| 梶原 健司（かじわら けんじ） | データ更新日：2024.04.18 |
教授 ／
マス・フォア・インダストリ研究所
応用理論研究部門
| 1. | Kazuki Hayashi, Yoshiki Jikumaru, Yohei Yokosuka, Kentaro Hayakawa, Kenji Kajiwara, Parametric generation of optimal structures through discrete exponential functions: unveiling connections between structural optimality and discrete isothermicity, Structural and Multidisciplinary Optimization, https://doi.org/10.1007/s00158-024-03767-1, 67, 41, 2024.02.  |
| 2. | Jun-ichi Inoguchi, Yoshiki Jikumaru, Kenji Kajiwara, Kenjiro T. Miura, Wolfgang K. Schief, Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles, Computer Aided Geometric Design, 10.1016/j.cagd.2023.102233, 105, 102233-102233, 2023.07, In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characterize them as invariant curves under the integrable flow on plane curves which is governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formulation developed here, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for the generation of discrete log-aesthetic curves for given G1 data based on the similarity geometry. Our method is able to generate S-shaped discrete curves with an inflection as well as C-shaped curves according to the boundary condition. The resulting discrete curves are regarded as self-adaptive discretization and thus high-quality even with a small number of points.. |
| 3. | Sebastian Elias Graiff-Zurita, Kenji Kajiwara and Toshitomo Suzuki, Fairing of Discrete Planar Curves by Integrable Discrete Analogue of Euler's Elasticae, International Journal of Mathematics for Industry, 10.1142/S2661335222500071, 14, 1, 225007, 2023.03. |
| 4. | Sebastian Elias Graiff-Zurita, Kenji Kajiwara, Kenjiro T. Miura, Fairing of planar curves to log-aesthetic curves, Japan Journal of Industrial and Applied Mathematics, 10.1142/S2661335222500071, 40, 1203-1219, 2023.03. |
| 5. | Sebastián Elías Graiff Zurita, Kenji Kajiwara, Toshitomo Suzuki, Fairing of discrete planar curves to integrable discrete analogue of Euler’s elasticae, International Journal of Mathematics for Industry, 10.1142/s2661335222500071, 2023.03, We construct a method for fairing a given discrete planar curve by using the integrable discrete analog of Euler’s elastica, which is a discrete version of the approximation algorithm presented by Brander et al. We first give a brief review of the integrable discrete analog of Euler’s elastica proposed by Bobenko and Suris, then we present a detailed account of the fairing algorithm, and we apply this method to an architectural problem of characterizing the keylines of Japanese handmade pantiles.. |
| 6. | Sebastián Elías Graiff Zurita, Kenji Kajiwara, Kenjiro T. Miura, Fairing of planar curves to log-aesthetic curves, Japan Journal of Industrial and Applied Mathematics, 10.1007/s13160-023-00567-w, 2023.03, Abstract We present an algorithm to fair a given planar curve by a log-aesthetic curve (LAC). We show how a general LAC segment can be uniquely characterized by seven parameters and present a method of parametric approximation based on this fact. This work aims to provide tools to be used in reverse engineering for computer-aided geometric design. Finally, we show an example of usage by applying this algorithm to the data points obtained from 3D scanning a model-car roof.. |
| 7. | 宇田川誠一，井ノ口順一，梶原健司, Sine-Gordon方程式の解法とその離散化, 日本大学医学部 一般教育研究紀要, 50, 7-24, 2022.12. |
| 8. | Shota Shigetomi, Kenji Kajiwara, Explicit formulas for isoperimetric deformations of smooth and discrete elasticae, JSIAM Letters, https://doi.org/10.14495/jsiaml.13.80, 13, 80-83, 2021.12, [URL]. |
| 9. | Shota Shigetomi, Kenji Kajiwara, Explicit formulas for isoperimetric deformations of smooth and discrete elasticae, JSIAM Letters, 10.14495/jsiaml.13.80, 13, 80-83, 2021.12. |
| 10. | Nalini Joshi, Kenji Kajiwara, Tetsu Masuda, Nobutaka Nakazono, Discrete power functions on a hexagonal lattice I: derivation of defining equations from the symmetry of the Garnier system in two variables, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 10.1088/1751-8121/ac11bd, 54, 33, 2021.08. |
| 11. | Nalini Joshi, Kenji Kajiwara, Tetsu Masuda, Nobutaka Nakazono, Discrete power functions on a hexagonal lattice I: Derivation of defining equations from the symmetry of the Garnier system in two variables, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8121/ac11bd, 54, 33, 335202-335202, 2021.07, The discrete power function on the hexagonal lattice proposed by Bobenko et al is considered, whose defining equations consist of three cross-ratio equations and a similarity constraint. We show that the defining equations are derived from the discrete symmetry of the Garnier system in two variables.. |
| 12. | Kenjiro T. Miura, R.U. Gobithaasan, Péter Salvi, Dan Wang, Tadatoshi Sekine, Shin Usuki, Jun-ichi Inoguchi and Kenji Kajiwara, epsilon kappa-Curves: controlled local curvature extrema, VISUAL COMPUTER, 10.1007/s00371-021-02149-8, 2021.05. |
| 13. | Miura, Kenjiro T, Gobithaasan, R. U, Salvi, Péter, Wang, Dan, Sekine, Tadatoshi, Usuki, Shin, Inoguchi, Jun-ichi, Kajiwara, Kenji, 𝜖𝜅 -Curves: controlled local curvature extrema, The Visual Computer, 10.1007/s00371-021-02149-8, 38, 8, 2723-2738, 2021.05, The kappa-curve is a recently published interpolating spline which consists of quadratic Bezier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or is an element of kappa-curves. kappa-curves have been implemented as the curvature tool in Adobe Illustrator (R) and Photoshop (R) and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bezier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using is an element of kappa-curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.. |
| 14. | Shizuo Kaji, Hyeongki Park, Kenji Kajiwara, Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves, Nonlinear Systems and Their Remarkable Mathematical Structures Volume 2, 356-381, 2020.01, A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of n-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous threefold symmetric Bricard6R linkage also known as the Kaleidocycle, which exhibits a characteristic "turning over" motion. We can model such a linkage as a discrete closed curve in ℝ3 with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. We describe certain motions of this object that are governed by the semi-discrete mKdV equations, where infinitesimally the motion of each vertex is confined in the osculating plane.. |
| 15. | Sebastián Elías Graiff-Zurita, Kenji Kajiwara, Fairing of discrete planar curves by discrete Euler's elasticae, JSIAM Letters, https://doi.org/10.14495/jsiaml.11.73, 11, 73-76, 2019.12, [URL]. |
| 16. | Sebastián Elías, Graiff Zurita, Kenji Kajiwara, Fairing of discrete planar curves by discrete Euler’s elasticae, JSIAM Letters, 10.14495/jsiaml.11.73, 11, 73-76, 2019.12, After characterizing the integrable discrete analogue of the Euler's elastica, we focus our attention on the problem of approximating a given discrete planar curve by an appropriate discrete Euler's elastica. We have decided to do the fairing process via a $L^2!$-distance minimization, because other approaches have presented numerical instabilities. The optimization problem was solved via a gradient-driven optimization method (IPOPT). This problem is non-convex and the result strongly depends on the initial guess. So, we have decided to use a discrete analogue of the algorithm provided by Brander et al., which gives an initial guess to the optimization method.. |
| 17. | Shizuo Kaji, Kenji Kajiwara, Hyeongki Park, Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves, Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 2 (CRC Press,2019), 356-381, 2019.12, A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of n-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous threefold symmetric Bricard6R linkage also known as the Kaleidocycle, which exhibits a characteristic "turning over" motion. We can model such a linkage as a discrete closed curve in ℝ3 with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. We describe certain motions of this object that are governed by the semi-discrete mKdV equations, where infinitesimally the motion of each vertex is confined in the osculating plane.. |
| 18. | Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Discrete Local Induction Equation, Journal of Integrable Systems, 10.1093/integr/xyz003, 4, 1, xyz003, 2019.07, The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrödinger equation. In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schrödinger equation. We also present explicit formulas for both smooth and discrete curves in terms of τ functions of the two-component KP hierarchy.. |
| 19. | Hyeongki Park, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Nozomu Matsuura, Yasuhiro Ohta, Isoperimetric deformations of curves on the Minkowski plane, Int. J. Geom. Methods Mod. Phys., 10.1142/S0219887819501007, 16, 7, 1950100, 2019.07, We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the τ functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class give regular solutions to the defocusing mKdV equation. Some pictures illustrating typical dynamics of the curves are presented.. |
| 20. | Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Discrete local induction equation, Journal of Integrable Systems, 10.1093/integr/xyz003, 4, 1, xyz003, 2019.06, [URL]. |
