This paper is intended as an attempt to set up the global smoothing for the periodic Benjamin equation. It is shown that for Hs（T） initial data with 8 〉 -1/2 and for any s 〈 s1〈 min{s ＋ 1,3s ＋ 1}, the differenc...This paper is intended as an attempt to set up the global smoothing for the periodic Benjamin equation. It is shown that for Hs（T） initial data with 8 〉 -1/2 and for any s 〈 s1〈 min{s ＋ 1,3s ＋ 1}, the difference of the evolution with the linear evolution is in Hs1 （T） for all times, with at most polynomial growing HS1 norm. Unlike Korteweg-de Vries （KdV） equation, there are less symmetries of the Benjamin system, especially for the resonant function. The new ingredient is that we need to deal with some new difficulties that are caused by the lack of symmetries.展开更多
Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space. ELSV formula connects the Hodge integrals with Hurwitz numbers, and the generatin...Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space. ELSV formula connects the Hodge integrals with Hurwitz numbers, and the generating function of Hurwitz numbers satisfies the cut-and-join equation. Therefore, it is natural to consider how to use the cut-and-join equation for Hurwitz numbers to compute Hodge integrals which appear in ELSV formula. In this paper, at first, we will review the method introduced in Goulden et al.＇s paper to get the λg conjecture for Hodge integral. Through some variables transformation, the generating function of Hurwitz number becomes a symmetric polynomial which satisfies a symmetrized cut-and-join equation. By comparing the coefficients of the lowest degree term of both sides in this equation, we can get the ,λg conjecture. Then, in a similar way, we obtain our main result in this paper： a recursive formula for Hodge integral of type contains only one ,λg-l-class. We also point out that our results are closely related to the degree 0 Virasoro conjecture for a curve.展开更多
The Schrodinger equation with hyperbolic potential V （ x ）=- Vosinh 2q （ x / d） / cosh 6 （ x / d） （q= 0, 1, 2, 3） is studied by transforming it into the confluent Heun equation. We obtain genera/symmetric and ...The Schrodinger equation with hyperbolic potential V （ x ）=- Vosinh 2q （ x / d） / cosh 6 （ x / d） （q= 0, 1, 2, 3） is studied by transforming it into the confluent Heun equation. We obtain genera/symmetric and antisymmetric polynomial solutions of the SchrSdinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction＇s nodes decreases with the increase of potentiaJ strengths, and the particle tends to the bottom of the potential well correspondingly.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11171026 and 11271175)National Natural Science Foundation of Shandong Province(Grant No.ZR2012AQ026)
文摘This paper is intended as an attempt to set up the global smoothing for the periodic Benjamin equation. It is shown that for Hs（T） initial data with 8 〉 -1/2 and for any s 〈 s1〈 min{s ＋ 1,3s ＋ 1}, the difference of the evolution with the linear evolution is in Hs1 （T） for all times, with at most polynomial growing HS1 norm. Unlike Korteweg-de Vries （KdV） equation, there are less symmetries of the Benjamin system, especially for the resonant function. The new ingredient is that we need to deal with some new difficulties that are caused by the lack of symmetries.
文摘Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space. ELSV formula connects the Hodge integrals with Hurwitz numbers, and the generating function of Hurwitz numbers satisfies the cut-and-join equation. Therefore, it is natural to consider how to use the cut-and-join equation for Hurwitz numbers to compute Hodge integrals which appear in ELSV formula. In this paper, at first, we will review the method introduced in Goulden et al.＇s paper to get the λg conjecture for Hodge integral. Through some variables transformation, the generating function of Hurwitz number becomes a symmetric polynomial which satisfies a symmetrized cut-and-join equation. By comparing the coefficients of the lowest degree term of both sides in this equation, we can get the ,λg conjecture. Then, in a similar way, we obtain our main result in this paper： a recursive formula for Hodge integral of type contains only one ,λg-l-class. We also point out that our results are closely related to the degree 0 Virasoro conjecture for a curve.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11047025,11075126 and 11031005the Ministry of Education Doctoral Program Funds under Grant Nos.20126101110004,20116101110017SRF for ROCS
文摘The Schrodinger equation with hyperbolic potential V （ x ）=- Vosinh 2q （ x / d） / cosh 6 （ x / d） （q= 0, 1, 2, 3） is studied by transforming it into the confluent Heun equation. We obtain genera/symmetric and antisymmetric polynomial solutions of the SchrSdinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction＇s nodes decreases with the increase of potentiaJ strengths, and the particle tends to the bottom of the potential well correspondingly.
