Narrow Results

The aim of this study is to give a deep investigation into the dynamics of the simplified modified Camassa–Holm equation (CHe) for shallow water waves. Taking advantage of the semi-inverse method, we develop the variational principle, based on which the Hamiltonian of the system is extracted. By means of the Galilean transformation, the governing equation is transformed into a planar dynamical system. Then, the bifurcation analysis is presented via employing the theory of the planar dynamical system. Correspondingly, the quasi-periodic and chaotic behaviors of the system are also discussed by introducing two different kinds of perturbed terms. Finally, the variational method is based on the variational principle and Ritz method, and the Kudryashov method is used to construct the diverse solitary wave solutions, which include the bright solitary, dark solitary, kink solitary and the bright–dark solitary wave solutions. The graphic depictions of the obtained diverse solitary wave solutions are presented to elucidate the physical properties. The findings of this research enable us to gain a deeper understanding of the nonlinear dynamic characteristics of the considered equation.

In this paper, we deal with the following fractional Kirchhoff–Schrödinger–Poisson system:

Suppose , . We deal with the existence or nonexistence conditions of ground states solution for the multi-singular elliptic problem:

In 1978, Rabinowitz proved the existence of a non-constant $T$-periodic solution for nonlinear Hamiltonian systems on ${\mathbf{\text{R}}}^{2n}$ with Hamiltonian function being super-quadratic at the infinity and zero for any given $T>0$. Since the minimal period of this solution may be $T/k$ for some positive integer $k$, he proposed the question whether there exists a solution with $T$ as its minimal period for such a Hamiltonian system. This is the so-called Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, this conjecture has been deeply studied by many mathematicians. But under the original structural conditions of Rabinowitz, the conjecture is still open when $n\ge 2$. In this paper, I give a brief survey on the studies of this conjecture and hope to lead to more interests on it.

We analyze the results published in this journal about two conditionally solvable quantum-mechanical models. We show that the authors failed to derive the spectrum correctly because they did not take into consideration all the roots of the truncation condition used for that aim and did not interpret them correctly.

A system consisting of an ion and an electron interacting through the Hulthén potential confined in an impenetrable spherical box, with the ion at the centre is considered. Superpotential which is the crucial quantity in supersymmetric quantum mechanics is proposed for the 1s and 2p states. Variational wave functions are thence derived. Energies are calculated from these for different values of the radius of box (R) and these are compared to the exact values; good agreement is shown to exist between the two. The variational wave functions are further employed to calculate the absorption oscillator strength for the 1s→2p transition and the dipole polarizability for different values of R.

Considering that gravitational force might deviate from Newton's inverse-square law and become much stronger in small scale, we present a method to detect the possible existence of extra dimensions in the ADD model. By making use of an effective variational wave function, we obtain the nonrelativistic ground energy of a helium atom and its isoelectronic sequence. Based on these results, we calculate gravity correction of the ADD model. Our calculation may provide a rough estimation about the magnitude of the corresponding frequencies which could be measured in later experiments.

The approximate energy expression for neutron matter is refined with respect to the tensor forces. The refined energy expression includes the main part of the three-body-cluster kinetic-energy terms caused by tensor correlations appropriately. Furthermore, the new energy expression guarantees necessary conditions on tensor structure functions. Significant improvement is seen in the numerical results of the variational calculation with the refined energy expression for neutron matter.

In this work, the energy eigenvalues for the confined Lennard–Jones potential are calculated through the Variational Method allied to the Supersymmetric Quantum Mechanics. Numerical results are obtained for different energy levels, parameters of the potential and values of confinement radius. In the limit, where this radius assumes great values, the results for the non-confined case are recovered.

The principle of stationary variance is advocated as a viable variational approach to quantum field theory (QFT). The method is based on the principle that the variance of energy should be at its minimum when the state of a quantum system reaches its best approximation for an eigenstate. While not too much popular in quantum mechanics (QM), the method is shown to be valuable in QFT and three special examples are given in very different areas ranging from Heisenberg model of antiferromagnetism (AF) to quantum electrodynamics (QED) and gauge theories.

For the general $D$-dimensional radial anharmonic oscillator with potential $V(r)=\frac{1}{g^2}\widehat{V}(gr)$ the perturbation theory (PT) in powers of coupling constant $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) is developed constructively in $r$-space and in $(gr)$-space, respectively. The Riccati–Bloch (RB) equation and generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate $r$-space and in $(gr)$-space, respectively, exploring the logarithmic derivative of wave function $y$. It is shown that PT in powers of $g$ developed in RB equation leads to Taylor expansion of $y$ at small $r$ while being developed in GB equation leads to a new form of semiclassical expansion at large $(gr)$: it coincides with loop expansion in path integral formalism. In complementary way PT for large $g$ developed in RB equation leads to an expansion of $y$ at large $r$ and developed in GB equation leads to an expansion at small $(gr)$. Interpolating all four expansions for $y$ leads to a compact function (called the Approximant), which should uniformly approximate the exact eigenfunction at $r\in [0,\infty )$ for any coupling constant $g\ge 0$ and dimension $D>0$. As a concrete application, the low-lying states of the cubic anharmonic oscillator $V=r^2+g{r}^3$ are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is $\lesssim 10^{-4}$ for $r\in [0,\infty )$ for coupling constant $g\ge 0$ and dimension $D=1,2,\dots .$ In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7–8 s.d. for $g\ge 0$ and $D=1,2,\dots .$

In our previous paper I (del Valle–Turbiner, 2019) a formalism was developed to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)=\frac{1}{g^2}\widehat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As a result, the Approximant was introduced — a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations, it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper, the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials $V(r)=r^2+g^{2(m-1)}{r}^{2m}$, $m=2,3$, respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8–12 figures for any $D=1,2,3,\dots$ and $g\ge 0$, while the relative deviation of the Approximant from the exact eigenfunction is less than $10^{-6}$ for any $r\ge 0$.

