Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

Compactification and signature transition in Kaluza-Klein spinor cosmology

We study the classical and quantum cosmology of a 4 + 1-dimensional space-time with a non-zero cosmological constant coupled to a self-interacting massive spinor field. We consider a spatially flat Robertson-Walker universe with the usual scale factor R (t) and an internal scale factor a (t) associated with the extra dimension. For a free spinor field the resulting equations admit exact solutions, whereas for a self-interacting spinor field one should resort to a numerical method for exhibiting their behavior. These solutions give rise to a degenerate metric and exhibit signature transition from a Euclidean to a Lorentzian domain. Such transitions suggest a compactification mechanism for the internal and external scale factors such that a ∼ R⁻¹ in the Lorentzian region. The corresponding quantum cosmology and the ensuing Wheeler-DeWitt equation have exact solutions in the mini-superspace when the spinor field is free, leading to wavepackets undergoing signature change. The question of stabilization of the extra dimension is also discussed.     

Baryon duality solutions with the harmonic-oscillator spectrum and the amplitude signs in πN→πΔ

By constructing exact exchange-degenerate solutions of baryons with the harmonic-oscillator spectrum for a set of reactions MB→M′B′, MB → M′D′ and MD → M′D′, we investigate the duality constraints on the amplitude signs in MB → M′D′. It is shown that the amplitude signs determined by duality are consistent with those predicted by the extended version of SU(6)_W symmetry with the universal dominance of the ΔL_Z = 0 or ΔL_Z = ±1 amplitude in the mesonic transitions of the baryons.     

Coherence relaxation during the diffusion of resonance radiation in media with a magnetic field

The kinetic Green's function method is used to obtain equations which describe the transport of resonance radiation in magnetic fields for arbitrary ratios between the natural widthγ, the Doppler width Δω _D, and the Zeeman splitting of the excited atomic levels. It is shown that as ΔωD/γ→0 the equations become very much simpler and in particular cases admit an exact solution. In particular, the decay of coherence in an infinite homogeneous space is characterized by five relaxation times, which are defined by a system of algebraic equations.     

New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational exp <Emphasis Type="BoldItalic">(−φ(η))</Emphasis>-expansion method

In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into the corresponding partial differential equation and the rational exp (?φ(η))-expansion method is implemented to find exact solutions of nonlinear equation. We find hyperbolic, trigonometric, rational and exponential function solutions using the above equation. The results of various studies show that the suggested method is very effective and can be used as an alternative for finding exact solutions of nonlinear equations in mathematical physics. A comparative study with the other methods gives validity to the technique and shows that the method provides additional solutions. Graphical representations along with the numerical data reinforce the efficacy of the procedure used. The specified idea is very effective, pragmatic for partial differential equations of fractional order and could be protracted to other physical phenomena.     

A probabilistic analysis of the difficulties of unifying quantum mechanics with the theory of relativity

A procedure is given for the transformation of quantum mechanical operator equations into stochastic equations. The stochastic equations reveal a simple correlation between quantum mechanics and classical mechanics: Quantum mechanics operates with “optimal estimations,” classical mechanics is the limit of “complete information.” In this connection, Schrödinger's substitution relationsp _x → -i? ?/?x, etc, reveal themselves as exact mathematical transformation formulas. The stochastic version of quantum mechanical equations provides an explanation for the difficulties in correlating quantum mechanics and the theory of relativity: In physics “time” is always thought of as a numerical parameter; but in the present formalism of physics “time” is described by two formally totally different quantities. One of these two “times” is a numerical parameter and the other a random variable. This last concept of time shows all the properties required by the theory of relativity and is therefore to be considered as the relativistic time.     

Three-body scattering in configuration space

The asymptotic forms of the wavefunction and Faddeev components in configuration space are shown to determine uniquely the solutions of the Schrödinger or Faddeev differential equations for 2 → (2, 3) and 3 → (2, 3) processes. An antisymmetrized form of the Faddeev differential equation for three equivalent fermions is given and its angular analysis is performed in the general case of local potentials with tensor interaction for neutron-deuteron scattering. We describe a numerical method for solving the corresponding boundary value problem and apply it to scattering and break-up at E_n^1ab = 14.4 MeV in the doublet S state for the four local potentials of Malfliet and Tjon, Reid, de Tourreil and Sprung, and de Tourreil, Rouben and Sprung. For the three realistic potentials, elastic scattering amplitudes differ by 5%, and amplitudes for break-up in the two-neutron state ¹S₀ differ by less than 4%.     

基于严格交换势的低能电子与H2分子碰撞振动激发散射截面的研究

严格交换势用于研究低能电子与H₂分子的弹性和非弹性散射截面,线性代数方法和R-矩阵传播子相结合求解基于振动密耦合方法的积分-微分耦合方程组,由此得到收敛的(0→0,0→1,0→2)散射微分截面和积分截面.理论计算结果与目前优秀的实验值和其他理论计算值进行了比较,表明基于振动密耦合方程的严格交换势在低能电子与H₂分子振动激发散射中有重要作用. 关键词： 严格交换势 2分子振动激发')" href="#">H₂分子振动激发 微分截面 积分截面     

On the superradiant phase transition for molecules in a quantized radiation field: the dicke maser model

A system of N two-level molecules coupled to finitely many modes of a quantized radiation field via a truncated dipolar interaction is investigated. The thermodynamic and correlation functions can be exactly computed in the limit N → ∞. The system exhibits a second order phase transition from normal to superradiance. Different effective Hamiltonians with linear Heisenberg equations of motion become asymptotically exact in the limit N → ∞.     

