APPLICATIONS OF THE P-R SCHEME FOR GENERALIZED NONLINEAR SCHRODINGER EQUATIONS IN SOLVING SOLITON SOLUTIONS

Spatial soliton solutions of a class of generalized nonlinear Schrodinger equations in N-space are discussed analytically and numerically. This achieved using a traveling wavemethod to formulate one-soliton solution and the P-R method is employed to the numerlcal solutions and the interactions between the solirons for the generalized nonlinear systems in Z-pace.The results presented show that the soliton phenomena are characteristics associated with the nonlinearhies of the dynamical systems.     

Different physical structures of solutions for a generalized Boussinesq wave equation

A technique based on the reduction of order for solving differential equations is employed to investigate a generalized nonlinear Boussinesq wave equation. The compacton solutions, solitons, solitary pattern solutions, periodic solutions and algebraic travelling wave solutions for the equation are expressed analytically under several circumstances. The qualitative change in the physical structures of the solutions is highlighted.     

Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations

A straightforward algorithm for the symbolic computation of generalized (higher‐order) symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the generalized symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi‐discrete lattice equations. With our Integrability Package, generalized symmetries are obtained for several well‐known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of generalized symmetries exists. The existence of a sequence of such symmetries is a predictor for integrability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Two-grid discretization schemes for nonlinear Schrödinger equations

We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of two-coupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction–diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.     

Second-order type-changing evolution equations with first-order intermediate equations

This paper presents a partial classification for C^∞ type-changing symplectic Monge-Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems.     

Integrable nonlinear evolution equations and dynamical systems in multidimensions

The investigation of nonlinear evolution equations and dynamical systems integrable in multidimensions constitutes at present our main research interest. Here we survey findings obtained recently as well as over time: solvable equations (both PDEs and ODEs) are reported, philosophical motivations and methodological approaches are outlined. For more detailed treatments, including the display and analysis of solutions, the interested reader is referred to the original papers.On leave while serving as Secretary General, Pugwash Conferences on Science and World Affairs, Geneva, London, Rome.     

A minimization method and applications to the study of solitons

Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior. In this paper, we prove a general, abstract theorem ( Theorem 26) which allows to prove the existence of a class of solitons. Such solitons are suitable minimizers of a constrained functional and they are called hylomorphic solitons. Then we apply the abstract theory to problems related to the nonlinear Schrödinger equation (NSE) and to the nonlinear Klein–Gordon equation (NKG).     

New features of soliton dynamics in 2+1 dimensions

Exponentially localized soliton solutions have been found recently for all the equations of the hierarchy related to the Zakharov-Shabat hyperbolic spectral problem in the plane. In particular theN ²-soliton solution of the Davey-Stewartson I equation is considered. It is shown that the boundaries fix the kinematics of solitons, while the dynamics of their mutual interaction is determined by the chosen initial condition. The interacting solitons can have, asymptotically, zero mass and can simulate quantum effects as inelastic scattering, fusion and fission, creation and annihilation.Work supported in part by M.U.R.S.T.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方法,即改进的广义射影Ricccati方程方法, 求解(2+1)维色散长波方程, 得到该方程的新的更一般形式的行波解, 包括扭状孤波解, 钟状解,孤子解和周期解. 并对部分新形式孤波解画图示意.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方 法,即改进的广义射影Ricccati方程方法,求解(2 1)维色散长波方程,得到该方程的新的 更一般形式的行波解,包括扭状孤波解,钟状解,孤子解和周期解．并对部分新形式孤波解画 图示意．     

Ill-posedness of nonlocal Burgers equations

Nonlocal generalizations of Burgers equation were derived in earlier work by Hunter [J.K. Hunter, Nonlinear surface waves, in: Current Progress in Hyberbolic Systems: Riemann Problems and Computations, Brunswick, ME, 1988, in: Contemp. Math., vol. 100, Amer. Math. Soc., 1989, pp. 185–202], and more recently by Benzoni-Gavage and Rosini [S. Benzoni-Gavage, M. Rosini, Weakly nonlinear surface waves and subsonic phase boundaries, Comput. Math. Appl. 57 (3–4) (2009) 1463–1484], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage [S. Benzoni-Gavage, Local well-posedness of nonlocal Burgers equations, Differential Integral Equations 22 (3–4) (2009) 303–320] under an appropriate stability condition originally pointed out by Hunter. In this article, it is shown that the latter condition is not only sufficient for well-posedness in Sobolev spaces but also necessary. The main point of the analysis is to show that when the stability condition is violated, nonlocal Burgers equations reduce to second order elliptic PDEs. The resulting ill-posedness result encompasses various cases previously studied in the literature.     

The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of “width refinement” which enlarges the width of strongly refined regions around the shock fronts; the optimal width is found by a numerical study. As a result we are able to improve efficiency by decreasing the number of adaption steps. The performance of the finite volume scheme is compared with several lower order methods.     

Inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Variational iteration method for solving cubic nonlinear Schrödinger equation

The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.     

Stability for steady states of Navier-Stokes-Poisson equations

In this paper, we study the stationary solution and nonlinear stability of Navier-Stokes-Poisson equations. Using variational method, we construct steady states of the N-S-P system as minimizers of a suitably defined energy functional, then show their dynamical stability against general, i.e. not necessarily spherically symmetric perturbation.     

Global solution to a singular integro-differential system related to the entropy balance

This paper is devoted to the mathematical analysis of a thermodynamic model describing phase transitions with thermal memory in terms of an entropy equation and a momentum balance for the microforces. The initial and boundary value problem is addressed for the related integro-differential system of partial differential equations (PDEs). Existence and uniqueness, continuous dependence on the data, and regularity results are proved for the global solution, in a finite time interval.     

NEW PERIODIC SOLUTIONS OF ITO＇S 5th-ORDER mKdV EQUATION AND ITO＇S 7th-ORDER mKdV EQUATION

Based on the modified Jocobi elliptic function expansion method and the modified extended tanh-function method, a new algebraic method is presented to obtain multiple travelling wave solutions for nonlinear wave equations. By using the method ,Ito‘s 5th-order and 7th-order mKdV equations are studied in detail and more new exact Jocobi elliptic function periodic solutions are found. With modulus m→1 or m→0, these solutions degenerate into corresponding solitary wave solutions, shock wave solutions and trigonometric function solutions.     

Exact Traveling Wave Solutions for Higher Order Nonlinear Schrödinger Equations in Optics by Using the (G'/G, 1/G)-expansion Method

The propagation of the optical solitons is usually governed by the nonlinear Schrödinger equations. In this article, the two variable (G'/G, 1/G)-expansion method is employed to construct exact traveling wave solutions with parameters of two higher order nonlinear Schrödinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations rediscovered from the traveling waves. Thismethod can be thought of as the generalization of well-known original (G'/G)-expansion method proposed by M. Wang et al. It is shown that the two variable (G'/G, 1/G)-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.     

Energy preserving integration of bi-Hamiltonian partial differential equations

The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discretization.     

The first integral method to some complex nonlinear partial differential equations

In this paper, the first integral method is used to construct exact solutions of the Hamiltonian amplitude equation and coupled Higgs field equation. The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones.