Hsiao-Fan Liu
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Hsiao-Fan Liu has a total of 9 co-authors. Here are the top co-authors by number of shared publications:
- Hung-Lin Chiu
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, R.O.C.
2 shared publications Chuan-Tsung Chan
Department of Applied Physics, Tunghai University, Taiwan
1 shared publication- Chung-Tsun Shieh
Department of Mathematics Tamkang University 25137 New Taipei City Taiwan
1 shared publication - Lung-Hui Chen
General Education Center Ming Chi University of Technology 24301 New Taipei City Taiwan
1 shared publication - Sergey Buterin
Saratov State University, Saratov, Russia
1 shared publication - Sin-Hua Lai
Fundamental Education Center, National Chin-Yi University of Technology, Taichung, Taiwan, R.O.C.
1 shared publication
Latest Publications
- 30 July 2026
- Published by Elsevier in Journal of Geometry and Physics
- Vol. 229
Darboux transformations for the derivative nonlinear Schrödinger type systems
- 30 July 2026
- Published by Elsevier in Journal of Geometry and Physics
- Vol. 229
Darboux transformations for the derivative nonlinear Schrödinger type systems
- Open Access
- 18 February 2025
- Published by SIGMA (Symmetry, Integrability and Geometry: Methods and Application) in Symmetry, Integrability and Geometry: Methods and Applications
- Vol. 21, 011
On Invariants of Constant $p$-Mean Curvature Surfaces in the Heisenberg Group $H_1$One primary objective in submanifold geometry is to discover fascinating and significant classical examples of $H_1$. In this paper which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant $p$-mean curvature surfaces, we have identified intriguing examples of such surfaces. Notably, we present a complete description of rotationally invariant surfaces of constant $p$-mean curvature and shed light on the geometric interpretation of the energy $E$ with a lower bound.- 18 February 2025
- Published by SIGMA (Symmetry, Integrability and Geometry: Methods and Application) in Symmetry, Integrability and Geometry: Methods and Applications
- Vol. 21, 011
Open AccessOn Invariants of Constant $p$-Mean Curvature Surfaces in the Heisenberg Group $H_1$One primary objective in submanifold geometry is to discover fascinating and significant classical examples of $H_1$. In this paper which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant $p$-mean curvature surfaces, we have identified intriguing examples of such surfaces. Notably, we present a complete description of rotationally invariant surfaces of constant $p$-mean curvature and shed light on the geometric interpretation of the energy $E$ with a lower bound. - Open AccessA characterization of constant $p$-mean curvature surfaces in the Heisenberg group $H_1$In Euclidean $3$-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature $K=-1$. Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, employing the fundamental theorem of surfaces in the Heisenberg group $H_{1}$, we show in this paper that the existence of a constant $p$-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second-order ODE (1.2), which is a kind of {\bf Li\'{e}nard equations}. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to (1.2) (or (1.5)), and hence use the types of the solution to divide constant $p$-mean curvature surfaces into several classes. As a result, after a kind of normalization, we obtain a representation of constant $p$-mean curvature surfaces and classify further all constant $p$-mean curvature surfaces. In Section 9, we provide an approach to construct $p$-minimal surfaces. It turns out that, in some sense, generic $p$-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in [3] (or see [7,8]).Open AccessA characterization of constant $p$-mean curvature surfaces in the Heisenberg group $H_1$In Euclidean $3$-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature $K=-1$. Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, employing the fundamental theorem of surfaces in the Heisenberg group $H_{1}$, we show in this paper that the existence of a constant $p$-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second-order ODE (1.2), which is a kind of {\bf Li\'{e}nard equations}. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to (1.2) (or (1.5)), and hence use the types of the solution to divide constant $p$-mean curvature surfaces into several classes. As a result, after a kind of normalization, we obtain a representation of constant $p$-mean curvature surfaces and classify further all constant $p$-mean curvature surfaces. In Section 9, we provide an approach to construct $p$-minimal surfaces. It turns out that, in some sense, generic $p$-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in [3] (or see [7,8]).
- 1 May 2022
- Published by Wiley in Mathematical Methods in the Applied Sciences
- Vol. 45 (16) , 9635-9652
Sturm–Liouville‐type operators with frozen argument and Chebyshev polynomials
In recent years, there appeared a growing interest in the inverse spectral theory for functional‐differential operators with frozen argument. Such operators are nonlocal and belong to the so‐called loaded differential operators, which frequently appear in mathematics as well as natural sciences and engineering. However, to various classes of nonlocal operators, classical methods of the inverse spectral theory are not applicable. For this reason, it is relevant to develop new methods and approaches for solving inverse problems for operators of this type. We establish a deep connection between the inverse problem for operators with frozen argument and Chebyshev polynomials of the first and the second kinds. Appearing to be of an interest from the point of view of the Chebyshev polynomials theory itself, this connection gives a new perspective method for studying inverse problems for operators with frozen argument. In particular, it allows one to completely describe all non‐degenerate and degenerate cases, that is, when the corresponding inverse problem is uniquely solvable or not, respectively. Moreover, it gives a convenient description of all isospectral potentials in the degenerate case, which is demonstrated by some illustrative examples.
- 1 May 2022
- Published by Wiley in Mathematical Methods in the Applied Sciences
- Vol. 45 (16) , 9635-9652
Sturm–Liouville‐type operators with frozen argument and Chebyshev polynomials
In recent years, there appeared a growing interest in the inverse spectral theory for functional‐differential operators with frozen argument. Such operators are nonlocal and belong to the so‐called loaded differential operators, which frequently appear in mathematics as well as natural sciences and engineering. However, to various classes of nonlocal operators, classical methods of the inverse spectral theory are not applicable. For this reason, it is relevant to develop new methods and approaches for solving inverse problems for operators of this type. We establish a deep connection between the inverse problem for operators with frozen argument and Chebyshev polynomials of the first and the second kinds. Appearing to be of an interest from the point of view of the Chebyshev polynomials theory itself, this connection gives a new perspective method for studying inverse problems for operators with frozen argument. In particular, it allows one to completely describe all non‐degenerate and degenerate cases, that is, when the corresponding inverse problem is uniquely solvable or not, respectively. Moreover, it gives a convenient description of all isospectral potentials in the degenerate case, which is demonstrated by some illustrative examples.
- 16 January 2022
- Published by Springer Nature in Journal of Fixed Point Theory and Applications
- Vol. 24 (1) , 1-19
Periodic Cauchy problem of Heisenberg ferromagnet and its geometric framework
- 16 January 2022
- Published by Springer Nature in Journal of Fixed Point Theory and Applications
- Vol. 24 (1) , 1-19
Periodic Cauchy problem of Heisenberg ferromagnet and its geometric framework
- 1 January 2020
- Published by International Press of Boston in Asian Journal of Mathematics
- Vol. 24 (3) , 483-500
The star mean curvature flow on 3-sphere and hyperbolic 3-space
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2020
- Published by International Press of Boston in Asian Journal of Mathematics
- Vol. 24 (3) , 483-500
The star mean curvature flow on 3-sphere and hyperbolic 3-space
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 15 October 2018
- Published by World Scientific Pub Co Pte Ltd in Random Matrices: Theory and Applications
- Vol. 07 (04)
A q-generalization of the Toda equations for the q-Laguerre/Hermite orthogonal polynomials
- 15 October 2018
- Published by World Scientific Pub Co Pte Ltd in Random Matrices: Theory and Applications
- Vol. 07 (04)
A q-generalization of the Toda equations for the q-Laguerre/Hermite orthogonal polynomials