Formal power series solutions in a parameter

This paper is a continuation of the previous paper (J. Differential Equations 165 (2000) 255). The main subject is the Gevrey property of formal solutions of an analytic ordinary differential equation in powers of a parameter. In one case, a given formal solution itself is of the Gevrey type, while, in another case, the existence of a formal solution implies the existence of formal solutions of the Gevrey types. These situations are explained systematically in this paper.     

验证形式可积系统解的完备性的机械化算法

在 Reid等人标准型概念的基础上 ,引进形式可积系统的概念 .并给出验证形式可积系统解的完备性的机械化算法 ,并用此方法证明了各向同性弹性力学方程组的胡海昌解的完备性问题 .     

Generalized extended tanh-function method and its application

In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like, period-form solutions of nonlinear evolution equations (NEEs). Compared with most of the existing tanh-function method, extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By using this method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Make use of the method, we study the (3 + 1)-dimensional potential-YTSF equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions’ soliton-like solutions, singular soliton-like solutions, periodic form solutions.     

Existence of analytic solutions to analytic nonlinear q-difference equations

In this paper, some analytic approaches are formulated for the existence of analytic solutions of analytic nonlinear difference equations. From the point of view of dynamical systems, analytic solutions of such kinds of equations can be generally expressed by formal power series of exponential variables, so we are interested in considering a difference equation as a q-difference equation via a suitable coordinate transformation. After stating analytic results for formal series solutions of nonlinear q-difference equations, we also derive some results for the existence of analytic solutions to autonomous rational difference equations.     

Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

In Kravchenko (2008) [8] it was shown that the tool introduced there and called the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we prove a fundamental property of this operator: it preserves the order of zeros and poles of generalized analytic functions and transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a “complicated” Vekua equation from positive formal powers and a generating sequence of a “simpler” Vekua equation. Similar results are obtained regarding the construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theories are considered and examples are given to illustrate the procedure.     

On the Gevrey order of formal solutions of nonlinear difference equations

We prove a Maillet type theorem for formal solutions of nonlinear difference systems, relating the Gevrey order of the formal solutions to the lowest level of an associated, linear difference operator.     

偏微分方程组的形式解与投影极限

以分层理论为基础,给出了偏微分方程组形式解的存在性定理以及形式解与Ehresmann链的投影极限的关系.     

非正则奇性全特征型偏微分方程的形式解(Ⅰ)

本文研究了一类含两个空间变量的具有非正则奇异性的全特征型偏微分方程的形式解.利用幂级数的比较系数法,得到了适当的形式幂级数解的Gevrey指标,并证明了形式解的存在唯一性.     

New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations

In this paper, by using the improved Riccati equations method, we obtain several types of exact traveling wave solutions of breaking soliton equations and Whitham-Broer-Kaup equations. These explicit exact solutions contain solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions. The method employed here can also be applied to solve more nonlinear evolution equations.     

Exact solutions of the Korteweg-de Vries equation with space and time dependent coefficients by the extended unified method

Recently the unified method for finding traveling wave solutions of nonlinear evolution equations was proposed by one of the authors. It was shown that, this method unifies all the methods being used to find these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg-de Vries equation with space-time dependent coefficients.     

一类具非线性边界条件的高阶方程的双参数奇摄动问题

研究了一类含双参数的非线性高阶微分方程的奇摄动问题.运用合成展开法构造了问题的形式渐近解,并运用微分不等式理论证明了原问题解的存在性及所得形式渐近解的一致有效性.     

Purposeful contexts for formal notation in a spreadsheet environment

This article addresses the early stages of children's introduction to the use of variables in formal algebraic notation. We conjecture that some of the difficulties encountered by children in this area may be accentuated by their lack of appreciation of the purpose or power of formal notation. A teaching approach is described that aims to situate the use of formal notation in meaningful contexts. A study of a teaching sequence based on children working with this approach, using graphical feedback in problem solutions, is presented.     

Gevrey character of formal solutions of differential—difference equations with polynomials coefficients

In this paper we prove that formal factorial series as well as formal power series in 1/x solutions of a differential—difference equation with polynomial coefficients are Gevrey of some order which can be determined from a suitable Newton polygon     

Analysis of Formal and Analytic Solutions for Singularities of the Vector Fractional Differential Equations

In this article,we study on the existence of solution for a singularities of a system of nonlinear fractional differential equations (FDE).We construct a formal power series solution for our considering FDE and prove convergence of formal solutions under conditions.We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function.     

Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

New solitary wave and periodic wave solutions for general types of KdV and KdV–Burgers equations

In this paper, we first introduced improved projective Riccati method by means of two simplified Riccati equations. Applying the improved method, we consider the general types of KdV and KdV–Burgers equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions are obtained. Some of them are found for the first time.     

Explicit exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order

In this paper, we improved a method presented previously (Phys. Lett. A 285 (2001) 355) by means of a proper transformation. Applying the improved method, we consider the generalized compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions, are obtained.     

A waiting time phenomenon for modulations of pattern in reaction–diffusion systems

In order to describe slow modulations in time and space of stable or slightly unstable spatially periodic stationary solutions of pattern forming reaction–diffusion systems, so-called phase diffusion equations and Cahn–Hilliard equations can be derived via multiple scaling analysis as formal approximation equations. In the case that these equations degenerate, waiting time phenomena are well known to occur. In this paper, we prove that such waiting time phenomena can also occur approximately in the original reaction–diffusion systems by proving estimates between the formal approximations and the exact solutions of the original systems.     

On k-summability of formal solutions for a class of partial differential operators with time dependent coefficients

We study the k-summability of divergent formal solutions for the Cauchy problem of a certain class of linear partial differential operators with time dependent coefficients. The problem is reduced to a k-summability property of formal solutions for a linear similar ordinary differential equation associated with the Cauchy problem.     

两类Chemotaxis模型解的性质

本文讨论了抛物-双曲和抛物-常微两个Chemotaxis模型初边值问题解的性质.利用形式级数展开的方法，得到全局解在微小扰动下，导致解在有限时间内爆破，并对爆破时间进行了估计.因此说明了这种模型空间齐性解的不稳定性。