Two-dimensional accessible solitons in PT-symmetric potentials

Two-dimensional parity-time (PT) symmetric potentials are introduced, which allow the existence of spatial solitons in the model of the strongly nonlocal nonlinear Schr?dinger equation. Two-dimensional accessible solitons are found in the form of solutions separating the radial amplitude, given in terms of Laguerre polynomials, a?phase function involving quadratic, linear, and constant phase shifts, and a specific azimuthal modulation function. Shape-preserving solitons are constructed from Laguerre?CGaussian functions containing the self-similar variable and an exponential form of the azimuthal modulation, containing sine and cosine functions, when a suitable PT-symmetric potential is chosen. Interesting soliton profiles and the corresponding PT-symmetric potentials are displayed for different values of the parameters.     

Dark and gray solitons of ($$$$)-dimensional nonlocal nonlinear media with periodic response function

By using the standard symmetry reduction method, some exact analytical solutions including gray solitons and gray soliton lattice solutions are derived for the (\(2+1\))-dimensional nonlinear optical media with periodic nonlocal response. Furthermore, dark/gray soliton solutions and dark soliton lattice solutions are found by means of hyperbolic function expansion method and elliptic function expansion method for the nonlocal nonlinear system, respectively. It is found that two critical points exist for soliton solutions, and the switching dynamics of solitons may be described by the critical points.     

Exact solitons,periodic peakons and compactons in an optical soliton model

Nonlinear Dynamics - To study the properties of one-dimensional spatial solitons in thermal nonlocal media, we investigate the dynamical behaviour and bifurcation of solutions of the planar systems...     

Controllable propagation path of imaginary value off-axis vortex soliton in nonlocal nonlinear media

Nonlinear Dynamics - Both imaginary value off-axis solitons and imaginary value off-axis vortex solitons are analytically obtained in nonlocal nonlinear media. The imaginary value off-axis solitons...     

Dark spatiotemporal optical solitary waves in self-defocusing nonlinear media

Dark three-dimensional spatiotemporal solitons or the “dark light bullets” in the self-defocusing nonlinear media with equal diffraction and dispersion lengths are demonstrated analytically. Our results show that the main characteristic of the dark light bullets can be described by the cylindrical Korteweg–de Vries (CKdV) equation. The dark wave packets are composed of the single-layer and multilayer toroidal rings. For the multilayer rings, there exist small inner rings enclosed in a large ring-shaped toroidal structure, when one chooses different orders of the soliton solutions of the CKdV equation. The radius of dark ring solitons increases with the propagation distance. Present results provide a feasible method for controlling the fundamental structure of these beams in the self-defocusing nonlinear media.     

Interactions among different types of nonlinear waves described by the Kadomtsev–Petviashvili equation

In nonlinear science, the interactions among solitons are well studied because the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlevé waves. In this paper, taking the Kadomtsev–Petviashvili (KP) equation as an illustration model, a new method is established to find interactions among different types of nonlinear waves. The nonlocal symmetries related to the Darboux transformation (DT) of the KP equation is localized after embedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions. It is shown that the essential and unique role of the DT is to add an additional soliton on a Boussinesq-type wave or a KdV-type wave, which are two basic reductions of the KP equation.     

非局部弹性理论概述及在当代材料背景下的一些进展

局部作用原理在发展经典连续介质力学的本构关系中起着重要的作用，由此导出的简单物质理论得到了广泛的应用．然而，随着科技的发展，各种具有微结构的新材料不断涌现，理论和实验表明，非局部理论可以更好地刻画这些材料的宏观力学行为．本文简要介绍了一些传统的非局部弹性理论，包括Eringen 理论、Kunin 理论、Mindlin 理论；阐述了针对复合材料发展的，具有时间-空间非局部特征的Willis 方程、最新的时间-空间耦合非局部弹性动力学理论以及近场动力学理论．时间-空间非局部理论反映了复合材料宏观性能固有的非局部特征，而具有空间非局部特征的近场动力学理论便于处理具有不连续性的问题．最后，本文讨论了非局部理论的发展中值得关注的一些问题.     

Controllable soliton transition and interaction in nonlocal nonlinear media

Nonlinear Dynamics - Two kinds of controllable evolutions of optical solitons are predicted in nonlocal nonlinear media by gradually tuning the characteristic length of response function. On the...     

