Fusion,fission, and annihilation of complex waves for the （2＋l）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system

With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ my ＋ nt ＋ Г（x,y, t） for the （2＋1）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system （GCBS） are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

Nonlocal symmetries,consistent Riccati expansion integrability,and their applications of the (2+1)-dimensional Broer–Kaup–Kupershmidt system

For the (2+1)-dimensional Broer–Kaup–Kupershmidt(BKK) system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion(CRE) is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.     

Integrability and Solutions of the(2+1)-dimensional Hunter–Saxton Equation

In this paper,the(2+1)-dimensional Hunter-Saxton equation is proposed and studied.It is shown that the(2+1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations.Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation,a non-isospectral Lax-pair of the(2+1)-dimensional Hunter–Saxton equation is derived.In addition,exact singular solutions with a finite number of corners are obtained.Furthermore,the(2+1)-dimensional μ-Hunter–Saxton equation is presented,and its exact peaked traveling wave solutions are derived.     

Constructing(2+1)-dimensional N=1 supersymmetric integrable systems from the Hirota formalism in the superspace

The N = 1 supersymmetric extensions of two integrable systems,a special negative Kadomtsev–Petviashvili(NKP)system and a(2+1)-dimensional modified Korteweg–de Vries(MKd V) system,are constructed from the Hirota formalism in the superspace.The integrability of both systems in the sense of possessing infinitely many generalized symmetries are confirmed by extending the formal series symmetry approach to the supersymmetric framework.It is found that both systems admit a generalization of W∞type algebra and a Kac-Moody–Virasoro type subalgebra.Interestingly,the first one of the positive flow of the supersymmetric NKP system is another N = 1 supersymmetric extension of the(2+1)-dimensional MKd V system.Based on our work,a hypothesis is put forward on a series of(2+1)-dimensional supersymmetric integrable systems.It is hoped that our work may develop a straightforward way to obtain supersymmetric integrable systems in high dimensions.     

A Few Integrable Dynamical Systems,Recurrence Operators,Expanding Integrable Models and Hamiltonian Structures by the r-Matrix Method

We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov–Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical system, heat equation, and so on, are followed to be generated. Then we expand some differential operators presented in the paper to deduce two types of expanding dynamical models. By taking the generalized Burgers dynamical system as an example, we deform its expanding model to get a half-expanding system, whose recurrence operator is derived from Lax representation, and its Hamiltonian structure is also obtained by adopting a new way. Finally, we expand the generalized Burgers dynamical system to the(2+1)-dimensional case whose Hamiltonian structure is derived by Poisson tensor and gradient of the Casimir function. Besides, a kind of(2+1)-dimensional expanding dynamical model of the(2+1)-dimensional dynamical system is generated as well.     

Soliton molecules and the CRE method in the extended mKdV equation

The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition, which is called velocity resonance. One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters. The soliton molecules including the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton are exhibited in figures 1–3, respectively.Meanwhile, the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevé method. The consistent Riccati expansion(CRE) method is applied to the extended mKdV equation. It demonstrates that the extended mKdV equation is a CRE solvable system. A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.     

Generalized symmetries of an N=1 supersymmetric Boiti–Leon–Manna–Pempinelli system

The formal series symmetry approach(FSSA), a quite powerful and straightforward method to establish infinitely many generalized symmetries of classical integrable systems, has been successfully extended in the supersymmetric framework to explore series of infinitely many generalized symmetries for supersymmetric systems. Taking the N= 1 supersymmetric Boiti–Leon–Manna–Pempinelli system as a concrete example, it is shown that the application of the extended FSSA to this supersymmetric system leads to a set of infinitely many generalized symmetries with an arbitrary function f(t). Some interesting special cases of symmetry algebras are presented, including a limit case f(t) = 1 related to the commutativity of higher order generalized symmetries.     

Soliton excitations and chaotic patterns for the (2+1)-dimensional Boiti-Leon-Pempinelli system

By improved projective equation approach and a linear variable separation approach, a new family of exact solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli (BLP) system is derived. Based on the derived solitary wave solution, some dromion and solitoff excitations and chaotic behaviours are investigated.     

Exact Solutions to (3+1) Conformable Time Fractional Jimbo–Miwa,Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations

Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.     

