PT symmetry in a fractional Schrödinger equation

We investigate the fractional Schrödinger equation with a periodic ‐symmetric potential. In the inverse space, the problem transfers into a first‐order nonlocal frequency‐delay partial differential equation. We show that at a critical point, the band structure becomes linear and symmetric in the one‐dimensional case, which results in a nondiffracting propagation and conical diffraction of input beams. If only one channel in the periodic potential is excited, adjacent channels become uniformly excited along the propagation direction, which can be used to generate laser beams of high power and narrow width. In the two‐dimensional case, there appears conical diffraction that depends on the competition between the fractional Laplacian operator and the ‐symmetric potential. This investigation may find applications in novel on‐chip optical devices.

     

Spectral splitting method for nonlinear Schrödinger equations with singular potential

We consider the time-dependent one-dimensional nonlinear Schrödinger equation with pointwise singular potential. By means of spectral splitting methods we prove that the evolution operator is approximated by the Lie evolution operator, where the kernel of the Lie evolution operator is explicitly written. This result yields a numerical procedure which is much less computationally expensive than multi-grid methods previously used. Furthermore, we apply the Lie approximation in order to make some numerical experiments concerning the splitting of a soliton, interaction among solitons and blow-up phenomenon.     

Unveiling the Link Between Fractional Schrödinger Equation and Light Propagation in Honeycomb Lattice

We suggest a real physical system — the honeycomb lattice — as a possible realization of the fractional Schrödinger equation (FSE) system, through utilization of the Dirac‐Weyl equation (DWE). The fractional Laplacian in FSE causes modulation of the dispersion relation of the system, which becomes linear in the limiting case. In the honeycomb lattice, the dispersion relation is already linear around the Dirac point, suggesting a possible connection with the FSE, since both models can be reduced to the one described by the DWE. Thus, we propagate Gaussian beams in three ways: according to FSE, honeycomb lattice around the Dirac point, and DWE, to discover universal behavior — the conical diffraction. However, if an additional potential is brought into the system, the similarity in behavior is broken, because the added potential serves as a perturbation that breaks the translational periodicity of honeycomb lattice and destroys Dirac cones in the dispersion relation.     

Solitary wave solutions as a signature of the instability in the discrete nonlinear Schrödinger equation

The effect of instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schrödinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves in discrete systems is a signature of the modulation instability is used. With the help of this notion we conjecture that instability effects on moving solitons can be qualitative estimated from the analytical solutions. Results from numerical simulations are presented to support this conjecture.     

A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrödinger equations

We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrödinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly.     

Soliton interaction in the coupled mixed derivative nonlinear Schrödinger equations

The bright one- and two-soliton solutions of the coupled mixed derivative nonlinear Schrödinger equations in birefringent optical fibers are obtained by using the Hirota's bilinear method. The investigation on the collision dynamics of the bright vector solitons shows that there exists complete or partial energy switching in this coupled model. Such parametric energy exchanges can be effectively controlled and quantificationally measured by analyzing the collision dynamics of the bright vector solitons. The influence of two types of nonlinear coefficient parameters on the energy of each vector soliton, is also discussed. Based on the significant energy transfer between the two components of each vector soliton, it is feasible to exploit the future applications in the design of logical gates, fiber directional couplers and quantum information processors.     

Generalized Darboux Transformations,Rogue Waves,and Modulation Instability for the Coherently Coupled Nonlinear Schrödinger Equations in Nonlinear Optics

Nonlinear optics plays a central role in the advancement of optical science and laser‐based technologies. The second‐order rogue‐wave solutions and modulation instability for the coherently coupled nonlinear Schrödinger equations with the positive coherent coupling in nonlinear optics are reported in this paper. Generalized Darboux transformations for such coupled equations are derived, with which the second‐order rational solutions for the purpose of modelling the rogue waves are obtained. With respect to the slowly‐varying complex amplitudes of two interacting optical modes, it is observed that 1) number of valleys of the second‐order rogue waves increases and peak value of the second‐order rogue wave decreases first and then increases; 2) single‐hump second‐order rogue wave turns into the double‐hump second‐order rogue wave; 3) single‐hump bright second‐order rogue wave turns into the dark second‐order rogue wave and finally becomes the three‐hump bright second‐order rogue wave. Meanwhile, baseband modulation instability through the linear stability analysis is seen.     

Symmetry Reductions,Group‐Invariant Solutions,and Conservation Laws of a (2+1)‐Dimensional Nonlinear Schrödinger Equation in a Heisenberg Ferromagnetic Spin Chain

Nonlinear spin excitations in ferromagnetic spin chains are studied for spintronic and magnetic devices including magnetic‐field sensors and for high‐density data storage. Here, (2+1)‐dimensional nonlinear Schrödinger equation is investigated, which describes the nonlinear spin dynamics for a Heisenberg ferromagnetic spin chain. Lie point symmetry generators and Lie symmetry groups of that equation are derived. Lie symmetry groups are related to the time, space, scale, rotation transformations, and Galilean boosts of that equation. Certain solutions, which are associated with the known solutions, are constructed. Based on the Lie symmetry generators, the reduced systems of such an equation are obtained. Based on the polynomial expansion and through one of the reduced systems, group‐invariant solutions are constructed. Soliton‐type group‐invariant solutions are graphically investigated and effects of the magnetic coupling coefficients, that is, α₁, α₂, α₃, and α₄, on the soliton's amplitude, width, and velocity are discussed. It is seen that α₁, α₂, α₃, and α₄ have no influence on the soliton's amplitude, but can affect the soliton's velocity and width. Lax pair and conservation laws of such an equation are derived.     

