Lump waves,solitary waves and interaction phenomena to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we investigated the $(2+1)$-dimensional Konopelchenko–Dubrovsky equation. The lump waves, solitary waves as well as interaction between lump waves and solitary waves are presented based on the Hirota bilinear form of this equation. It is worth noting that the rational solutions are obtained by taking a long wave limit, and we also discussed the lump solutions and rogue wave solutions. Moreover, all these solutions are presented via 3-dimensional plots and density plots with choosing some special parameters to show the dynamic graphs.     

On the weakly-bound (1,1)-states in the three-body ddμ and dtμ muonic ions

The both total and binding energies of the $(1,1)$-states in the weakly-bound three-body muonic ddμ and dtμ ions are determined to high numerical accuracy. The binding energy of the $(1,1)$-state in the muonic dtμ ion is evaluated as $\epsilon (dt\mu )=?0.66033003831(30)\text{eV}$, while for the same state in the muonic ddμ ion we have found that $\epsilon (dd\mu )=?1.9749806166970(30)\text{eV}$. These energies are the most accurate numerical values obtained for these systems and they are sufficient for all current and future experimental needs.     

I–V characteristics and conductance of strained SWCNTs

We present a new procedure to investigate the I–V characteristics and the conductance for strained SWCNTs. These electronic transport properties have been studied theoretically at zero temperature for zig-zag, armchair and chiral SWCNTs under the effect of the uniaxial tension and torsional strain. The analytical expression of the energy spectrum in the tight binding approximation has been used to calculate the induced current and the conductance through Landauer–Büttiker formalism. It is shown that the conductance for unstrained CNTs at initial values of the voltage can take discrete values which are equal to zero and 4 ($e^2/h$) for semiconducting and conducting SWCNTs respectively. The emergence of the kinks in the I–V characteristics is due to the discrete electronic spectrum in the SWCNTs. The location and number of kinks are changeable under the effect of strain process. The conductance in a strained armchair $(5,5)$ CNT decreases to zero under torsional strain, consequently, it will transform the conducting SWCNTs at a threshold value of strain to a semiconducting SWCNT. In contrast, by applying the uniaxial tension on the armchair $(5,5)$ CNT, the conductance does not change absolutely. There is a different behavior can be observed by applying the strain on zig-zag $(10,0)$ CNT, where the conductance decreases rapidly and slightly under the influence of uniaxial tension and torsional strain, respectively. We found that the conductance of chiral $(10,9)$ CNT is not significantly affected by applying the strain under consideration. More interestingly, the band structure of chiral $(10,9)$ CNT under uniaxial tension and torsional strain have been investigated within the tight binding approximation.     

Asymptotic behavior of two-electron expectation values in two-electron excited states

Singly-excited states of the two-electron atom cease being bound when $Z\to 1$ (from above), the outer orbital becoming infinitely diffuse. The asymptotic relations$\underset{Z\to 1}{\lim }?{(Z?1)}^k〈{(1sns)}^{1,3}S\left|r_{12}^k\right|{(1sns)}^{1,3}S〉=〈(n?1)s^{(0)}\left|r^k\right|(n?1)s^{(0)}〉,$ where $k=?1,1,2,3,?$, are demonstrated to hold. Here, $(n?1)s^{(0)}$ is a hydrogenic s orbital with principal quantum number $(n?1)$. New, more nuanced light is shed on the already challenged dogma that the Pauli principle keeps the electrons further apart in the triplet than in the corresponding singlet.     

Duality between the SU(2) lattice gauge model and bosonic t − J model

We investigate the duality between the $SU(2)$ lattice gauge model and the bosonic $t-J$ model. We construct the relations between the gauge field operators and particle operators, and map the low-energy regime of the $SU(2)$ lattice gauge model to a $U(1)$ bosonic $t-J$ model coupled with a $U(1)$ gauge field. The mapped model can be interpreted as a bosonic $t-J$ model with particle-hole symmetry, or a mean-field form of the bosonic $t-J$ model with the coexistence of a two-particle pairing and four particle-pairing. The duality between the lattice gauge model and the bosonic $t-J$ model provides a direct connection between gauge theory and strongly correlated systems.     

