A Generalized Lyapunov-Sylvester Computational Method for Numerical Solutions of NLS Equation with Singular Potential

In the present paper a numerical method is developed to approximate the solution of two-dimensional Nonlinear Schrodinger equation in the presence of a sin- gular potential. The method leads to generalized Lyapunov-Sylvester algebraic opera- tors that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and sta- ble using the Lyapunov criterion, lax equivalence theorem and the properties of the generalized Lyapunov-Sylvester operators.

New Exact Solutions to NLS Equation and Coupled NLS Equations

A transformation is introduced on the basis of the projective Riccati equations, and it is applied as an intermediate in expansion method to solve nonlinear Schrǒdinger (NLS) equation and coupled NLS equations. Manykinds of envelope travelling wave solutions including envelope solitary wave solution are obtained, in which some arefound for the first time.

Integrable Variable-coefficient Coupled Cylindrical NLS Equations and Their Explicit Solutions

Based on the generalized dressing method, we propose integrable variable coefficient coupled cylin-drical nonlinear SchrSdinger equations and their Lax pairs. As applications, their explicit solutions and their reductions are constructed.

Attractors and Dimensions for Discretizations of a NLS Equation with a Non-local Nonlinear Term

In this paper we consider a semi-dicretized nonlinear Schrdinger (NLS) equation with local integral nonlinearity. It is proved that for each mesh size, there exist attractors for the discretized system. The bounds tor the Hausdorff and fractal dimensions of the discrete attractors are obtained, and the various bounds are independent of the mesh sizes. Furthermore. numerical experiments are given and many interesting phenomena are observed such as limit cycles, chaotic attractors and a so-called crisis of the chaotic attractors.

The Fifth Order Peregrine Breather and Its Eight-Parameter Deformations Solutions of the NLS Equation

We construct here explicitly new deformations of the Peregrine breather of order 5 with 8 real parameters.This gives new families of quasi-rational solutions of the NLS equation and thus one can describe in a more precise way the phenomena of appearance of multi rogue waves. With this method, we construct new patterns of different types of rogue waves. We get at the same time, the triangular configurations as well as rings isolated. Moreover, one sees appearing for certain values of the parameters, new configurations of concentric rings.

NLS equation wronskians Peregrine breather rogue waves

The Peregrine breather of order eleven（P_（11） breather） solution to the focusing one-dimensional nonlinear Schrdinger equation（NLS） is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given： when all parameters are equal to 0 we recover the famous P_（11） breather. We obtain new families of quasi-rational solutions to the NLS equation in terms of explicit quotients of polynomials of degree 132 in x and t by a product of an exponential depending on t. We study these solutions by giving patterns of their modulus in the（x; t） plane, in function of the different parameters.

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper, we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear SchrSdinger equation （NLS） with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions, one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results, some previous one- and two- soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one- and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.

PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear SchrSdinger （NLS） equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.

An Integrable Symplectic Map Related to Discrete Nonlinear Schrdinger Equation

The method of nonlinearization of spectral problem is developed and applied to the discrete nonlinear Schr6dinger （DNLS） equation which is a reduction of the Ablowitz-Ladik equation with a reality condition. A new integable symplectic map is obtained and its integrable properties such as the Lax representation, r-matrix, and invariants are established.

Approximate symmetry reduction for perturbed nonlinear Schrdinger equation

We investigate the one-dimensional nonlinear SchrSdinger equation with a perturbation of polynomial type. The approximate symmetries and approximate symmetry reduction equations are obtained with the approximate symmetry perturbation theory.

A Novel Method for Solving Nonlinear Schrödinger Equation with a Potential by Deep Learning

The improved physical information neural network algorithm has been proven to be used to study integrable systems. In this paper, the improved physical information neural network algorithm is used to study the defocusing nonlinear Schrödinger (NLS) equation with time-varying potential, and the rogue wave solution of the equation is obtained. At the same time, the influence of the number of network layers, neurons and the number of sampling points on the network performance is studied. Experiments show that the number of hidden layers and the number of neurons in each hidden layer affect the relative L2-norm error. With fixed configuration points, the relative norm error does not decrease with the increase in the number of boundary data points, which indicates that in this case, the number of boundary data points has no obvious influence on the error. Through the experiment, the rogue wave solution of the defocusing NLS equation is successfully captured by IPINN method for the first time. The experimental results of this paper are also compared with the results obtained by the physical information neural network method and show that the improved algorithm has higher accuracy. The results of this paper will be contributed to the generalization of deep learning algorithms for solving defocusing NLS equations with time-varying potential.

A high-order accurate wavelet method for solving Schrdinger equations with general nonlinearity

A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a Galerkin procedure is developed for the spatial discretization of the generalized nonlinear Schr6dinger （NLS） equa- tions, and a system of ordinary differential equations for the time dependent unknowns is obtained. Then, the classical fourth-order explicit Runge-Kutta method is used to solve this semi-discretization system. To justify the present method, several widely considered problems are solved as the test examples, and the results demonstrate that the proposed wavelet algorithm has much better accuracy and a faster convergence rate in space than many existing numerical methods.

