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Periodic solutions of Allen–Cahn system with the fractional Laplacian
Nonlinear Analysis ( IF 1.4 ) Pub Date : 2020-07-25 , DOI: 10.1016/j.na.2020.112061
Zhuoran Du , Changfeng Gui

We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian (xx)su(x)+F(u(x))=0in R,where u(x)=(u(x),v(x)). The function F:R2R is a smooth double-well potential. We prove the existence of periodic solutions with large period T by using variational methods. Moreover, we draw a conclusion that the second component of periodic solution is identical to zero if the origin is a saddle point of F, whereas the second component is not identical to zero if the origin is a local maximum point of F. A Hamiltonian identity for periodic solutions is also established.



中文翻译:

分数拉普拉斯算子的Allen-Cahn系统的周期解

我们考虑以下与分数拉普拉斯算子相关的非线性系统的周期解 -XXsüX+FüX=0在 [R哪里 üX=üXvX。功能F[R2[R是平滑的双阱势。我们证明了周期大的周期解的存在Ť通过使用变体方法。此外,我们得出一个结论,如果原点是的鞍点,则周期解的第二个分量等于零。F,而如果原点是的局部最大值,则第二个分量不等于零 F。还建立了周期解的哈密顿恒等式。

更新日期:2020-07-25
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