| 摘要: |
| 主要研究一个反应扩散方程组和两个反应扩散方程式的无穷多个多脉冲驻波解的存在性和不稳定性.首先, 研究某个特征值问题的特征值和特征函数来建立谱不稳定性.为此目的,引入Evans函数 (即复解析函数).Evans函数的定义域是整个右半复平面, 其左边界是一条竖直直线,并且位于虚轴的左边.一个非常重要点是一个复数是特征值问题的特征值,当且仅当那个复数是Evans函数的一个零点.然后把线性化稳定性标准 (即驻波脉冲解的非线性稳定性,线性稳定性和谱稳定性之间的等价性)和Evans函数的性质结合在一起来建立多脉冲解的不稳定性. |
| 关键词: 非线线性系统 多脉冲驻波解 线性化稳定性标准 谱稳定性 Evans函数 |
| DOI： |
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| 基金项目: |
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| Evans functions for multiple standing pulse solutions of a nonlinear system of reaction diffusion equations |
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ZHANG Linghai
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Department of Mathematics, Lehigh University
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| Abstract: |
| The main purpose is to accomplish the existence and instability of infinitely many multiple standing pulse solutions of a nonlinear system of reaction diffusion equations and some nonlinear scalar reaction diffusionequations. We will study the eigenvaluesand eigen functions of some eigen value problems to establish the spectral instability. To do this, we will introduceEvans functions(complex analytic functions). The Evans functions are defined in some right half complex plane ?, the left boundary is a vertical straight line located to the left of the imaginary axis. It turns out that a complex number λ0 is an eigenvalue of the eigenvalue problem if and only if λ0 is a zero of an Evans function. Then we will coupletogether the linearized stability criterion(the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) to accomplish the nonlinear instability of the multiple standing pulse solutions.Overall,thereexistinfinitelymanymultiplestandingpulsesolutionstoboththesystemandthescalar equation: for any natural number m > 1, there is a multiple standing pulse solution with m peaks. All of the multiple standing pulse solutions are unstable. |
| Key words: nonlinear system of reaction diffusion equations multiple standing pulse solutions existence instability Evans functions |
