Generalized Extended tanh-function Metho d for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper,the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations.We choose Fisher's equation,the nonlinear schr(o|¨)dinger equation to illustrate the validity and advantages of the method.Many new and more general traveling wave solutions are obtained.Furthermore,this method can also be applied to other nonlinear equations in physics.     

Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method

The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov （KPP） equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL NONLINEAR DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation,which extend some results of the previous papers.     

SEMICLASSICAL LIMIT TO THE GENERALIZED NONLINEAR SCHR?DINGER EQUATION

In this paper, we investigate the semiclassical limit of the generalized nonlinear Schrdinger equation for initial data with Sobolev regularity. Also, we will analyze the structure of the fluid dynamical system with quantum effect corresponding to the semiclassical limit of the generalized nonlinear Schrdinger equation.     

A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION

In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to this problem is proved. The method developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.     

Existence of Classical Solutions to a Stationary Simplified Quantum Energy-Transport Model in 1-Dimensional Space

The existence of classical solutions to a stationary simplified quantum energytransport model for semiconductor devices in 1-dimensional space is proved.The model consists of a nonlinear elliptic third-order equation for the electron density,including a temperature derivative,an elliptic nonlinear heat equation for the electron temperature,and the Poisson equation for the electric potential.The proof is based on an exponential variable transformation and the Leray-Schauder fixed-point theorem.     

The Asymptotic Behavior of the Stochastic Nonlinear Schr(o)dinger Equation With Multiplicative Noise

The nonlinear Schr(o)dinger equation is one of the basic models for nonlinear waves. In some circumstances, randomness has to be taken into account and it often occurs through a random potential. Here, we consider the following equation     

ON THE INSTABILITY OF SOLUTIONS TO A NONLINEAR VECTOR DIFFERENTIAL EQUATION OF FOURTH ORDER

This paper presents a new result related to the instability of the zero solution to a nonlinear vector differential equation of fourth order.Our result includes and improves an instability result in the previous literature,which is related to the instability of the zero solution to a nonlinear scalar differential equation of fourth order.     

OSCILLATION FOR NONLINEAR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES

Through the use of generalized Riccati transformation techniques, we establish some oscillation criteria for one type of nonlinear dynamic equation on time scales. Several examples, including a semilinear dynamic equation and a nonlinear Emden-Fowler dynamic equation, are also given to illustrate these criteria and to improve the results obtained in some references.     

SUPERPOSITION FORMULAE OF A FIFTH ORDER KdV EQUATION AND ITS MODIFIED EQUATION

The Backlund transformation (BT) for a fifth order KdV equation is presented in the bilinear form.Furthermore,a nonlinear superposition formula related to the BT obtained above is proved rigorously.By the way,a nonlinear superposition formula of a modified fifth order KdV equation is also given.     

TRAVELING WAVE SOLUTIONS AND THEIR STABILITY OF NONLINEAR SCHRDINGER EQUATION WITH WEAK DISSIPATION

In this paper,several new constant-amplitude and variable-amplitude wave solutions(namely,traveling wave solutions) of a generalized nonlinear Schrdinger equation are investigated by using the extended homogeneous balance method,where the balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation,respectively.In addition,stability analysis of those solutions are also conducted by regular phase plane technique.     

DEEP REDUCTION OF DARBOUX TRANSFORMATION TO A3×3 SPECTRAL PROBLEM

gi. IntroductionWave propagation in optical fibers is governed by the higher order nonlinear Schrodinger(HNLS) equation. In receDt years, many authors have analyzed the HNLS equation fromdtherellt poillts of view[1'2]. How to find the exact solutions of the equation is an illterestingwork.In 13], the following higher-order nonlinear Schr6dinger equationwas discussed. It was found in 14] that when FI: PZ: P3 = 1: 6: 3, this equation can betransformed to.Consider a 3 x 3 spectral problem…     

Notes on Global Existence for the Nonlinear Schrdinger Equation Involves Derivative

In this paper, we consider the scattering for the nonlinear Schr¨odinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schr¨odinger equation with nonlinear term |u|2involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for p ≥2d+32d-1and d ≥ 2, which is lower than the Strauss exponents.     

Deep Reduction of Darboux Transformation to a $3\times3$ Spectral Problem

The Darboux transformation of a 3 × 3 spectral problem which is associated with the higherorder nonlinear SchrSdinger equation is given. Some solutions of the higher-order nonlinear Schroedinger equation are provided by taking different “seeds“.     

SOME NONLINEAR OPERATOR EQUATIONS AND THEIR APPROXIMATE SOLUTIONS

Consider nonlinear operator equation: Ax Fx=α, where A is a dense closed linear operator, F is a nonlinear operator. In this paper, we shall give the criterion about solution existence of this equation and the convergence criterion of Galerkin approximate solutions.     

A NUMERICAL METHOD FOR DETERMINING THE OPTIMAL EXERCISE PRICE TO AMERICAN OPTIONS

American Options can be exercised prior to the date of expiration,the valuation of American options then constitutes a free boundary value problem.How to determine the free boundary,i.e. the optimal exercise price,is a key problem.In this paper,a nonlinear equation is given.The free boundary can be obtained by solving the nonlinear equation and the numerical results are better.     

A priori estimates for classical solutions of fully nonlinear elliptic equations

For the fully nonlinear uniformly elliptic equation F(D2u) = 0, it is well known that the viscosity solutions are C2,α if the nonlinear operator F is convex (or concave). In this paper, we study the classical solutions for the fully nonlinear elliptic equation where the nonlinear operator F is locally C1,β a.e. for any 0 < β < 1. We will prove that the classical solutions u are C2,α. Moreover, the C2,α norm of u depends on n,F and the continuous modulus of D2u.     

SOME NONLINEAR OPERATOR EQUATIONS AND THEIR APPROXIMATE SOLUTIONS

Consider nonlinear operator equation, Ax+Fx=α, where A is a dense closed linear operator, F is a nonlinear operator. In this paper, we shall give the criterion about solution existence of this equation and the convergence criterion of Galerkln approximate solutions.