Nonlinear coherent perfect photon absorber in asymmetrical atom–nanowires coupling system

Coherent perfect absorption provides a method of light-controlling-light and has practical applications in optical communications. Recently, a cavity-based nonlinear perfect photon absorption extends the coherent perfect absorber(CPA)beyond the linear regime. As nanowire-based system is a more competitive candidate for full-optical device, we introduce a nonlinear CPA in the single two-level atom–nanowires coupling system in this work. Nonlinear input–output relations are derived analytically, and three contributions of atomic saturation nonlinearity are explicit. The consociation of optical nonlinearity and destructive interference makes it feasible to fabricate a nonlinear monoatomic CPA. Our results also indicate that a nonlinear system may work linearly even when the incoming lights are not weak any more. Our findings show promising applications in full-optical devices.     

Chaotic behaviors of the （2＋1）-dimensional generalized Breor-Kaup system

With the help of the Maple symbolic computation system and the projective equation approach,a new family of variable separation solutions with arbitrary functions for the(2+1)-dimensional generalized Breor-Kaup(GBK) system is derived.Based on the derived solitary wave solution,some chaotic behaviors of the GBK system are investigated.     

Electronic structures and optical properties of HfO_2–TiO_2 alloys studied by first-principles GGA + U approach

The phase diagram of HfO_2–TiO_2 system shows that when Ti content is less than 33.0 mol%, HfO_2–TiO_2 system is monoclinic; when Ti content increases from 33.0 mol% to 52.0 mol%, it is orthorhombic; when Ti content reaches more than 52.0 mol%, it presents rutile phase. So, we choose the three phases of HfO_2–TiO_2 alloys with different Ti content values. The electronic structures and optical properties of monoclinic, orthorhombic and rutile phases of HfO_2–TiO_2 alloys are obtained by the first-principles generalized gradient approximation(GGA) +U approach, and the effects of Ti content and crystal structure on the electronic structures and optical properties of HfO_2–TiO_2 alloys are investigated. By introducing the Coulomb interactions of 5 d orbitals on Hf atom(U_1~d), those of 3 d orbitals on Ti atom(U_2~d), and those of 2 p orbitals on O atom(Up) simultaneously, we can improve the calculation values of the band gaps, where U_1~d, U_2~d, and Up values are 8.0 eV, 7.0 eV, and 6.0 eV for both the monoclinic phase and orthorhombic phase, and 8.0 eV, 7.0 eV, and 3.5 eV for the rutile phase. The electronic structures and optical properties of the HfO_2–TiO_2 alloys calculated by GGA +U_1~d(U_1~d= 8.0 eV) +U_2~d(U_2~d= 7.0 eV) +U~p(U~p= 6.0 eV or 3.5 eV) are compared with available experimental results.     

Quasi-classical trajectory study of H+LiH(v=0,1,2,j=0)→Li+H_2 reaction on a new global potential energy surface

Quasi-classical trajectory(QCT) calculations are reported for the H+LiH(v = 0–2, j = 0)→Li+H_2 reaction on a new ground electronic state global potential energy surface(PES) of the LiH_2 system. Reaction probability and integral cross sections(ICSs) are calculated for collision energies in the range of 0 eV–0.5 eV. Reasonable agreement is found in the comparison between present results and previous available theoretical results. We carried out statistical analyses with all the trajectories and found two main distinct reaction mechanisms in the collision process, in which the stripping mechanism(i.e., without roaming process) is dominated over the collision energy range. The polarization dependent differential cross sections(PDDCSs) indicate that forward scattering dominates the reaction due to the dominated mechanism. Furthermore,the reactant vibration leads to a reduction of the reactivity because of the barrierless and attractive features of PES and mass combination of the system.     

Multi-soliton solutions of a two-component Camassa–Holm system:Darboux transformation approach

We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate the CH2 system to a negative flow of the Broer–Kaup or twoboson hierarchy.The solutions of this negative flow are given in terms of Wronskians via Darboux transformation.Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.     

Instantaneous solitons and fractal solitons for a (2+1)-dimensional nonlinear system

By an improved projective equation approach and a linear variable separation approach, a new family of exact solutions of the (2+1)-dimensional Broek--Kaup system is derived. Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as instantaneous solitons and fractal solitons are investigated.     

