Tokyo Tech News
Published: March 31, 2009
The Allen-Cahn equation (Nagumo equation) is a basic reaction-diffusion equation for describing two-phase problems in chemistry and in biochemistry. Researchers had been aware of one-dimensional traveling waves and of axially symmetric conical waves. Now, Taniguchi has reported the discovery of axially nonsymmetrical propagation. He describes achieving pyramidal traveling wave solutions in the Allen-Cahn equation in a mathematically rigorous manner.
A pyramidal traveling wave is among the simplest of truly three-dimensional traveling waves. It can deliver information about a cross-sectional polygon, as well as information about a one-dimensional traveling wave. Taniguchi's discovery provides basic mathematical background in support of studying three-dimensional propagations in three-dimensional chemical and biochemical media.
The contour surface of a pyramidal traveling wave solution is in the shape of a reverse pyramid. That pyramid's cross section is a polygon inscribed with a circle. Taniguchi has proved that the pyramidal traveling wave is robust under specified fluctuations. Thus has he demonstrated stable axially nonsymmetrical propagation in the Allen-Cahn equation.
SIAM Journal on Mathematical Analysis 39, no. 1, pp. 319–344 (2007).
Department of Mathematical and Computing Sciences, Graduate School of Information Science and Engineering
The contour surface of a pyramidal traveling wave.
Graduate School of Information Science and Engineering Mathematical and Computing Sciences