NOVEL INTEGRABLE HAMILTONIAN HIERARCHIES WITH SIX POTENTIALS

This paper aims to construct six-component integrable hierarchies from a kind of matrix spectral problems within the zero curvature formulation.Their Hamiltonian formulations are furnished by the trace identity,which guarantee the commuting property of infinitely many symmetries and conserved Hamiltonian functionals.Illustrative examples of the resulting integrable equations of second and third orders are explicitly computed.

An Eight Component Integrable Hamiltonian Hierarchy from a Reduced Seventh-Order Matrix Spectral Problem

We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.

Financial Rogue Waves

We analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic,which is nonlinear wave alternative of the Black-Scholes model.These rogue wave solutions may be used to describe thepossible physical mechanisms for rogue wave phenomenon in financial markets and related fields.

Finite Symmetry Transformation Groups and Some Exact Solutions to(2+1)-Dimensional Cubic Nonlinear Schrdinger Equantion

Making use of the direct method proposed by Lou et al. and symbolic computation, finite symmetry transformation groups for a （2＋ l）-dimensional cubic nonlinear Schrodinger （NLS） equation and its corresponding cylindrical NLS equations are presented. Nine related linear independent infinitesimal generators can be obtained from the finite symmetry transformation groups by restricting the arbitrary constants in infinitesimal forms. Some exact solutions are derived from a simple travelling wave solution.

Assessment of theoretical approaches to derivation of internal solitary wave parameters from multi-satellite images near the Dongsha Atoll of the South China Sea

This study assesses the accuracy and the applicability of the Korteweg-de Vries(KdV)and the nonlinear Schr?dinger(NLS)equation solutions to derivation of dynamic parameters of internal solitary waves(ISWs)from satellite images.Visible band images taken by five satellite sensors with spatial resolutions from 5 m to 250 m near the Dongsha Atoll of the northern South China Sea(NSCS)are used as a baseline.From the baseline,the amplitudes of ISWs occurring from July 10 to 13,2017 are estimated by the two approaches and compared with concurrent mooring observations for assessments.Using the ratio of the dimensionless dispersive parameter to the square of dimensionless nonlinear parameter as a criterion,the best appliable ranges of the two approaches are clearly separated.The statistics of total 18 cases indicate that in each 50%of cases,the KdV and the NLS approaches give more accurate estimates of ISW amplitudes.It is found that the relative errors of ISW amplitudes derived from two theoretical approaches are closely associated with the logarithmic bottom slopes.This may be attributed to the nonlinear growth of ISW amplitudes as propagating along a shoaling thermocline or topography.The test results using three consecutive satellite images to retrieve the ISW propagation speeds indicate that the use of multiple satellite images(>2)may improve the accuracy of retrieved phase speeds.Meanwhile,repeated multi-satellite images of ISWs can help to determine the types of ISWs if mooring data are available nearby.

Exact Analytical Solutions in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length

In the paper, the generalized Riccati equation rational expansion method is presented. Making use of the method and symbolic computation, we present three families of exact analytical solutions of Bose-Einstein condensates with the time-dependent interatomic interaction in an expulsive parabolic potential. Then the dynamics of two anlytical solutions are demonstrated by computer simulations under some selectable parameters including the Feshbach-managed nonlinear coefficient and the hyperbolic secant function coefficient.

Asymptotic stability of solitons to 1D nonlinear Schrodinger equations in subcritical case

We prove the asymptotic stability of solitary waves to 1D nonlinear Schrodinger equations in the subcritical case with symmetry and spectrum assumptions.One of the main ideas is to use the vector fields method developed by S.Cuccagna,V.Georgiev,and N.Visciglia[Comm.Pure Appl.Math.,2013,6:957-980]to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term.Meanwhile,we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrodinger equations obtained by G.Staffilani[Duke Math.J.,1997,86(1):109-142]to control the high moments of the solutions emerging from the vector fields method.

The nonlinear evolution of rogue waves generated by means of wave focusing technique

Generating the rogue waves in offshore engineering is investigated,first of all,to forecast its occurrence to protect the offshore structure from being attacked,to study the mechanism and hydrodynamic properties of rouge wave experimentally as well as the rouge/structure interaction for the structure design.To achieve these purposes demands an accurate wave generation and calculation.In this paper,we establish a spatial domain model of fourth order nonlinear Schrdinger(NLS) equation for describing deep-water wave trains in the moving coordinate system.In order to generate rogue waves in the experimental tank efficiently,we take care that the transient water wave(TWW) determines precisely the concentration of time/place.First we simulate the three-dimensional wave using TWW in the numerical tank and modeling the deepwater basin with a double-side multi-segmented wave-maker in Shanghai Jiao Tong University(SJTU) under the linear superposing theory.To discuss its nonlinearity for guiding the experiment,we set the TWW as the initial condition of the NLS equation.The differences between the linear and nonlinear simulations are presented.Meanwhile,the characteristics of the transient water wave,including water particle velocity and wave slope,are investigated,which are important factors in safeguarding the offshore structures.