SU(3) spin–orbit-coupled Bose–Einstein condensate confined in a harmonic plus quartic trap

We consider a SU(3) spin–orbit coupled Bose–Einstein condensate confined in a harmonic plus quartic trap.The ground-state wave functions of such a system are obtained by minimizing the Gross–Pitaevskii energy functional, and the effects of the spin-dependent interaction and spin–orbit coupling are investigated in detail.For the case of ferromagnetic spin interaction, the SU(3) spin–orbit coupling induces a threefold-degenerate plane wave ground state with nontrivial spin texture.For the case of antiferromagnetic spin interaction, the system shows phase separation for weak SU(3) spin–orbit coupling, where three discrete minima with unequal weights in momentum space are selected, while hexagonal honeycomb lattice structure for strong SU(3) SOC, where three discrete minima with equal weights are selected.     

Stability,electronic structures,and mechanical properties of Fe–Mn–Al system from first-principles calculations

The stability, electronic structures, and mechanical properties of the Fe–Mn–Al system were determined by firstprinciples calculations. The formation enthalpy and cohesive energy of these Fe–Mn–Al alloys are negative and show that the alloys are thermodynamically stable. Fe_3Al, with the lowest formation enthalpy, is the most stable compound in the Fe–Mn–Al system. The partial density of states, total density of states, and electron density distribution maps of the Fe–Mn–Al alloys were analyzed. The bonding characteristics of these Fe–Mn–Al alloys are mainly combinations of covalent bonding and metallic bonds. The stress-strain method and Voigt–Reuss–Hill approximation were used to calculate the elastic constants and moduli, respectively. Fe_(2.5)Mn_(0.5)Al has the highest bulk modulus, 234.5 GPa. Fe_(1.5)Mn_(1.5)Al has the highest shear modulus and Young's modulus, with values of 98.8 GPa and 259.2 GPa, respectively. These Fe–Mn–Al alloys display disparate anisotropies due to the calculated different shape of the three-dimensional curved surface of the Young's modulus and anisotropic index. Moreover, the anisotropic sound velocities and Debye temperatures of these Fe–Mn–Al alloys were explored.     

Structural,electronic,and magnetic properties in FeAlAu_n(n=1–6)clusters:A first-principles study

The geometries,electronic and magnetic properties of the trimetallic clusters Fe Al Aun(n = 1–6) are systematically investigated using density functional theory(DFT).A number of new isomers are obtained to probe the structural evolutions.All doped clusters show larger relative binding energies than pure Aun+2partners,indicating that doping with Fe and Al atoms can stabilize the Aun clusters.The highest occupied molecular orbital–lowest unoccupied molecular orbital(HOMO–LUMO) gaps,vertical ionization potentials and vertical electron affinities are also studied and compared with those of pure gold clusters.Magnetism calculations demonstrate that the magnetic moments of Fe Al Aun clusters each show a pronounced odd–even oscillation with the number of Au atoms.     

Multi-Soliton Solutions for the Coupled Fokas–Lenells System via Riemann–Hilbert Approach

We aim to construct multi-soliton solutions for the coupled Fokas–Lenells system which arises as a model for describing the nonlinear pulse propagation in optical fibers. Starting from the spectral analysis of the Lax pair, a Riemann–Hilbert problem is presented. Then in the framework of the Riemann–Hilbert problem corresponding to the reflectionless case, N-soliton solutions to the coupled Fokas–Lenells system are derived explicitly.     

Coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions

In this paper, coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions are addressed. Because of the discontinuity of the activation function, Filippov solution of the neural network is defined. Based on Brouwer's fixed point theorem and definition of Mittag–Leffler stability, sufficient criteria are established to ensure the existence of (2k + 3)~n (k ≥ 1) equilibrium points, among which (k + 2)~n equilibrium points are locally Mittag–Leffler stable. Compared with the existing results, the derived results cover local Mittag–Leffler stability of both fractional-order and integral-order recurrent neural networks. Meanwhile discontinuous networks might have higher storage capacity than the continuous ones. Two numerical examples are elaborated to substantiate the effective of the theoretical results.