The Role Of Painleve II In Predicting New Liquid Crystal Self-Assembly Mechanisms
The Role Of Painleve II In Predicting New Liquid Crystal Self-Assembly Mechanisms
复制标题
Painleve II 在预测新液晶自组装机制中的作用
DOI：
--
复制
发表时间：
2017
影响因子：
2.5
通讯作者：
W. Troy
中科院分区：
文献类型：
--
作者：
W. Troy
We prove the existence of a new class of solutions, called shadow kinks, of the Painleve II equation d2wdz2=2w3+zw+α,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${{frac {{
m d}^{2} w}{{
m d}z^{2}}}=2w^{3} +zw+alpha,}$$end{document} where α<0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${alpha < 0}$$end{document} is a constant. Shadow kinks are sign changing solutions which satisfy w(z)∼--z/2asz→-∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${ w(z) sim -{sqrt {-z/2}} {
m as} z o - infty}$$end{document} and w(z)∼-αzasz→∞.documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${w(z) sim -{frac {alpha}{z}} {
m as} z o infty.}$$end{document} These solutions play a critical role in the prediction of a new class of topological defects, one dimensional shadow kinks and two dimensional shadow vortices, in light-matter interaction experiments on nematic liquid crystals. These new defects are physically important since it has recently been shown (Wang et al. in Nat Mater 15:106–112, 2016) that topological defects are a “template for molecular self-assembly” in liquid crystals. Connections with the modified KdV equation are also discussed.
We prove the existence of a new class of solutions, called shadow kinks, of the Painleve II equation d2wdz2=2w3+zw+α,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${{frac {{
m d}^{2} w}{{
m d}z^{2}}}=2w^{3} +zw+alpha,}$$end{document} where α<0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${alpha < 0}$$end{document} is a constant. Shadow kinks are sign changing solutions which satisfy w(z)∼--z/2asz→-∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${ w(z) sim -{sqrt {-z/2}} {
m as} z o - infty}$$end{document} and w(z)∼-αzasz→∞.documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${w(z) sim -{frac {alpha}{z}} {
m as} z o infty.}$$end{document} These solutions play a critical role in the prediction of a new class of topological defects, one dimensional shadow kinks and two dimensional shadow vortices, in light-matter interaction experiments on nematic liquid crystals. These new defects are physically important since it has recently been shown (Wang et al. in Nat Mater 15:106–112, 2016) that topological defects are a “template for molecular self-assembly” in liquid crystals. Connections with the modified KdV equation are also discussed.