Solution of Partial Derivative Equations of Poisson and Klein-Gordon with Neumann Conditions as a Generalized Problem of Two-Dimensional Moments

It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.

N-Soliton Solutions of General Nonlinear Schrdinger Equation with Derivative

The bilinear equation of the genera/nonlinear Schrodinger equation with derivative （GDNLSE） and the N-soliton solutions are obtained through the dependent variable transformation and the Hirota method, respectively. The bilinear equation of the nonlinear Schrodinger equation with derivative （DNLSE） and its multisoliton solutions are given by reduction.

Solitary Wave Solutions of a Generalized Derivative Nonlinear Schrdinger Equation

With the aid of a class of nonlinear ordinary differential equation （ODE） and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Sehrodinger equation （GDNLSE） have been found out, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of GDNLSE satisfy certain constraint conditions. For more general GDNLSE, the similar results are also given.

Finite Parameter Solutions to the Manakov and the Derivative Manakov Equations

Finite-dimensional integrable Hamiltonian systems, obtained through the non- linearization of the 3×3 spectral problems associated with the Manakov and the derivative Manakov equations, are investigated. A generating function method is used to give a simple and effective way to prove the involutivity of integrals. Finite-parameter solutions of the Manakov and the derivative Manakov equations are calculated based on the commutative systems of ordinary differential equations with these integrals as Hamiltonians.

Direct Perturbation Method for Derivative Nonlinear Schrdinger Equation

We extend Lou＇s direct perturbation method for solving the nonlinear Schrodinger equation to the case of the derivative nonlinear Schrodinger equation （DNLSE）. By applying this method, different types of perturbation solutions are obtained. Based on these approximate solutions, the analytical forms of soliton parameters, such as the velocity, the width and the initial position, are carried out and the effects of perturbation on solitons are analyzed at the same time. A numerical simulation of perturbed DNLSE finally verifies the results of the perturbation method.

The Interaction and Degeneracy of Mixed Solutions for Derivative Nonlinear Schrodinger Equation

The mixed solutions of the derivative nonlinear Schrödinger equation from the trivial seed (zero solution) are derived by using the determinant representation. By adjusting the interaction and degeneracy of mixed solutions, it is possible to obtain different types of solutions: phase solutions, breather solutions, phase-breather solutions and rogue waves.

Peregrine Rogue Waves Generated by the Interaction and Degeneration of Soliton-Like Solutions: Derivative Nonlinear Schrödinger Equation

We study the Peregrine rogue waves within the framework of Derivative Nonlinear Schrödinger equation, which is used to describe the propagation of Alfven waves in plasma physics and sub-picosecond or femtosecond pulses in nonlinear optics. The interaction and degeneration of two soliton-like solutions and its relations for the breather solution have been analyzed. The Peregrine rogue waves have been considered from the two kinds of formation processes: it can be generated through the limitation of the infinitely large period of the breather solutions, and it can be interpreted as the soliton-like solutions with different polarities. As a special example, a special Peregrine rogue wave is generated by a breather solution and phase solution, which is given by the trivial seed (zero solution).

Adaptive Maxwell’s Equations Derived Optimization and Its Application in Antenna Array Synthesis

In this paper,a self-adaptive method for the Maxwell’s Equations Derived Optimization(MEDO)is proposed.It is implemented by applying the Sequential Model-Based Optimization(SMBO)algorithm to the iterations of the MEDO,and achieves the automatic adjustment of the parameters.The proposed method is named as adaptive Maxwell’s equations derived optimization(AMEDO).In order to evaluate the performance of AMEDO,eight benchmarks are used and the results are compared with the original MEDO method.The results show that AMEDO can greatly reduce the workload of manual adjustment of parameters,and at the same time can keep the accuracy and stability.Moreover,the convergence of the optimization can be accelerated due to the dynamical adjustment of the parameters.In the end,the proposed AMEDO is applied to the side lobe level suppression and array failure correction of a linear antenna array,and shows great potential in antenna array synthesis.