The variational method is used to study the hard confinement of a two-particle quantum system in two potential models, the Cornell potential and the global potential, with Dirichlet-type boundary conditions at various cut-off radii. The trial wave function is constructed as the product of the $1S$ free hydrogen atom wave function or $1S$ harmonic oscillator wave function times a linear cut-off function of the form $(r-z)$ to ensure hard entrapment within a sphere of radius z. The behavior of $|\psi {|}^2$, the wave function at the origin (WFO) and the mean radius $〈r〉$ are computed for different situations and compared for the two potential models.

In this paper,the ground state energies of hydrogen-like impurity in a lens-shaped quantum dot (GaAs/In_1-xGa_xAs) under vertical magnetic field have been discussed by using effective mass approximation and variational method. It gives that for a lens-shaped quantum dot, due to the asymmetry of the vertical and lateral bound potentials, the electronic ground state energies are related not only with the deviation distance but also with the deviation direction; for the spherical quantum dot, the ground state energy is only related with the distance of the impurity deviation, neither with vertical nor lateral deviation. And with the increasing of the magnetic field, the ground state energy is increasing.

The ground-state energies of large polaron E_p and bipolaron E_B in three-dimensional lightly doped cuprates are calculated variationally taking into account the short- and long-range electron-phonon interactions and Coulomb correlation in the continuum model and adiabatic approximation. The binding energy of a large bipolaron and its stability region are determined as a function of the ratio of dielectric constants η = ε_∞/ε₀. It is found that the large bipolaron is stable in a broad region of η.

A variational calculation is given within the effective mass approximation of the binding energy of a hydrogenic donor impurity located on the axis of an infinitely long circular quantum well wire. A uniform magnetic field is applied along the axis of the wire. The different effective masses of the wire and the barrier are taken into consideration. This has not been done hitherto in the presence of the magnetic field. New analytical expressions for the electron energy levels and the binding energies of the hydrogenic impurity in the ground and the first four excited states have been derived.

Moreover, the form of the binding energy reported in earlier works in the special case of zero magnetic field has been amended. A new form of the trial wavefunction has also been introduced. It has the advantage of satisfying the required boundary conditions in the case of different masses of the wire and barrier, and thus it resembles the exact solution in this respect. The new form of the trial wavefunction has been applied to the case of the ground state. It has improved the results of binding energy considerably as they tend to approach the exact solution from below.

In this paper, we systematically investigate the periodic solutions of the Rössler equations up to certain topological length. To overcome the difficulties for a return map that is multivalued and non-invertible in the nonlinear system, we propose a new approach that establishes one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is numerically stable for cycle searching, and two-orbit fragments can be used as basic building blocks to initialize the system. The topological classification based on the whole orbit structure seems more effective than partitioning the Poincaré surface of section. The current research supplies an interesting framework for a systematic classification of periodic orbits in a chaotic flow.

In this paper, the exciton binding energy in a cubic semiconductor nanostructure quantum dot has been investigated by using variational method. Results show that in cubic quantum dot with a size of 50 Å, will have the most exciton binding energy.

In this study, we investigate the evolution of vortex in harmonically trapped two-component coupled Bose–Einstein condensate with quintic-order nonlinearity. We derive the vortex solution of this two-component system based on the coupled quintic-order Gross–Pitaevskii equation model and the variational method. It is found that the evolution of vortex is a metastable state. The radius of vortex soliton shrinks and expands with time, resulting in periodic breathing oscillation, and the angular frequency of the breathing oscillation is twice the value of the harmonic trapping frequency under infinitesimal nonlinear strength. At the same time, it is also found that the higher-order nonlinear term has a quantitative effect rather than a qualitative impact on the oscillation period. With practical experimental setting, we identify the quasi-stable oscillation of the derived vortex evolution mode and illustrated its features graphically. The theoretical results developed in this work can be used to guide the experimental observation of the vortex phenomenon in ultracold coupled atomic systems with quintic-order nonlinearity.

Exciton states are essential to comprehend the basal photoelectric properties in metal halide perovskites (MHPs) and provide reference for their future research, in which the exciton binding energy (EBE), determining the balance of the populations between excitons and free carriers, plays an important role in defining the optoelectronic utilization of MHPs. Thereby, we theoretically study the effects of bound potentials, due to the exciton coupling with the longitudinal optical (LO) phonon, between the electron and hole of the exciton on the EBE applying the variational method by using different effective potentials and two trail wavefunctions. We find that the EBE of this kind of materials is not only related to the chemical composition, but also remains inseparable from the space size, dielectric constant and LO-phonon energy, moreover, these correlations are better described by Barentzen potential. In addition, the results also show that the effects of carriers-LO-phonon coupling can explain the relationships between the EBE and exciton active range and effective Bohr radius to a certain extent, and can analyze their intrinsic correlation among these factors. These findings enable us to explain some experimental results and provide some help to understand optical electric dynamics in MHPs.