High-order breather,M-kink lump and semi-rational solutions of potential Kadomtsev-Petviashvili equation

N-kink soliton and high-order synchronized breather solutions for potential Kadomtsev-Petviashvili equation are derived by means of the Hirota bilinear method,and the limit process of high-order synchronized breathers are shown.Furthermore,M-lump solutions are also presented by taking the long wave limit.Additionally,a family of semi-rational solutions with elastic collision are generated by taking a long-wave limit of only a part of exponential functions,their interaction behaviors are shown by three-dimensional plots and contour plots.     

Soliton,breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

The nonlinear Schro¨dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schro¨dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schro¨dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.     

Experimental demonstration of excitation and propagation of intrinsic localized modes in a mass–spring chain

Excitation and propagation of nonlinear localized oscillations are demonstrated experimentally by using a mass–spring chain to emulate those in the celebrated Fermi–Pasta–Ulam chain of β type. Two types of experiments are performed, one corresponding to a boundary-value problem and the other to an initial-value one. In the former experiment, one end of the chain is driven sinusoidally in the direction of the chain above a cut-off frequency. In the latter experiment, only a number of weights centered around the weight in the middle of the chain are held displaced initially and released suddenly. The experimental results show quantitatively good agreements with the numerical ones. Two videos linked also show visually mobile and stationary localized oscillations.     

Breather resonant phase locking by an external perturbation

We study the effect of the resonant phase locking in the problem of the sine-Gordon equation breather under the action of a small oscillating external force with slowly varying frequency. We obtain equations determining the time evolution of the parameters of the perturbed breather. We describe the regular asymptotic procedure of averaging such equations and show that the averaged equations in the leading order already well describe the phenomenon of resonant phase locking in which the breather oscillations are strongly excited. We obtain necessary and sucient conditions for the phase locking relating the rate of the perturbation frequency variation and its amplitude to the initial data of the breather. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 356–367, August, 2007.     

由谱数据构造实双反对称矩阵

讨论实完全反对称矩阵的一个特秆值反问题．研究了实完全反对称矩阵的一些特征性质，构造一个实反对称矩阵使其各阶顺序主子矩阵具有指定的特征值．证明了：给定满足一定分隔条件的两组数，存在一个实完全反对称矩阵，使其各阶中心主子矩阵具有相应的特征值．     

The dual curvature tensors and dynamics of gravitomagnetic matter

Gravitomagnetic charge that can also be referred to as the dual mass or magnetic mass is the topological charge in gravity theory. A gravitomagnetic monopole at rest can produce a stationary gravitomagnetic field. Due to the topological nature of gravitomagnetic charge, the metric of spacetime where the gravitomagnetic matter is present will be nonanalytic. In this paper both the dual curvature tensors (which can characterize the dynamics of gravitational charge/monopoles) and the antisymmetric gravitational field equation of gravitomagnetic matter are presented. We consider and discuss the mathematical formulation and physical properties of the dual curvature tensors and scalar, antisymmetric source tensors, dual spin connection (including the low‐motion weak‐field approximation), dual vierbein field as well as dual current densities of gravitomagnetic charge. It is shown that the dynamics of gravitomagnetic charge can be founded within the framework of the above dual quantities. In addition, the duality relationship in the dynamical theories between the gravitomagnetic charge (dual mass) and the gravitoelectric charge (mass) is also taken into account in the present paper.     

An exact symplectic solution for the dynamic analysis of shear deformable antisymmetric angle-ply laminated plates

From the mixed variational principle, by the selection of the state variables and its dual variables, the Hamiltonian canonical equation for the dynamic analysis of shear deformable antisymmetric angle-ply laminated plates is derived, leading to the mathematical frame of symplectic geometry and algorithms, and the exact solution for the arbitrary boundary conditions is also derived by the adjoint orthonormalized symplectic expansion method. Numerical results are presented with the emphasis on the effects of length/thickness ratio, arbitrary boundary conditions, degrees of anisotropy, number of layers, ply-angles and the corrected coefficients of transverse shear.     

Passive optimal control of an antisymmetric angle-ply laminate

Using the maximum principle of Ref. 1, a procedure to find numerical solutions of certain optimal control problems is given. As an application of this procedure, the optimal control of an antisymmetric angle-ply laminate is worked out in detail. Numerical solutions are given in the form of graphs.     

Lump and mixed rogue-soliton solutions to the 2+1 dimensional Ablowitz-Kaup-Newell-Segur equation

In this paper, the 2+1 dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation which obtained from the potential Boiti-Leon-Manna-Pempi nelli (pBLMP) equation, is introduced. Through the bilinear method and ansatz technique, the rational solutions consisting of rogue wave and lump soliton solutions are constructed, where we discuss the condition of guaranteeing the positiveness and analyticity of the lump solutions. The collection of a quadratic function with an exponential function describing rational-exponential solutions is proved, the interaction consisting of one lump and one soliton with fission and fusion phenomena. The second kind of interaction comprises the line rogue wave and soliton solution, which is inelastic. With the usage of the extended homoclinic test approach, the homoclinic breather-wave solution is derived. The characteristics of these various solutions are exhibited and illustrated graphically.     

Synchronization and antisynchronization of N‐coupled fractional‐order complex chaotic systems with ring connection

This paper is devoted to investigate synchronization and antisynchronization of N‐coupled general fractional‐order complex chaotic systems described by a unified mathematical expression with ring connection. By means of the direct design method, the appropriate controllers are designed to transform the fractional‐order error dynamical system into a nonlinear system with antisymmetric structure. Thus, by using the recently established result for the Caputo fractional derivative of a quadratic function and a fractional‐order extension of the Lyapunov direct method, several stability criteria are derived to ensure the occurrence of synchronization and antisynchronization among N‐coupled fractional‐order complex chaotic systems. Moreover, numerical simulations are performed to illustrate the effectiveness of the proposed design.