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Fractional differential equations are more and more used in modeling memory（history-dependent,nonlocal,or hereditary） phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY OPERATOR OF A FRACTIONAL ORDER

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid-liquid crystalline （LC） polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie-Ericksen theory is described by the first Rivlin-Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion-extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extru- date of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a seriesof new anisotropic non-Newtonian fluid problems can be addressed.

N-Fold Darboux Transformation for a Nonlinear Evolution Equation

In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.

Global asymptotical stability in a rational difference equation

In this paper we prove a global attractivity result for the unique positive equilibrium point of a difference equation,which improves and generalizes some known ones in the existing literature.Especially,our results completely solve an open problem and some conjectures proposed in[1,2,3,4].

A Degenerate KAM Theorem for Partial Differential Equations with Unbounded Perturbations

In this paper,an infinite dimensional KAM theorem with unbounded perturbations and double normal frequencies is established under qualitative non-degenerate conditions.This is an extension of the degenerate KAM theorem with bounded perturbations by Bambusi,Berti,Magistrelli,and us.As applications,for derivative nonlinear Schrödinger equation with periodic boundary conditions,quasi-periodic solutions around constant solutions are obtained.

Soliton Solution of the DNLS Equation Based on Hirota's Bilinear Derivative Transform

Based on the method of Hirota＇s bilinear derivative transform, the derivative nonlinear Schrodinger equation with vanishing boundary condition has been directly solved. The oneand two-soliton solutions are given as two typical examples in the illustration of the general procedures and the concrete cut-off technique of the series-form solution, and the n-soliton solution is also attained by induction method. Our study shows their equivalence to the existing soliton solutions by a simple parameter transformation. The methodological importance of bilinear derivative transform in dealing with an integrable nonlinear equation has also been emphasized. The evolution of one and two-soliton solution with respect to time and space has been discussed in detail. The collision among the solitons has been manifested through an example of two-soliton case, revealing the elastic essence of the collision and the invariance of the soliton form and characteristics.

Exponential Attractor of the 3D Derivative Ginzburg-Landau Equation

In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution. Furthermore, the existence of a global attractor and an exponential attractor with finite dimensions are proved.

Truncation Analysis for the Derivative Schrodinger Equation

The truncation equation for the derivative nonlinear Schrodinger equation has been dis- cussed in this paper. The existence of a special heteroclinic orbit has been found by using geometrical singular perturbation theory together with Melnikov's technique.

High-order numerical method for the derivative nonlinear Schrodinger equation

In this work,a fourth-order numerical scheme in space and two second-order numerical schemes in both time and space are proposed for the derivative nonlinear Schrodinger equation.We verify the mass conservation for the two-level implicit scheme.The influence on the soliton solution by adding a small random perturbation to the initial condition is discussed.The numerical experiments are given to test the accuracy order for different schemes,respectively.We also test the conservative property of mass and Hamiltonian for these schemes from the numerical point of view.

Long-time Asymptotic Behavior for the Derivative Schrödinger Equation with Finite Density Type Initial Data

In this paper,the authors apply■steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data iqt+qxx+1(lq|^(2)q)_(x)=0,q(x,0)=q0(x),where lim x→±∞ qo(x)=g0(x)=q±and|q±|=1.Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert problem.They compute the long time asymptotic expansion of the solution in differeit space-time regions.For the regionζ=x/t with|ζ+2|<1,the long time asymptotic is given by q(x,t)=T(∞)^(-2)q^(r)Λ(x,t)+O(t^(-3/4)),in which the leading term is N(I)solitons,the second term is a residual error from a■equation.For the regionζ+2|>1,the long time asymptotic is given by q(x,t)=t(∞)^(-2)q^(r)Λ(x,t)-t^(-1/2)if11+O(t^(-3/4)) in which the leading term is N(I)solitons,the second t^(-1/2)order term is soliton-radiation interactions and the third term is a residual error from a■equation.These results are verification of the soliton resolution conjectuore for the derivative Schrödinger equation.In their case of finite density type initial data,the phase functionθ(z)is more complicated that in finite mass initial data.Moreover,two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis,respectively.

On Skew Triangular Matrix Rings

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equations. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.