| 21. | Hyeongki Park, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Nozomu Matsuura, Yasuhiro Ohta, Isoperimetric deformations of curves on the Minkowski plane, International Journal of Geometric Methods in Modern Physics, 10.1142/S0219887819501007, 16, 1950100(20 pages), 2019.06. |
| 22. | Hyeongki Park, Kenji Kajiwara, Takashi Kurose, Nozomu Matsuura, Defocusing mKdV flow on centroaffine plane curves, JSIAM Letters, 10.14495/jsiaml.10.25, 10, 25-28, 2018.07, [URL]. |
| 23. | Hyeongki Park, Kenji Kajiwara, Takashi Kurose, Nozomu Matsuura, Defocusing mKdV flow on centroaffine plane curves, JSIAM Letters, 10.14495/jsiaml.10.25, 10, 25-28, 2018.07, We show that plane curves in the centroaffine geometry admit a flow which is described by the defocusing modified KdV equation. We establish a correspondence between this flow and the KdV flow in the equicentroaffine geometry. We also present an explicit formula for theKdV flow in terms of the τ function.. |
| 24. | Dimetre Triadis, Philip Broadbridge, Kenji Kajiwara, Ken Ichi Maruno, Integrable Discrete Model for One-Dimensional Soil Water Infiltration, Studies in Applied Mathematics, 10.1111/sapm.12208, 140, 4, 483-507, 2018.05, [URL], We propose an integrable discrete model of one-dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection–diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time-dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self-adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naïve discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.. |
| 25. | Dimetre Triadis, Philip Broadbridge, Kenji Kajiwara, Ken-Ichi Maruno, Integrable Discrete Model for One-Dimensional Soil Water Infiltration, Studies in Applied Mathematics, 10.1111/sapm.12208, 140, 4, 483-507, 2018.05, We propose an integrable discrete model of one-dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection–diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time-dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self-adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naïve discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.. |
| 26. | Kenjiro T. Miura, Sho Suzuki, R. U. Gobithaasan, Shin Usuki, Jun ichi Inoguchi, Masayuki Sato, Kenji Kajiwara, Yasuhiro Shimizu, Fairness metric of plane curves defined with similarity geometry invariants, Computer-Aided Design and Applications, 10.1080/16864360.2017.1375677, 15, 2, 256-263, 2018.03, [URL], A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bézier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on σ and G1 constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose σ-curve to introduce symmetry concept for the log-aesthetic curve.. |
| 27. | Jun ichi Inoguchi, Kenji Kajiwara, Kenjiro T. Miura, Masayuki Sato, Wolfgang K. Schief, Yasuhiro Shimizu, Log-aesthetic curves as similarity geometric analogue of Euler's elasticae, Computer Aided Geometric Design, 10.1016/j.cagd.2018.02.002, 61, 1-5, 2018.03, [URL], In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler–Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.. |
| 28. | Kenjiro T. Miura, Sho Suzuki, R. U. Gobithaasan, Shin Usuki, Jun-ichi Inoguchi, Masayuki Sato, Kenji Kajiwara, Yasuhiro Shimizu, Fairness metric of plane curves defined with similarity geometry invariants, Computer-Aided Design and Applications, 10.1080/16864360.2017.1375677, 15, 2, 256-263, 2018.03, A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bézier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on σ and G1 constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose σ-curve to introduce symmetry concept for the log-aesthetic curve.. |
| 29. | Jun-ichi Inoguchi, Kenji Kajiwara, Kenjiro T. Miura, Masayuki Sato, Wolfgang K. Schief, Yasuhiro Shimizu, Log-aesthetic curves as similarity geometric analogue of Euler's elasticae, Computer Aided Geometric Design, 10.1016/j.cagd.2018.02.002, 61, 1-5, 2018.03, In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler–Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.. |
| 30. | 井ノ口 順一, 梶原 健司, 三浦 憲二郎, 朴 炯基, Schief Wolfgang, 相似幾何における弾性曲線とその離散化・CAGDとの関連について, 非線形波動研究の新潮流 -理論とその応用-, 61-68, 2018.03. |
| 31. | Nalini Joshi, Kenji Kajiwara, Tetsu Masuda, Nobutaka Nakazono, Yang Shi, Geometric description of a discrete power function associated with the sixth Painlevé equation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 10.1098/rspa.2017.0312, 473, 2207, 2017.11, [URL], In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with W (3A(1) 1 ) symmetry. By constructing the action of W (3A(1) 1 ) as a subgroup of W (D(1) 4 ), i.e. the symmetry group of PVI, we show how to relate W (D(1) 4 ) to the symmetry group of the lattice. Moreover, by using translations in W (3A(1) 1 ), we explain the odd–even structure appearing in previously known explicit formulae in terms of the t function.. |