Spectral properties of the period-doubling lattice: exact renormalization group study

Using an exact real space renormalization group technique, we have studied the electronic and vibrational properties of a class of one dimensional aperiodic lattice model, for which the substitution rules are A → AB and B → AA. The spectra are self-similar and exhibit hierarchical structures. Analytical and numerical analyses reveal the existence of extended eigenstates.     

All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation listing

This paper derives all 36 analytical solutions of the energy eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor for polynomial degrees 1 through 4 using classical algebraic theory. By the use of double-parameterization the full general solution sets are illustrated in a compact, symmetric, structural, and usable form that is valid for asymmetry parameter $\eta \in \left({- \infty , + \infty}\right)$ . These results are useful for code developers in the area of Perturbed Angular Correlation (PAC), Nuclear Quadrupole Resonance (NQR) and rotational spectroscopy who want to offer exact solutions whenever possible, rather that resorting to numerical solutions. In addition, by using standard linear algebra methods, the characteristic equations of all integer and half-integer spins I from 0 to 15, inclusive are represented in a compact and naturally parameterized form that illustrates structure and symmetries. This extends Nielson’s?[1] listing of characteristic equations for integer spins out to I?=?15, inclusive.     

Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.     

Loop dynamics for gauge theories: A numerical algorithm

The loop equations defining the lattice U(N) gauge theory are recalled and a formal solution is presented (for N = 1 and N → ∞). Wilson's expectation function W(C) for a loop C is expressed as a matrix element of a resolvent in the space of all the loops between C and 0 (no loop). It is shown that such a solution provides a new numerical algorithm to compute physical quantities. This is based on a Ulam-von Neumann type of method for computing inverse matrix elements by introducing importance sampling for the paths in the space of the matrix indices, which in this case are the loops. As a result W(C) is obtained by summing over important paths in the loop space connecting C to 0. A Monte Carlo program is presented, for the N → ∞ case, where a very simple form for the importance sampling is introduced so that the computer time for each step in the construction of the path is minimized. The rates for successful paths (i.e. path C → 0 within a given finite number of steps) are computed for D = 2 and D = 4. Both rates and computer time involved encourage us to attempt a large scale calculation. Here the numerical studies of the convergence and of the fluctuations are presented only for D = 2. Convergence is rather fast, but specially in the weak-coupling region rare and large fluctuations appear thus suggesting that a better tuning for the importance sampling is needed.     

Computing exact solutions for conformable time fractional generalized seventh-order KdV equation by using $${\left( {{\varvec{G}}}^{\prime }/{{\varvec{G}}}\right) }$$-expansion method

In this paper, the authors have established the \(\left( G^{\prime }/G\right)\)-expansion method to find exact solutions for conformable time fractional generalized seventh-order KdV equation (FGKdV7). This method is an effective method in finding exact traveling wave solutions of nonlinear evolution equations in mathematical physics. The effectiveness of this manageable method has been shown by applying it to several particular cases of the FGKdV7. The present approach has the potential to be applied to other nonlinear fractional differential equations. All of the numerical calculations in the present study have been performed on a PC applying some programs written in Mathematica.     

The correlation energies and nonlinear optical absorptions of an exciton in a disc-like quantum dot

Using exact diagonalization techniques, the low-lying states of an exciton, and the linear and nonlinear optical absorptions in a disc-like quantum dot are theoretically studied. The numerical results for the typical GaAs material show the so-called quantum size effect. Also, our study is restricted on the transition between the S state (L = 0) and the P state (L = 1). The optical absorption coefficients are greatly enhanced because of the induced size confinement. Meantime, we find that the total optical absorption coefficient is about two times bigger than that obtained by without considering exciton effects. Additionally, the optical absorption saturation intensity can be controlled by the incident optical intensity I.     

Stochastic theory of nonlinear rate processes with multiple stationary states

A comparison has been made between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states. We have shown, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of the deterministic equations. The presence of fluctuations in the stochastic formulation leads to true equilibrium solutions, i.e. solutions which are independent of initial conditions as t → ∞. We show that the stochastic formulation leads to two distinct time scales for relaxation. The mean time for the reaction system to reach the quasi-stationary states from any initial state is of $\text{O}$(N⁰) where N is a measure of the size of the reaction system. The mean time for relaxation from a quasi-stationary state to the true equilibrium state is $\text{O}$(e^N). The results obtained from the stochastic formulation as regards the number and location of the quasi-stationary states are in complete agreement with the deterministic results.     

Proton structure functions in the quasielastic limit

The evolution equation for the nonsinglet distribution of partons is solved in the leading order of perturbative QCD. It is shown that an exact analytic solution to the evolution equations can be found in the quasielastic limit. The Q ² evolution of the structure function for x → 1 is in good agreement with experimental data.     

Unified compact ADI methods for solving nonlinear viscous and nonviscous wave equations

In this paper, two unified alternating direction implicit (ADI) methods, based on the combination of fourth-order compact difference for the approximations of the second spatial derivatives with approximation factorization of difference operators, are presented for solving a two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations, respectively. By the discrete energy method, it is shown that their solutions converge to exact solutions with an order of two in time and four in space in L²- and H¹-norms. Finally, numerical findings testify the computational efficiency of the algorithms.