential and ±gM, where M is the norm, gets added to the total energy. In the realm of stability analysis, we find (linearly) stable solutions in the case with repulsive self-interactions which also are stable to self-similar deformations. For attractive interactions, however, the minima at the edges of the potential get deeper and a barrier in the center forms as we increase the norm. This leads to instabilities at a critical value of M (related to the number of particles in the BEC). Comparing the stability criteria from Derrick’s theorem and Bogoliubov-de Gennes analysis stability results, we find that both predict stability for repulsive self-interactions and instability at a critical mass M for attractive interactions. However, the numerical analysis gives a much lower critical mass. The numerical analysis shows further that the initial instabilities violate the symmetry x → -x assumed by Derrick’s theorem.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

A solvable model of a nonlinear extension of quantum mechanics

We introduce a particular nonlinear generalization of quantum mechanics which has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics problem. Here, we hope that this simple example will elucidate some of the issues of interpreting nonlinear generalization of quantum mechanics that have been put forth to resolve questions about quantum measurement theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the 'mass' (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. Here 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 all κ as long as their mass is less than a critical value M*(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 curve M*(κ) versus κ lies 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, for κ < 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 dimension d.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Stability of trapped solutions of a nonlinear Schrödinger equation with a nonlocal nonlinear self-interaction potential

This work focuses on the study of the stability of trapped soliton-like solutions of a (1 + 1)-dimensional nonlinear Schrödinger equation (NLSE) in a nonlocal, nonlinear, self-interaction potential of the form [|Ψ(x,t)| 2 +|Ψ(-x,t)| 2 ] κ where κ is an arbitrary nonlinearity parameter. Although the system with κ = 1 (i.e. fully integrable case) was first reported by Yang (2018 Phys. Rev. E 98 042202), here in the present work, we extend this model to the one in which κ is arbitrary. This allows us to compare the stability properties of the now trapped solutions to previously found solutions of the more usual NLSE with κ ≠ 1 which are moving soliton solutions. We show that there is a simple, one-component, nonlocal Lagrangian and corresponding action governing the dynamics of the system. Using a collective coordinate method derived from the action as well as assuming the validity of Derrick's theorem, we find that these trapped solutions are stable for 0 < κ < 2 and unstable when κ > 2. At the critical value of κ, i.e. κ = 2, the solution can either collapse or blowup linearly in time when q 0 = 0, where q 0 is the center of the initial density ρ(x, t = 0) = ψ*ψ of the solution. For q 0 ≠ 0 the displaced solution collapses. When κ > 2 initial small displacements from the origin also lead to collapse of the wave function. This phenomenon is not seen in the usual NLSE.

collective coordinates↗

DOE OSTI2021

Behavior of solitary waves of coupled nonlinear Schrödinger equations subjected to complex external periodic potentials with odd-$\mathcal{PT}$ symmetry

In this work, we discuss the response of both moving and trapped solitary wave solutions of a two-component nonlinear Schrödinger system in 1 + 1 dimensions to an odd-$\mathcal{PT}$ external periodic complex potential. The dynamical behavior of perturbed solitary waves is explored by conducting numerical simulations of the nonlinear system and using a collective coordinate variational approximation. We present case examples corresponding to choices of parameter values and initial conditions involved therein. The results of the collective coordinate approximation are compared against numerical simulations where we observe qualitatively good agreement between the two. Unlike the case for a single-component solitary wave in a complex periodic $\mathcal{PT}$-symmetric potential, the collective coordinate equations do not have a small oscillation regime, and initially the height of the two components changes in opposite directions often causing instability. We find that the dynamic stability criteria we have used in the one-component case are a good indicator for the onset of dynamic instabilities in the present setup.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Solitary wave solutions of the 2+1 and 3+1 dimensional nonlinear Dirac equation constrained to planar and space curves

Here, we study the effect of curvature and torsion on the solitons of the nonlinear Dirac equation considered on planar and space curves. Since the spin connection is zero for the curves considered here, the arc variable provides a natural setting to understand the role of curvature. We obtain for various curves in two and three dimensions the transformation of the Dirac equation to the 1 + 1 dimensional flat space Dirac equation written in terms of the arc length. Depending on the curvature, the soliton profile either narrows or expands. Our results may be applicable to yet-to-be-synthesized curved quasi-one dimensional Bose condensates.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Neutrino condensation from a new Higgs-like interaction

In this work, we revisit the scenario of having a new interaction between neutrinos and a Higgs-like scalar suggested by short-baseline neutrino anomalies. We find that there are two possible attractive channels in the resulting effective 4-Fermi theory which lead to a neutrino condensation in cosmic neutrinos and the creation of a neutrino superfluid at low temperatures and finite chemical potential. We find that at the minimum of the effective potential V the condensates are mostly made up of pairs of left-left + right-right composites, with a tiny admixture of left-right + right-left composites which differs from the scenario of Kapusta which focused on the energetically unfavored left-right + right-left scenario. We obtain an approximate equation for the gap Δ 2 = 4 m H ( μ - m ) Exp [ - π 2 G eff μ μ 2 - m 2 ] , where m H is the mass of the Higgs scalar, and G eff is the effective (renormalized) 4- Fermi interaction. Recent constraints on cosmological neutrinos suggest the possibility that log 10 ( G eff MeV 2 ) = - 1.72 which would lead to an exponentially small gap.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Stability and response of trapped solitary wave solutions of coupled nonlinear Schrödinger equations in an external, $\mathcal{PT}$- and supersymmetric potential

In this work, we present trapped solitary wave solutions of a coupled nonlinear Schrödinger (NLS) system in 1 + 1 dimensions in the presence of an external, supersymmetric and complex $\mathcal{PT}$-symmetric potential. The Schrödinger system this work focuses on possesses exact solutions whose existence, stability, and spatio-temporal dynamics are investigated by means of analytical and numerical methods. Two different variational approximations are considered where the stability and dynamics of the solitary waves are explored in terms of eight and twelve time-dependent collective coordinates (CCs). We find regions of stability for specific potential choices as well as analytic expressions for the small oscillation frequencies in the CC approximation. Our findings are further supported by performing systematic numerical simulations of the NLS system.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

Here, the damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter $\kappa$ is analyzed, when the external force is periodic in space and given by $\mathcal{f(x) = r}$ cos(Kx), both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force λ = 2π/K , and the soliton width $l_s$, which is a function of the nonlinearity $\kappa$ as well as the initial frequency ω 0 of the solitary wave, there is a transition (at fixed ω 0 ) from trapped to unbound behavior of the soliton, which depends on the parameters $\mathcal{r}$ and K of the external force and the nonlinearity parameter $\kappa$. We previously studied this phenomena when $\kappa$ = 1 where we showed that for λ $\gg l_s$ the soliton oscillates in an effective potential, while forλ $\ll l_s$ it moves uniformly as a free particle. In this paper we focus on the $\kappa$ dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of $\mathcal{r}$ and K when $\kappa$ = 1/2,1,2 at fixed value of the frequency ω 0 . Since the solitary wave gets narrower for fixed ω 0 as a function of $\kappa$, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase $\kappa$.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