Modulation instability analysis and soliton solutions of an integrable coupled nonlinear Schrödinger system

Nonlinear Dynamics - In this paper, an integrable coupled nonlinear Schrödinger system is investigated, which is derived from the integrable Kadomtsev–Petviashvili system, and can be...     

Spectral method and Bayesian parameter estimation for the space fractional coupled nonlinear Schrödinger equations

Nonlinear Dynamics - In a lot of dynamic processes, the fractional differential operators not only appear as discrete fractional, but they also have a continuous nature in some sense. In the...     

Dynamics of diverse data-driven solitons for the three-component coupled nonlinear Schrödinger model by the MPS-PINN method

Nonlinear Dynamics - We improve the physical information neural network by adding multiple parallel subnets to predict seven types of soliton dynamics, such as one soliton, two solitons and soliton...     

Soliton interactions for coupled nonlinear Schrödinger equations with symbolic computation

Soliton interactions for the coupled nonlinear Schrödinger equations, governing the propagation of envelopes of electromagnetic waves in birefringent optical fibers, are investigated with symbolic computation. Based on the Hirota method, analytic two- and three-soliton solutions for this model are derived. Relevant interaction properties are discussed. Stationary bound vector solitons with the periodic attraction and repulsion are obtained. Soliton intensity could be reduced if the nonlinearity in optical fibers is enlarged, while the soliton period could be prolonged as the group velocity dispersion in the anomalous dispersion regime of optical fibers increases. Through the asymptotic analysis for the two-soliton solutions, interactions between two solitons are proven to be elastic. Besides, parallel soliton transmission systems without soliton interactions are presented. Moreover, interactions between the regular and bound vector solitons are studied. Dual complex structures and triple-soliton bound states are presented. Results could be of certain value to the studies on the soliton control and optical switching technologies.     

A differential quadrature algorithm for nonlinear Schrödinger equation

Numerical solutions of a nonlinear Schrödinger equation is obtained using the differential quadrature method based on polynomials for space discretization and Runge–Kutta of order four for time discretization. Five well-known test problems are studied to test the efficiency of the method. For the first two test problems, namely motion of single soliton and interaction of two solitons, numerical results are compared with earlier works. It is shown that results of other test problems agrees the theoretical results. The lowest two conserved quantities and their relative changes are computed for all test examples. In all cases, the differential quadrature Runge–Kutta combination generates numerical results with high accuracy.     

Complex excitations for the derivative nonlinear Schrödinger equation

Nonlinear Dynamics - The Darboux transformation (DT) formulae for the derivative nonlinear Schrödinger (DNLS) equation are expressed in concise forms, from which the multi-solitons, n-periodic...     

The bound-state soliton solutions of a higher-order nonlinear Schrödinger equation for inhomogeneous Heisenberg ferromagnetic system

Nonlinear Dynamics - The inverse scattering of a higher-order nonlinear Schrödinger equation for inhomogeneous Heisenberg ferromagnetic system with zero boundary condition is calculated by an...     

Spatial frequency range analysis for the nonlinear Schrödinger equation

An analysis of the spatial frequency ranges for the nonlinear Schrödinger equation (NLS), subject to initial conditions with Gaussian and band-limited spatial frequency spectra, is presented in this paper. The analysis is based on a Volterra series representation of the NLS equation. This study reveals the relationship between the spatial frequency ranges of the solution, along with the evolution of the system, and the spatial frequency ranges of the initial conditions, and extends previous results in linear and nonlinear finite dimensional systems. The analysis also reveals a variety of nonlinear phenomena including self-phase modulation, cross-phase modulation and Raman effects modelled using the NLS equation.     

Stationary solutions for nonlinear dispersive Schrödinger’s equation

This paper carries out the integration of the nonlinear dispersive Schrödinger’s equation by the aid of Lie group analysis. The stationary solutions are obtained. The two types of nonlinearity that are studied in this paper are power law and dual-power law so that the cases of Kerr law and parabolic law nonlinearity fall out as special cases.     

On the multiple-attractor coexsting system with parameter uncertainties using generalized cell mapping method

In this paper the generalized cell mapping (GCM) method is used to study multiple-attractor coexisting system with parameter uncertainties. The effects that the uncertain parameters has on the global properties of the system are presented. And It is obtained that the attractor with much smaller value of protect thickness, will disappear firstly with the degree of the uncertainty of parameter increasing. Project supported by the National Natural Science Foundation of China (19672046)     

DYNAMICAL CHARACTER FOR A PERTURBED COUPLED NONLINEAR SCHRODINGER SYSTEM

The dynamical character for a perturbed coupled nonlinear Schrodinger system with periodic boundary condition was studied. First, the dynamical character of perturbed and unperturbed systems on the invariant plane was analyzed by the spectrum of the linear operator. Then the existence of the locally invariant manifolds was proved by the singular perturbation theory and the fixed-point argument.     

