双线性神经网络方法和非线性波的研究

很多意义重大的自然科学和工程技术问题都可归结为NLEEs的研究。为了求解NLEEs人们提出了众多有效的方法，但诸方法具有各自的适用范围。所以开发新的方法，研究NLEEs的精确解有着重要的理论和实际意义。本项目将基于符号计算,研究双线性神经网络方法和NLEEs的非线性波及其相互作用解。具体研究内容有：1.双线性神经网络方法是我们最近提出的方法。将进一步完善该方法的理论并优化算法，提高方法的有效性和通用性。利用该方法构造NLEEs的多种丰富的精确解。2.基于直接方法，利用符号计算构造NLEEs的丰富的非线性波和相互作用解，并分析其特性。3.求解超定非线性代数方程组的时候，无法实现人工计算，所以必须借助于符号计算。我们将实现各种求解问题的机械化，即编写行之有效、通用性较强的程序包。为了解决利用符号计算软件无法求解非线性代数方程组的问题，我们将研究并行计算和分类讨论等有效措施，提高计算效率。

Many significant natural science and engineering problems can be attributed to the study of NLEEs. In order to solve NLEEs, many effective methods have been proposed, but each method has its own scope of application. Therefore, it has important theoretical and practical significance to develop new methods and study the exact solution of NLEEs. In this project, we will study the bilinear neural network method and the nonlinear waves of NLEEs based on the symbolic calculation. The specific research contents are as follows: 1. Bilinear neural network method is a new method recently proposed by us. The theory of the method will be further improved and the algorithm will be optimized to improve the effectiveness and universality of the method. This method is used to construct a variety of exact solutions of NLEEs. 2. Based on the direct methods, we will construct abundant nonlinear wave and interaction solutions of NLEEs by using symbolic calculation, and analyze their characteristics. 3. When overdetermined nonlinear algebraic equations are solved, manual calculation cannot be realized, so symbolic calculation is necessary. We will realize the mechanization of all kinds of problem solving, that is, to write effective and universal package. In order to solve the problem that the nonlinear algebraic equations cannot be solved by using symbolic computing software, we will study the effective measures such as parallel computation and classification discussion to improve the computational efficiency.