Thermodynamic calculation of spin scaling functions

Critical phenomena theory centers on the scaled thermodynamic potential per spin $?(\beta ,h)=|t{|}^{p}Y(h|t{|}^{?q})$, with inverse temperature $\beta =1/T$, $h=?\beta H$, ordering field H, reduced temperature $t=t(\beta )$, critical exponents p and q, and function $Y(z)$ of $z=h|t{|}^{?q}$. I discuss calculating $Y(z)$ with the information geometry of thermodynamics. Scaled solutions are found to obtain with three admissible functions $t(\beta )$: 1) $t=e^{?J\beta }$, 2) $t={\beta }^{?1}$, and 3) $t={\beta }_C?\beta$, where J and ${\beta }_C$ are constants. For $p=q$, information geometry yields $Y(z)=\sqrt{1+z^2}$, consistent with the one-dimensional (1D) ferromagnetic Ising model.     

Resonance studies using the contour deformation method in the complex momentum plane

A contour deformation method (CDM) in the complex momentum plane has been successfully extended and implemented to probe resonances in atomic and molecular systems. Specifically, solution of the Schrödinger equation is performed in momentum space with momentum deformed on a contour in the complex plane. The bound, resonant, and complex continuum states could be directly revealed from the eigenvalues of the Schrödinger equation in the complex momentum plane. The calculations of shape resonances in electron scattering with Na⁺ in Debye plasmas (one channel), and in the charge transfer process ${\mathrm{H}}^{?}(1s^2)+\mathrm{Li}(1s^{2}2s)$ ($1^2{\Sigma }^{+}$) $\to \mathrm{H}(1s)+{\mathrm{Li}}^{?}(1s^{2}2s^2)$ ($2^2{\Sigma }^{+}$) (coupled channels) are given as illustrative examples. It is shown that calculated results from CDM agree very well with those extracted from the eigenphase sum of scattering theories. The effectiveness of CDM is also demonstrated by comparing its results with those obtained by the complex rotation scaling and exterior complex scaling methods. The convergence of CDM results can be obtained by increasing the momentum integration region and the number of integration points. The studied examples demonstrate that CDM could be a powerful tool for studies of resonances in complex atomic and molecular systems.     

New findings for two new type sine hyperbolic potentials

We propose two new type sine hyperbolic potentials $V(x)=a^2{\sinh }^2?(x)?k{\tanh }^2?(x)$ and $V(x)=c^2{\sinh }^4?(x)?k{\tanh }^2?(x)$. They may become single- or double-well potentials depending on the potential parameters $a,c$ and k. We find that its exact solutions can be written as the confluent Heun functions $H_c(\alpha ,\beta ,\gamma ,\delta ,\eta ;z)$, in which the energy level E is involved inside the parameter η. The properties of the wave functions, which is strongly relevant for the potential parameters $a,c$ and k, are illustrated.     

Composition law for the Cole-Cole relaxation and ensuing evolution equations

Physically natural assumption says that any relaxation process taking place in the time interval $[t_0,t_2]$, $t_2>t_0\ge 0$ may be represented as a composition of processes taking place during time intervals $[t_0,t_1]$ and $[t_1,t_2]$ where $t_1$ is an arbitrary instant of time such that $t_0\le t_1\le t_2$. For the Debye relaxation such a composition is realized by usual multiplication which claim is not valid any longer for more advanced models of relaxation processes. We investigate the composition law required to be satisfied by the Cole-Cole relaxation and find its explicit form given by an integro-differential relation playing the role of the time evolution equation. The latter leads to differential equations involving fractional derivatives, either of the Caputo or the Riemann-Liouville senses, which are equivalent to the special case of the fractional Fokker-Planck equation satisfied by the Mittag-Leffler function known to describe the Cole-Cole relaxation in the time domain.