Dynamics of repelling soliton collisions in coupled Schrödinger equations

The system of Coupled Nonlinear Schrödinger's Equations (CNLSE) is solved numerically by means of a conservative difference scheme. Values of the cross-modulation parameter, α₂, are chosen to induce repelling collisions. The resulting waveforms are fitted to soliton profiles and studied via internal parameters including phase velocity to gain insight into the dynamics of the repelling collisions. A decaying oscillation of the post-collision profiles is observed and investigated in terms of varying α₂ and relative initial velocity of colliding profiles.     

Characteristic fractional step finite difference method for nonlinear section coupled system

For the section coupled system of multilayer dynamics of fluids in porous media, a parallel scheme modified by the characteristic finite difference fractional steps is proposed for a complete point set consisting of coarse and fine partitions. Some tech- niques, such as calculus of variations, energy method, twofold-quadratic interpolation of product type, multiplicative commutation law of difference operators, decomposition of high order difference operators, and prior estimates, are used in theoretical analysis. Optimal order estimates in 12 norm are derived to show accuracy of the second order approximation solutions. These methods have been used to simulate the problems of migration-accumulation of oil resources.     

Parallel Multisplitting AOR method for solving a class of system of nonlinear algebraic equations

ＰＡＲＡＬＬＥＬＭＵＬＴＩＳＰＬＩＴＴＩＮＧＡＯＲＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＡＣＬＡＳＳＯＦＳＹＳＴＥＭＯＦＮＯＮＬＩＮＥＡＲＡＬＧＥＢＲＡＩＣＥＱＵＡＴＩＯＮＳＢａｉＺｈｏｎｇｚｈｉ（白中治）（ＩｎｓｔｉｔｕｉｅｏｆＭａｔｈｅｍａｔｉｃｓＦｕ...     

Global solution for coupled nonlinear Klein-Gordon system

The globed solution for a coupled nonlinear Klein-Gordon system in two-dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.     

A two scaled numerical method for global analysis of high dimensional nonlinear systems

This paper first analyzes the features of two classes of numerical methods for global analysis of nonlinear dynamical systems, which regard state space respectively as continuous and discrete ones. On basis of this understanding it then points out that the previously proposed method of point mapping under cell reference (PMUCR), has laid a frame work for the development of a two scaled numerical method suitable for the global analysis of high dimensional nonlinear systems, which may take the advantages of both classes of single scaled methods but will release the difficulties induced by the disadvantages of them. The basic ideas and main steps of implementation of the two scaled method, namely extended PMUCR, are elaborated. Finally, two examples are presented to demonstrate the capabilities of the proposed method.     

The perturbation finite element method for solving problems with nonlinear materials

The perturbation method is one of the effective methods for so-lving problems in nonlinear continuum mechanics.It has been de-veloped on the basis of the linear analytical solutions for the o-riginal problems.If a simple analytical solution cannot be ob-tained.we would encounter difficulties in applying this method tosolving certain complicated nonlinear problems.The finite ele-ment method appears to be in its turn a very useful means for sol-ving nonlinear problems,but generally it takes too much time incomputation.In the present paper a mixed approach,namely,theperturbation finite element method,is introduced,which incorpo-rates the advantages of the two above-mentioned methods and enablesus to solve more complicated nonlinear problems with great savingin computing time.Problems in the elastoplastic region have been discussed anda numerical solution for a plate with a central hole under tensionis given in this paper.     

A multi-objective optimal PID control for a nonlinear system with time delay

It is generally difficult to design feedback controls of nonlinear systems with time delay to meet time domain specifications such as rise time, overshoot, and tracking error. Furthermore, these time domain specifications tend to be conflicting to each other to make the control design even more challenging. This paper presents a cell mapping method for multi-objective optimal feedback control design in time domain for a nonlinear Duffing system with time delay. We first review the multi-objective optimization problem and its formulation for control design. We then introduce the cell mapping method and a hybrid algorithm for global optimal solutions. Numerical simulations of the PID control are presented to show the features of the multi-objective optimal design.