Free vibration of thermo-elastic microplate based on spatiotemporal fractional-order derivatives with nonlocal characteristic length and time

A fractional-order thermo-elastic model taking into account the small-scale effects of the thermo-elastic coupled behavior is developed to study the free vibration of a higher-order shear microplate. The nonlocal strain gradient theory is modified with the introduction of the fractional-order derivatives and the nonlocal characteristic length. The Fourier heat conduction is replaced by the non-Fourier heat conduction with the introduction of the fractional order and the memory characteristic tim...     

On a theory of nonlocal elasticity of bi-Helmholtz type and some applications

A theory of nonlocal elasticity of bi-Helmholtz type is studied. We employ Eringen’s model of nonlocal elasticity, with bi-Helmholtz type kernels, to study dispersion relations, screw and edge dislocations. The nonlocal kernels are derived analytically as Green functions of partial differential equations of fourth order. This continuum model of nonlocal elasticity involves two material length scales which may be derived from atomistics. The new nonlocal kernels are nonsingular in one-, two- and three-dimensions. Furthermore, the nonlocal elasticity of bi-Helmholtz type improves the one of Helmholtz type by predicting a dispersion relationship with zero group velocity at the end of the first Brillouin zone. New solutions for the stresses and strain energy of screw and edge dislocations are found.     

Soliton solutions and Bäcklund transformations of a (2 + 1)-dimensional nonlinear evolution equation via the Jaulent–Miodek hierarchy

In this paper, a (2 + 1)-dimensional nonlinear evolution equation generated via the Jaulent–Miodek hierarchy is investigated. Based on the Bell polynomials and Hirota method, bilinear forms and Bäcklund transformations are derived. One- and two-soliton solutions are constructed via symbolic computation. Soliton solutions are obtained through the Bäcklund transformations. We can get three types by choosing different parameters: the kink, bell-shape, and anti-bell-shape solitons. Propagation of the one soliton and elastic interactions between the two solitons are discussed graphically. After the interaction of the two bell-shape or anti-bell-shape solitons, solitonic shapes and amplitudes keep invariant except for some phase shifts, while after the interaction of the kink soliton and anti-bell-shape soliton, the anti-bell-shape soliton turns into a bell-shape one, and the kink soliton keeps its shape, with their amplitudes unchanged.     

Orbital Stability of Periodic Peakons to a Generalized μ-Camassa–Holm Equation

In this paper, we study the orbital stability of the periodic peaked solitons of the generalized μ-Camassa–Holm equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Camassa–Holm equation and the modified Camassa–Holm equation. It is also integrable with the Lax-pair and bi-Hamiltonian structure and admits the single peakons and multi-peakons. By constructing an inequality related to the maximum and minimum of solutions with the conservation laws, we prove that, even in the case that the Camassa–Holm energy counteracts in part the modified Camassa–Holm energy, the shapes of periodic peakons are still orbitally stable under small perturbations in the energy space.     

Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations

Nonlinear Dynamics - A study of high-order solitons in three nonlocal nonlinear Schrödinger equations is presented. These include the $$\mathcal {PT}$$ -symmetric, reverse-time, and...     

Chimera states in ensembles of bistable elements with regular and chaotic dynamics

We consider ensembles of bistable elements with nonlocal interaction. It is shown that the bistability of units in the case of nonlocal interaction leads to the formation of chimera structures of a special type, which we have called double-well chimeras. Their distinctive feature consists in the formation of incoherence clusters with an irregular distribution of elements between two attractive sets existing in an individual element (two “potential wells”). Ensembles of different bistable units are considered, namely ensembles of cubic maps, FitzHugh–Nagumo oscillators in the regime of two stable equilibrium points and Chua’s circuits. The spatiotemporal behavior of the ensembles is studied in the cases of regular and chaotic dynamics in time, and different types of chimera structures are revealed.     

Bright hump solitons for the higher-order nonlinear Schrödinger equation in optical fibers

Under investigation is the higher-order nonlinear Schrödinger equation with the third-order dispersion (TOD), self-steepening (SS) and self-frequency shift, which can be used to describe the propagation and interaction of ultrashort pulses in the subpicosecond or femtosecond regime. Through the introduction of an auxiliary function, bilinear form is derived. Bright one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. From the one-soliton solutions, we present the parametric regions for the existence of single- and double-hump solitons, and find that they are affected by the coefficients of the group velocity dispersion (GVD) and TOD. Besides, propagation of the one single- or double-hump soliton is observed. We analytically obtain the amplitudes for the single- and double-hump solitons, and calculate the interval between the two peaks for the double-hump soliton. Moreover, soliton amplitudes are related to the coefficients of the GVD, TOD and SS, while the interval between the two peaks for the double-hump soliton is dependent on the coefficients of the GVD and TOD. Interactions are seen between the (i) two single-hump solitons, (ii) two double-hump solitons, and (iii) single- and double-hump solitons. Those interactions are proved to be elastic via the asymptotic analysis.     