| 32. | Nalini Joshi, Kenji Kajiwara, Tetsu Masuda, Nobutaka Nakazono, Yang Shi, Geometric description of discrete power function associated with the sixth Painlevé equation, Proc. R. Soc. A 473 20170312, 10.1098/rspa.2017.0312, 473, 2207, 2017.11. |
| 33. | 野見山雅之，筧三郎，梶原健司, 表面張力入りの Hele-Shaw 問題, 九州大学応用力学研究所研究集会報告, 26AO, S2, 176-181, 2017.05. |
| 34. | Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada, Geometric Aspects of Painlevé Equations, J. Phys. A: Math. Theoret., 10.1088/1751-8121/50/7/073001, 50, 7, 073001, 2017.02, In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev'e equations, with a particular emphasis on the discrete Painlev'e equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on $mathbb{P}^1 imesmathbb{P}^1$ and classified according to the degeration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeomtric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.. |
| 35. | Vladimir Bazhanov, Patrick Dorey, Kenji Kajiwara, Kanehisa Takasaki, Call for papers Special issue on fifty years of the Toda lattice, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8121/aa748f, 50, 31, 2017.01, [URL]. |
| 36. | Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada, Geometric aspects of Painlevé equations, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8121/50/7/073001, 50, 7, 2017.01, [URL], In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev equations, with a particular emphasis on the discrete Painlev� equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P1xP1. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.. |
| 37. | Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, dNLS flow on discrete space curves, Mathematics for Industry, 10.1007/978-981-10-1076-7, 24, 137-149, 2016.06. |
| 38. | Kenji Kajiwara, Toshinobu Kuroda, Nozomu Matsuura, Isogonal deformation of discrete plane curves and discrete Burgers hierarchy, Pacific Journal of Mathematics for Industry, 10.1186/s40736-016-0022-z, 8:3, 2016.03. |
| 39. | Kenji Kajiwara, Toshinobu Kuroda, Nozomu Matsuura, Isogonal deformation of discrete plane curves and discrete Burgers hierarchy, Pacific Journal of Mathematics for Industry, 10.1186/s40736-016-0022-z, 8, 1, 1-14, 2016.03, We study deformations of plane curves in the similarity geometry. It is known that continuous deformations of smooth curves are described by the Burgers hierarchy. In this paper, we formulate the discrete deformation of discrete plane curves described by the discrete Burgers hierarchy as isogonal deformations. We also construct explicit formulas for the curve deformations by using the solution of linear diffusion differential/difference equations.. |
| 40. | Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, dNLS Flow on Discrete Space Curves, arXiv:1509.08076, 2015.11, arXiv version of the paper on MI Lecture Note.. |
| 41. | Kenji Kajiwara, Saburo Kakei, Toda lattice hierarchy and Goldstein-Petrich flows for plane curves, Commentarii Mathematici Univ. St. Pauli, 63, 29-45, 2015.10. |
| 42. | Hisashi Ando, Mike Hay, Kenji Kajiwara, Tetsu Masuda, Explicit formula and extension of the discrete power function associated with the circle patterns of Schramm type, Mathematics for Industry, 10.1007/978-4-431-55007-5_15, 18, 19-32, 2015.09. |
| 43. | Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, dNLS Flow on Discrete Space Curves, MI Lecture Note, 64, 93-102, 2015.09, The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schödinger equation (NLS). In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schrödinger equation (dNLS). We also present explicit formulas for both NLS and dNLS flows in terms of the tau function of the 2-component KP hierarchy.. |
| 44. | 梶原健司, 筧三郎, Toda lattice hierarchy and Goldstein-Petrich flows for plane curves, Comment. Math. Univ. St. Pauli, 64, 29-45, 2015.05, A relation between the Goldstein-Petrich hierarchy for plane curves and the Toda lattice hierarchy is investigated. A representation formula for plane curves is given in terms of a special class of τ-functions of the Toda lattice hierarchy. A representation formula for discretized plane curves is also discussed.. |
| 45. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Discrete Isoperimetric Deformation of Discrete Curves, Mathematics for Industry, 10.1007/978-4-431-55007-5_15, 4, 111-122, 2014.09. |
