广义Newell-Whitehead-Segel方程的通解
General Solution For Generalised Newell-Whitehead-Segel Equations
摘要：
在这本专著中，研究了两组抛物型微分方程，每一组都具有非线性介质响应。这些方程通常被称为“Newell-Whitehead-Segel方程”，它模拟了各种各样的非线性物理，机械和生物系统。非线性介质响应可以从许多角度来看待，例如，来自介质的记忆响应，由此介质“记住”早期影响;反应性响应，由此介质对输入（例如，化学反应性、湍流和许多其他情况）积极响应;这些方程经常出现在生物科学中，当对种群动态进行建模时，无论种群是基因组的，例如等位基因，或环境中的动物物种;最后，这些方程组通常用于模拟来自可兴奋细胞介质的神经反应。所提供的解决方案是一个非常普遍的性质，事实上，其中，一个规范的解决方案，给出了一类非线性抛物型偏微分方程的非线性介质响应表示为p次迭代卷积或p次乘法响应。正则解集的优点是这些解涉及经典的代表形式，如格林函数或格林热核，并帮助研究人员进一步复杂化，分析和理解这些类型的非线性系统的系统行为。
英文摘要：
In this monograph, two sets of parabolic differential equations are studied, each with nonlinear medium response. The equations are generally referred to as "Newell-Whitehead-Segel equation," which model a wide variety of nonlinear physical, mechanical and biological systems. Nonlinear medium response can be viewed in many perspectives, such as, memory response from the medium, whereby, the medium "remembers" earlier influences; reactive responses, whereby, the medium is actively responsive to input, such as, chemical reactivity, turbulence and many other circumstances; these equations arise often in the biological sciences when modeling population dynamics, whether the population be genomic, such as, alleles, or animal species in the environment; finally, these sets of equations are often employed to model neurological responses from excitable cellular media. The solutions provided are of a very general nature, indeed, whereby, a canonical set of solutions are given for a class of nonlinear parabolic partial differential equations with nonlinear medium response expressed as either a p-times iterative convolution or p-times multiplicative response. The advantage of canonical solution sets are these solutions involve classic representative forms, such as, Green's function or Green's heat kernel and aid researchers in further complication, analysis and understanding of the systemic behavior of these types of nonlinear systems.
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