Multisoliton solutions in terms of double Wronskian determinant for a generalized variable-coefficient nonlinear Schrödinger equation from plasma physics, arterial mechanics, fluid dynamics and optical communications

In this paper, the multisoliton solutions in terms of double Wronskian determinant are presented for a generalized variable-coefficient nonlinear Schrödinger equation, which appears in space and laboratory plasmas, arterial mechanics, fluid dynamics, optical communications and so on. By means of the particularly nice properties of Wronskian determinant, the solutions are testified through direct substitution into the bilinear equations. Furthermore, it can be proved that the bilinear Bäcklund transformation transforms between (N − 1)- and N-soliton solutions.     

薛定谔猫态光场与二能级原子相互作用系统的量子特性

在非旋波近似下,通过采用相干态正交化展开的方法,研究了薛定谔猫态光场与二能级原子相互作用系统中,原子的布局数和光场的反聚束效应,并与旋波近似下的结果进行了对比.在旋波近似与非旋波近似下,讨论了初始光场强度、相干态间的相位角以及失谐量对原子布局数和光场反聚束效应的影响;在非旋波近似下,讨论了强弱耦合情况下光场的反聚束效应.研究结果表明:旋波近似与非旋波近似下,原子的布局数随着初始光场强度的不同,表现出不同的特性;当初始光场强度较小时,旋波近似与非旋波近似下,原子的布局数表现出相同的特性;随着初始光场强度的增大,旋波近似下,原子的布局数将表现出坍塌现象.耦合强度较大时,光场的聚束与反聚束效应在非旋波近似与旋波近似下有较大的区别;非旋波近似下,随着初始光场强度的增大,光场一直处于聚束效应状态;而旋波近似下,光场的聚束效应与反聚束效应交替出现.     

Asymptotics of eigenvalues of the Schrödinger operator with a strong δ-interaction on a loop

In this paper we investigate the operator H_β=−Δ−βδ(·−Γ) in , where β>0 and Γ is a closed C⁴ Jordan curve in . We obtain the asymptotic form of each eigenvalue of H_β as β tends to infinity. We also get the asymptotic form of the number of negative eigenvalues of H_β in the strong coupling asymptotic regime.     

Pauli blocking in degenerate plasmas and the separable potential approach

The Mott effect describes the dissolution of bound states in a dense partially ionized plasma. It occurs when the ionization potential depression, owing to effects of correlation and degeneracy, compensates the binding energy of the bound state. At high densities and moderate temperatures, the Pauli blocking becomes important and influences significantly the degree of ionization in the region of degenerate plasmas. A quantum statistical approach is used where the total density is decomposed in an uncorrelated, “free” part and correlations, as a consequence of the cluster decomposition of the self‐energy. The contribution of correlations to the total density is given by bound states and continuum correlations. Exact solutions for a separable potential are compared to perturbation theory and numerical solutions of the in‐medium Schrödinger equation. The in‐medium scattering phase shifts are evaluated, and the role of continuum correlations is discussed. The Pauli blocking of bound states and the density of states are considered for warm dense matter conditions.     

Crystal field analysis of the ground and excited state absorption of a Cr<Superscript>4+</Superscript> ion in LiAlO<Subscript>2</Subscript> and LiGaO<Subscript>2</Subscript> crystals

The exchange charge model of crystal field theory has been used to analyze the ground and excited state absorption of tetrahedrally coordinated Cr⁴⁺ ion in lithium aluminum oxide LiAlO₂ (γ-phase) and lithium dioxogallate LiGaO₂. The parameters of the crystal field acting on the Cr⁴⁺ ion are calculated from the crystal structure data, taking into account the crystal lattice ions located at distances up to 12.744 Å in LiGaO₂ and 13. 180 Å in LiAlO₂. The obtained energy level schemes were compared with experimental ground and excited state absorption spectra and literature data on the application of other crystal field models (the angular overlap model and Racah theory) to the considered crystals; a good agreement with experimental data is demonstrated.     

Exact and explicit solutions to the discrete nonlinear Schrödinger equation with a saturable nonlinearity

We analyze the discrete nonlinear Schrödinger equation with a saturable nonlinearity through the (G^′/G)-expansion method to present some improved results. Three types of analytic solutions with arbitrary parameters are constructed; hyperbolic, trigonometric, and rational which have not been explicitly computed before.     

Thermal reaction of electron deficient 4‐oxo‐4H‐pyrazole 1,2‐dioxide with cycloheptatriene: the first examples of the formation of an endo‐[4π + 6π]‐cycloadduct and an intramolecular 1,3‐dipolar reaction leading to a heterocage

The reaction of 3,5‐bis(methoxycarbonyl)‐4‐oxo‐4H‐pyrazole 1,2‐dioxide (1a) with 1,3,5‐cycloheptatriene (2b) gave a mixture of the novel endo‐[4 + 6]‐cycloadduct (4ab), anti‐exo‐[4 + 2]‐cycloadduct (5ab), and the heterocage (6ab) derived from the intramolecular 1,3‐dipolar cycloaddition reaction of the syn‐endo‐[4 + 2]‐cycloadduct. Analogous endo‐[4 + 6] selectivity in 1,3‐dipolar cycloadditions has not been reported previously. The X‐ray analysis indicates that 6ab has a very long N_sp3–N_sp3 bond distance of 1.617(4) Å. The cycloaddition behaviour is discussed on the basis of transition‐state structures optimized at the B3LYP/6‐31G(d) level of theory, from which predictions of the peri‐, regio‐, and stereoselectivities agreed well with the experimental results. Copyright © 2012 John Wiley & Sons, Ltd.