Weakly nonlocal envelope solitary waves: numerical calculations for the Klein-Gordon (φ) equation

“Weakly nonlocal” solitary waves differ from ordinary solitary waves by possessing small amplitude, oscillatory “wings” that extend indefinitely from the large amplitude “core”. Such generalized solitary waves have been discovered in capillarygravity water waves, particle physics models, and geophysical Rossby waves. In this work, we present explicit calculations of weakly nonlocal envelope solitary waves. Each is a sine wave modulated by a slowly-varying “envelope” that itself propagates at the group velocity. Our example is the cubically nonlinear Klein-Gordon equation, which is a model in particle physics (φ⁴ theory) and in electrical engineering (with a different sign). Both cases have weakly nonlocal“breather” solitons. Via the Lorentz invariance, each breather generates a one-parameter family of nonlocal envelope solitary waves. The φ⁴ breather was described and calculated in earlier work. This generates envelope solitons which have “wings” that are (mostly) proportional to the second harmonic of the sinusoidal factor. In this article, we calculate breathers and envelope solitary waves for the second, electrical engineering case. Since these, unlike the φ⁴ waves, contain only odd harmonics, the envelope solitary waves are nonlocal only via the third harmonic.     

On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

We consider a diffuse interface model which describes the motion of an ideal incompressible mixture of two immiscible fluids with nonlocal interaction in two-dimensional bounded domains. This model consists of the Euler equation coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence of globally defined weak solutions as well as well-posedness results for strong/classical solutions.     

Controllable behaviors of nonautonomous solitons on background of continuous wave and cnoidal wave in $$\mathcal {}$$-symmetric dimer with inhomogeneous effect

We obtain exact \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric nonautonomous soliton solutions on background waves. These solutions indicate that dispersion and nonlinear coefficients influence form factors of nonautonomous solitons such as amplitude, width and center; however, linear coupling coefficient and gain/loss parameter only influence phase of solitons. Based on these solutions, the controllable behaviors such as postpone, sustainment and restraint on continuous wave background in an exponential decreasing dispersion system are discussed. Moreover, the propagation behaviors of solitons on the cnoidal wave background in different dispersion systems are also studied.     

TWISTING STATICS AND DYNAMICS FOR CIRCULAR ELASTIC NANOSOLIDS BY NONLOCAL ELASTICITY THEORY

The torsional static and dynamic behaviors of circular nanosolids such as nanoshafts, nanorods and nanotubes are established based on a new nonlocal elastic stress field theory. Based on a new expression for strain energy with a nonlocal nanoscale parameter, new higher-order governing equations and the corresponding boundary conditions are first derived here via the variational principle because the classical equilibrium conditions and/or equations of motion can- not be directly applied to nonlocal nanostructures even if the stress and moment quantities are replaced by the corresponding nonlocal quantities. The static twist and torsional vibration of circular, nonlocal nanosolids are solved and discussed in detail. A comparison of the conventional and new nonlocal models is also presented for a fully fixed nanosolid, where a lower-order governing equation and reduced stiffness are found in the conventional model while the new model reports opposite solutions. Analytical solutions and numerical examples based on the new nonlocal stress theory demonstrate that nonlocal stress enhances stiffness of nanosolids, i.e. the angular displacement decreases with the increasing nonlocal nanoscale while the natural frequency increases with the increasing nonlocal nanoscale.     

Amplification,reshaping, fission and annihilation of optical solitons in dispersion-decreasing fiber

Amplification, reshaping, fission and annihilation of optical solitons can be applied in fiber lasers, all-optical switching devices and optical communications. In this paper, for the variable coefficient high-order nonlinear Schrödinger equation, analytic two- and three-soliton solutions are derived by the Hirota bilinear method. Optical solitons propagation in the dispersion-decreasing fibers is investigated theoretically. The influence of corresponding parameters is discussed based on obtained solutions. By choosing properly parameters, optical solitons are amplified and reshaped stably in a long distance. Besides, the number of amplified solitons can be chosen as required. Moreover, a novel phenomenon that three solitons can split into four solitons or merge into two solitons has been proposed. Results may be helpful to realize the amplification, reshaping, fission and annihilation of solitons, and will be valuable to the applications of optical amplifier, all-optical switching and optical self-routing.