Stationary layered solutions in ${\Bbb R}^2$ for an Allen–Cahn system with multiple well potential
Stationary layered solutions in ${\Bbb R}^2$ for an Allen–Cahn system with multiple well potential
复制标题
DOI：
10.1007/s005260050071
复制
发表时间：
1997-05
影响因子：
2.1
通讯作者：
S. Alama;L. Bronsard;C. Gui
中科院分区：
文献类型：
--
作者：
S. Alama;L. Bronsard;C. Gui
我们研究了具有多势垒势的椭圆组的整体解。我们寻求在两种意义上“异宿”的解：对于每个固定的解，它们连接(在)的一对恒定的全局极小值，并且当它们连接一对不同的一维驻波解时。这些解描述了在光滑相边界曲线附近的反应扩散系统解的局部结构。这些异宿解的存在证明了标量方程和矢量值Allen-Cahn方程之间的意外差异，即在矢量情况下，跃迁轮廓可以沿界面切向变化。我们还考虑了具有“鞍形”几何的整个定常解，它描述了光滑界面交叉点附近的解的结构。
We study entire solutions onof the elliptic systemwhereis a multiple-well potential. We seek solutionswhich are “heteroclinic,” in two senses: for each fixedthey connect (at) a pair of constant global minima of, and they connect a pair of distinct one dimensional stationary wave solutions when. These solutions describe the local structure of solutions to a reaction-diffusion system near a smooth phase boundary curve. The existence of these heteroclinic solutions demonstrates an unexpected difference between the scalar and vector valued Allen–Cahn equations, namely that in the vectorial case the transition profiles may vary tangentially along the interface. We also consider entire stationary solutions with a “saddle” geometry, which describe the structure of solutions near a crossing point of smooth interfaces.