Mathematical Sciences: Pattern Formation in Higher Order Differential Equations

DMS9622307特洛伊这项研究的主要目标是了解位于二级相变点附近的双稳系统中图案形成的机制。这种系统的一个例子是，当物理上重要的参数(例如，压力或金属合金的成分)通过临界值时，磁性材料经历从铁磁性到螺旋状的相变。这种相变也发生在液晶和浓缩肥皂溶液中。这些系统的一个原型模型由一个四阶偏微分方程组成，该方程是经典的二阶Fisher-Kolmogorov方程的推广。一个参数伽马乘以四阶空间导数，当伽马等于零时，方程简化为Fisher-Kolmogorov方程。因此，在物理学文献中，该模型被命名为扩展的Fisher Kolmogorov(EFK)方程。根据伽马的选择，数值研究表明，这组解可能会变得极其复杂。这些解包括称为“扭结”的单调过渡层、同宿轨道、周期和非周期解、混沌和行波。为了证明这类解的存在性，提出者和荷兰莱顿大学的L.A.Peletier正在使用他们最近发展的一种新的分析方法。该方法已成功地证明了扭结、周期解和混沌的存在。作者正在用这种新方法来研究波前的形成。除了EFK方程，提出者还研究了更广泛的一类方程。特别令人感兴趣的是悬索桥的模型，如金门大桥。该模型也是四阶的，但非线性项使得方程的分析与EFK模型的分析有根本的不同。与EFK方程一样，提出者正在研究悬索桥模型，以了解复杂图案形成的机理。%这项研究的主要目的是了解位于二阶相变点附近的双稳系统中图案形成的机制。这种系统的一个例子是，当物理上重要的参数(例如，压力或金属合金的成分)通过临界值时，磁性材料经历从铁磁性到螺旋状的相变。这种相变也发生在液晶和浓缩肥皂溶液中。这些系统的一个原型模型由一个四阶偏微分方程组成，该方程是经典的二阶Fisher-Kolmogorov方程的推广。一个参数伽马乘以四阶空间导数，当伽马等于零时，方程简化为Fisher-Kolmogorov方程。因此，在物理学文献中，该模型被命名为扩展的Fisher Kolmogorov(EFK)方程。根据伽马的选择，数值研究表明，这组解可能会变得极其复杂。这些解包括称为“扭结”的单调过渡层、同宿轨道、周期和非周期解、混沌和行波。为了证明这类解的存在性，提出者和荷兰莱顿大学的L.A.Peletier正在使用他们最近发展的一种新的分析方法。该方法已成功地证明了扭结、周期解和混沌的存在。作者正在用这种新方法来研究波前的形成。除了EFK方程，提出者还研究了更广泛的一类方程。特别令人感兴趣的是悬索桥的模型，如金门大桥。该模型也是四阶的，但非线性项使得方程的分析与EFK模型的分析有根本的不同。与EFK方程一样，提出者正在研究悬索桥模型，以了解复杂图案形成的机理。*--=835724635==_--

DMS9622307 Troy The primary goal of this research is to obtain an understanding of the mechanisms responsible for pattern formation in bistable systems which are near a second order phase transition point. An example of such a system is a magnetic material which undergoes a phase transition from ferromagnetic to helicoidal as a physically important parameter (e.g. pressure, or the composition of metal alloys) passes through a critical value. Such phase transitions also occur in liquid crystals, and in concentrated soap solutions. A prototype model for these systems consists of a fourth order partial differential equation derived as a generalization of the classical, second order Fisher-Kolmogorov equation. A parameter gamma multiplies the fourth order spatial derivative, and for gamma equal to zero the equation reduces to the Fisher-Kolmogorov equation. Thus, the model has been named the Extended Fisher Kolmogorov (EFK) equation in the physics literature. Depending on the choice of gamma, numerical studies indicate that the set of solutions can become extremely complicated. Such solutions include monotone transition layers known as ``kinks'', homoclinic orbits, periodic and aperiodic solutions, chaos, and travelling waves. To prove the existence of these kinds of solutions, the proposer and L.A. Peletier of the University of Leiden, Holland, are using a new analytical method which they have recently developed. The method has proved successful for proving the existence of kinks, periodic solutions and chaos. The proposer is now apply- ing this new method to study the formation of wave fronts. In addition to the EFK equation, the proposer is also studying a wider class of equations. Of particular interest is a model of suspension bridges such as the Golden Gate bridge. The model is also fourth order, but the nonlinear terms cause the analysis of the equation to be fundamentally different from that of the EFK model. As with the EFK equation, the proposer is studying the suspension bridge model in order to understand the mechanisms responsible for complicated pattern formation. %%% The primary goal of this research is to obtain an understanding of the mechanisms responsible for pattern formation in bistable systems which are near a second order phase transition point. An example of such a system is a magnetic material which undergoes a phase transition from ferromagnetic to helicoidal as a physically important parameter (e.g. pressure, or the composition of metal alloys) passes through a critical value. Such phase transitions also occur in liquid crystals, and in concentrated soap solutions. A prototype model for these systems consists of a fourth order partial differential equation derived as a generalization of the classical, second order Fisher-Kolmogorov equation. A parameter gamma multiplies the fourth order spatial derivative, and for gamma equal to zero the equation reduces to the Fisher-Kolmogorov equation. Thus, the model has been named the Extended Fisher Kolmogorov (EFK) equation in the physics literature. Depending on the choice of gamma, numerical studies indicate that the set of solutions can become extremely complicated. Such solutions include monotone transition layers known as ``kinks'', homoclinic orbits, periodic and aperiodic solutions, chaos, and travelling waves. To prove the existence of these kinds of solutions the proposer and L.A. Peletier of the University of Leiden, Holland, are using a new analytical method which they have recently developed. The method has proved successful for proving the existence of kinks, periodic solutions and chaos. The proposer is now apply- ing this new method to study the formation of wave fronts. In addition to the EFK equation, the proposer is also studying a wider class of equations. Of particular interest is a model of suspension bridges such as the Golden Gate bridge. The model is also fourth order, but the nonlinear terms cause the analysis of the equation to be fundamentally different from that of the EFK model. As with the EFK equation, the proposer is studying the suspension bridge model in order to understand the mechanisms responsible for complicated pattern formation. *** --=====================_835724635==_--