Mathematical Sciences: Pattern Formation in Higher Order Differential Equations

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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==_--