反应扩散PDE和耦合ODE系统的空间一致性态

Travelling waves in a PDE-ODE coupled system with nonlinear diffusion

We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value. Stable TW solutions for such systems have previously been observed numerically as well as in biological experiments on the growth of cellulolytic biofilms. In this work, we provide an analytical justification for these observations and prove existence and stability results for TW solutions of such models. Using the TW ansatz and a first integral, the system is re…

走向无限和超越：一些ODE和PDE案例研究

To Infinity and Beyond: Some ODE and PDE Case Studies

当数学/计算问题达到无穷大时，将分析和/或数值计算扩展到无穷大就成为一个众所周知的挑战。我们认为，在适当的奇异变换（原则上可以在计算中检测到），它成为可能超越无穷大的另一边，与解决方案再次成为良好的行为和计算继续正常。在我们的集中，常微分方程（ODE）的例子中，这种无穷大的交叉可以瞬间发生;在空间分布的偏微分方程（PDE）水平上，无穷大的交叉甚至可能持续有限的时间，需要引入概念（和计算）缓冲区，在其中连续（局部）检测和执行适当的奇异变换。这些观察（和相关工具）可以为在更广泛的发展方程中绕过无穷大（从而超越无穷大）的系统方法奠定基础;它们还有望有意义地无缝执行相关计算。沿着我们的分析路径，我们通过复杂化提出了一个正则化过程，并探讨其对动力学的影响;我们还讨论了一组紧化变换及其直观含义。

超越真实爆炸：Masuda迂回与复全息性

Beyond real blow-up: Masuda detours and complex holonomy

本文通过复时间迂回方法，研究复全息性在避免真实爆炸奇点中的作用，并探讨其在抛物型PDEs和ODEs中的应用。

Reduction approach to the dynamics of interacting front solutions in a bistable reaction-diffusion system and its application to heterogeneous media

The dynamics of pulse solutions in a bistable reaction-diffusion system are studied analytically by reducing partial differential equations (PDEs) to finite-dimensional ordinary differential equations (ODEs). For the reduction, we apply the multiple-scales method to the mixed ODE-PDE system obtained by taking a singular limit of the PDEs. The reduced equations describe the interface motion of a pulse solution formed by two interacting front solutions. This motion is in qualitatively good agreement with that observed for the original PDE system. Furthermore, it is found that the reduction not o…

无参数分叉：一些ODE和PDE例子

Bifurcations without parameters: some ODE and PDE examples

标准分歧理论涉及向量场族$dx/dt = f(x,λ)$、$x \in \R^n$，涉及一个或多个常数实参数$λ$。作为一个微分方程对$(x,λ)$，我们观察到一个叶状的总相空间常数$λ$。通常，还强加了平凡的固定解$x=0$的存在：$0 = f(0,λ)$。与此相反，没有参数的分叉通过一个常数参数$λ$丢弃叶理。相反，我们考虑系统$dx/dt = f(x,y), dy/dt = g(x,y)$。标准分歧理论则对应于特殊情况$y=λ, g=0$。为了只保留平凡解$x=0$，我们只需要所有$y$的$0 = f(0,y) = g(0,y)$。一个丰富的动力学现象出现时，正常双曲平凡固定流形$x=0$失败，由于零或纯虚特征值的雅可比矩阵$f_x(0,y)$。具体来说，我们解决的情况下，由于一个简单的特征值零，一个简单的纯虚对（没有参数的Hopf分叉），一个双特征值零（没有参数的Takens-Bogdanov分叉），并由于一个双特征值零与额外的时间反演对称性的正常双曲失败。这一结果是与安德烈·阿芬迪科夫、詹姆斯·C.亚历山大和斯特凡·利布舍尔

On singular Frobenius for second order linear partial differential equations

The main subject of this paper is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the method of Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily…

PDE-Based Optimization for Advection Diffusion Equation in 2D Domain

In this paper, we propose a PDE-based optimization motivated by the problem of microfluidic heat transfer to finding the optimal incompressible velocity fields in 2D domain. To solve this optimization model, we use spectral method to discretize the model to obtain an ODE based optimization. This way significantly reduces the complexity of the discretization optimization model, and gives a more accurate approximation of the original PDE based optimization. Some theoretical results are obtained.

定常旋转玻色-爱因斯坦凝聚体中涡旋位形的表征

On the characterization of vortex configurations in the steady rotating Bose--Einstein condensates

受原子玻色-爱因斯坦凝聚体（BECs）实验的启发，我们比较了一个系统的常微分方程（ODE）的预测一个和两个单独的涡在旋转的BECs与偏微分方程（PDE）的动力学。特别是，我们描述了由于立方排斥相互作用和稳定旋转的对称谐波陷阱中的轨道稳定的涡旋配置。对常微分方程系统进行了详细的分析，并对偏微分方程模型进行了数值逼近。这两个模型之间的良好协议是建立在半经典的（Escheras-Fermi）的限制，对应于在大的化学势的BEC。