Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symm...Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here.展开更多
In the 2014 movie, Spotlight, religion, represented by the Catholic Church, has an expected place for the community--the City of Boston, Massachusetts. And, the community of Boston, represented by the institution of a...In the 2014 movie, Spotlight, religion, represented by the Catholic Church, has an expected place for the community--the City of Boston, Massachusetts. And, the community of Boston, represented by the institution of a free press, has a corresponding expectation of the Church. In this paper, I explore these expectations as they are identified in the Oscar winning film, Spotlight.展开更多
We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other app...We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other applications）. Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the ttunter-Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameteressentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact soIutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter-Saxton equation.展开更多
基金The project supported by National Natural Science Foundations of China under Grant Nos. 10735030, 10475055, and 90503006； the Natural Science Research Plan in Shaanxi Province under Grant No. SJ08A09； the Research Fund of Postdoctoral of China under Grant No. 20070410727；the Research Found of Shaanxi Normal University
文摘Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here.
文摘In the 2014 movie, Spotlight, religion, represented by the Catholic Church, has an expected place for the community--the City of Boston, Massachusetts. And, the community of Boston, represented by the institution of a free press, has a corresponding expectation of the Church. In this paper, I explore these expectations as they are identified in the Oscar winning film, Spotlight.
文摘We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other applications）. Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the ttunter-Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameteressentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact soIutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter-Saxton equation.
