标题： 关于通过非线性变换将费舍尔-柯尔莫哥洛夫-彼得罗夫斯基-皮斯反应扩散偏微分方程转化为精确可解类的研究 显示英文标题

标题： On study of transition fronts of Fisher-KPP type reaction-diffusion PDEs by non-linear transformations into exactly solvable class

摘要： 种群进化过程中涉及增长和扩散过程的时空动力学可以通过称为反应扩散系统的偏微分方程（PDEs）类别进行建模。在本工作中，我们开发了一种非线性变换方法，该方法将原始的非线性Fisher-KPP类PDE转换为一个可以精确求解的类别。然后我们证明了所提出的非线性变换方法本质上保留了非线性动力系统解的松弛行为至渐近值。我们还表明，这些特定的变换非常易于通过热核和双变量埃尔米特多项式的解析近似得到精确的闭式解。通过这种方法，我们计算了传播波的前缘速度和形状，并展示了非线性变换如何影响短期和长期时期的这些参数。作为应用，我们专注于解决与进化动力学相关的Fisher-KPP类型的PDE的相关情况，根据合子条件为突变基因分配适应度。我们在以下情况下计算了速度的松弛，即Fisher情况、杂合子适应度较低的情况、杂合子适应度较高的情况，以及最后的一般非线性情况。我们还通过使用所提出的方法计算的精确解验证了之前的猜想。 显示英文摘要

摘要： Spatio-temporal dynamics of the evolution of population involving growth and diffusion processes can be modeled by class of partial diffusion equations (PDEs) known as reaction-diffusion systems. In this work, we developed a nonlinear transformations method that converts the original nonlinear Fisher-KPP class of PDEs into an exactly solvable class. We then demonstrated that the proposed nonlinear transformation method intrinsically preserves the relaxation behavior of the solutions to asymptotic values of the non-linear dynamical system. We also show that these particular transforms are very amenable to yield an exact closed form solution in terms of the heat kernel and analytical approximations through the two variable Hermite polynomials. With this proposed method, we calculated the front velocity and shape of the propagating wave and showed how the non-linear transformation affects these parameters for both short and long epochs. As applications, we focus on solving pertinent cases of the Fisher-KPP type of PDEs relating to the evolutionary dynamics by assigning fitness to the mutant gene according to zygosity conditions. We calculated the relaxation of velocity with the parameters of the initial conditions in the following cases, namely, the Fisher, the heterozygote inferior fitness, the heterozygote superior fitness, and finally a general nonlinearity case. We also verified previous conjectures through the exact solutions computed using the proposed method.