abstract

The texture of phase-space and bifurcation diagrams of two-dimensional discrete maps describing a lattice of interacting oscillators, confined in bi-stable potentials with deformable double-well shapes, are examined. There are considered two bistable potentials that belong to a family of hyperbolic nonlinear on-site potentials whose double-well shapes can be tuned differently: one has a variable barrier height and the other has variable minima positions. However, the two hyperbolic double-well potentials reduce to the well-known canonical \(\phi ^4\) field, familiar in the studies of structural phase transitions in centro-symmetric crystals and bistable processes in biophysics. It is shown that although the parametric maps are area-preserving, their routes to chaos display different characteristic features: the first map exhibits a cascade of period-doubling bifurcations with respect to the potential amplitude, but period-halving bifurcations with respect to the shape deformability parameter. On the other hand, the first bifurcation of the second map always coincides with the first pitchfork bifurcation of the \(\phi ^4\) map. However, an increase of the deformability parameter shrinks the region between successive period-doubling bifurcations. The two opposite bifurcation cascades characterizing the first map, and the shrinkage of regions between successive bifurcation cascades which is characteristic of the second map, suggest a non-universal character of the Feigenbaum-number sequences associated with the two discrete parametric double-well maps.

abstract

Relaxation properties (specifically time-rates) of the Smoluchowski diffusion process on a line, in a confining potential \(U(x) \sim x^m\), \(m=2n \geq 2\), can be spectrally quantified by means of the affiliated Schrödinger semigroup \(\exp (-t\hat {H})\), \(t\geq 0\). The inferred (dimensionally rescaled) motion generator \(\hat {H}= - {\mit \Delta } + {\cal {V}}(x)\) involves a potential function \({\cal {V}}(x)= ax^{2m-2} - bx^{m-2}\), \(a=a(m)\), \(b=b(m)\!\gt \!0\), which for \(m\!\gt \!2\) has a conspicuous higher degree (superharmonic) double-well form. For each value of \(m\gt 2\), \( \hat {H}\) has the zero-energy ground-state eigenfunction \(\rho _*^{1/2}(x)\), where \(\rho _*(x) \sim \exp -[U(x)]\) stands for the Boltzmann equilibrium PDF of the diffusion process. A peculiarity of \(\hat {H}\) is that it refers to a family of the quasi-exactly solvable Schrödinger-type systems, whose spectral data are either residual or analytically unavailable. As well, no numerically assisted procedures have been developed to this end. Except for the ground-state zero eigenvalue and incidental trial-error outcomes, the lowest positive-energy levels (and energy gaps) of \(\hat {H}\) are unknown. To overcome this obstacle, we develop a computer-assisted procedure to recover an approximate spectral solution of \(\hat {H}\) for \(m\gt 2\). This task is accomplished for the relaxation-relevant low part of the spectrum. By admitting larger values of \(m\) (up to \(m=104\)), we examine the spectral “closeness” of \(\hat {H}\), \(m\gg 2\) on \(R\) and the Neumann Laplacian \({\mit \Delta }_{\cal {N}}\) in the interval \([-1,1]\), known to generate the Brownian motion with two-sided reflection.

abstract

We try to give a theoretical analysis of \(2\nu \beta ^{-}\beta ^{-}\) decay to the final excited \(2^{+}\) states by considering SU(4) symmetry restoration within the framework of quasi-particle random phase approximation (QRPA). Pyatov’s method is used to restore the symmetry violations stemming from the mean-field approximation. A comparison of the calculated decay rates with other calculations and the corresponding experimental data is given.

abstract

The fusion cross sections and barrier distributions of stable Zr isotope targets \(^{90,92,94,96}\)Zr with \(^{30}\)Si projectile nuclei are investigated theoretically via the Energy-dependent Woods–Saxon Potential (EDWSP) and Coupled Channel (CC) models. In our calculations, we have taken the bombarding energy range of the projectile–target interaction to be around the Coulomb barrier for all reactions. All theoretical accounts have been worked via the NRV Knowledge Base, the CCFULL code, and Wong’s formula. We detailed interrogated the repercussions of phonon number in projectile and target nuclei on heavy-ion fusion cross sections and barrier distributions. Over this investigation, we presented that the EDWSP and CC models, and all computation bases operated to elucidate the fusion cross-section data and barrier distributions are decent. Our theoretical investigation proves the importance of inspecting heavy-ion fusion reactions with theory-based research and encourages new experimental investigations that are not yet included in the literature.