| 46. | 井ノ口順一, 梶原健司, 松浦望, 太田泰広, 離散 mKdV および離散サイン・ゴルドン方程式による空間離散曲線の変形, RIMS Kokyuroku Bessatsu, B47, 1-22, 2014.06. |
| 47. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Discrete models of isoperimetric deformation of plane curves, Mathematics for Industry, doi:10.1007/978-4-431-55060-0_7, 5, 89-100, 2014.06. |
| 48. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Discrete mKdV and discrete sine-Gordon flows on discrete space curves, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 10.1088/1751-8113/47/23/235202, 47, 23, 235202, 2014.06, In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric condition. The curve is reconstructed from the deformation data by using the Sym-Tafel formula. The isoperimetric equidistant deformation of the space curves does not preserve the torsion in general. However, it is possible to construct the torsion-preserving deformation by tuning the deformation parameters. Further, it is also possible to make an arbitrary choice of the deformation described by the discrete mKdV equation or by the discrete sine-Gordon equation at each step. We finally show that the discrete deformation of discrete space curves yields the discrete K-surfaces.. |
| 49. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Kobe University, Discrete mKdV and discrete sine-Gordon flows on discrete space curves, Journal of Physics A: Mathematical and Theoretical, doi:10.1088/1751-8113/47/23/235202, 47, 23, 235202, 2014.05. |
| 50. | Hisashi Ando, Mike Hay, Kenji Kajiwara, Tetsu Masuda, An explicit formula for the discrete power function associated with circle patterns of Schramm type, Funkcialaj Ekvacioj, 57, 2014, 1-41, 2014.04. |
| 51. | Hisashi Ando, Mike Hay, Kenji Kajiwara, Tetsu Masuda, An Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type, FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, 10.1619/fesi.57.1, 57, 1, 1-41, 2014.04, We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric tau functions for the sixth Painleve equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface. Moreover, we show that the discrete power function is an immersion when the real part of the exponent is equal to one.. |
| 52. | 井ノ口順一, 梶原健司, 松浦望, 太田泰広, 空間離散曲線の等周変形と離散 K 曲面, 九州大学応用力学研究所研究集会報告, 25AO-S2, 1-7, 2014.03. |
| 53. | 筧三郎, 梶原健司, 変形 KdV 階層による平面曲線の運動と戸田階層, 九州大学応用力学研究所研究集会報告, 25AO-S2, 8-13, 2014.03. |
| 54. | Kenji Kajiwara, Nobutaka Nakazono, Hypergeometric solutions to the symmetric q-Painlevé equations, International Mathematical Research Notices, 10.1093/imr/rnt237, 2013, 2013.11. |
| 55. | B.F. Feng, J. Inoguchi, Kenji Kajiwara, K. Maruno, Y. Ohta, Integrable discretizations of the Dym equation, Frontiers of Mathematics in China, 10.1007/s11464-013-0321-y, 8, 5, 1017-1029, 2013.10. |
| 56. | Bao-Feng Feng, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Yasuhiro Ohta, Integrable discretizations of the Dym equation, FRONTIERS OF MATHEMATICS IN CHINA, 10.1007/s11464-013-0321-y, 8, 5, 1017-1029, 2013.10, Integrable discretizations of the complex and real Dym equations are proposed. N-soliton solutions for both semi-discrete and fully discrete analogues of the complex and real Dym equations are also presented.. |
| 57. | K. Maruno, B.F. Feng, J. Inoguchi, K. Kajiwara, Y. Ohta, Semi-discrete analogues of the elastic beam equation and the short pulse equation, Proceedings of 2013 International Symposium on Nonlinear Theory and its Applications, 278-281, 2013.09, Two integrable nonlinear differential- difference systems, semi-discrete analogues of the Wadati-Konno-Ichikawa elastic beam equation and the short pulse equation, are constructed by using a geometric approach.. |
| 58. | K. Maruno, B.F. Feng, J. Inoguchi, K. Kajiwara, Y. Ohta, Semi-discrete analogues of the elastic beam equation and the short pulse equation, Proceedings of 2013 International Symposium on Nonlinear Theory and its Applications, 278-281, 2013.07. |
| 59. | 梶原 健司, 平面曲線の等周変形の離散モデルと離散可積分系, 科学・技術への研究課題への数学アプローチ–数学モデリングの基礎と展開, 46, 2013.03. |
| 60. | 梶原 健司, 三谷浩将, 筧三郎, 離散曲線のダイナミクスと離散可積分系, 立教大学数理物理学研究センター Lecture Note, 1, 1-47, 2013.02. |
| 61. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Motion and Bäcklund transformations of discrete plane curves, Kyushu Journal of Mathematics, doi:10.2206/kyushujm.66.303, 66, 2, 303-324, 2012.10, We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms of the τ function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.. |
| 62. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Motion and Bäcklund Transformations of Discrete Plane Curves, Kyushu Journal of Mathematics, 10.2206/kyushujm.66.303, 66, 2, 303-324, 2012.09, We construct explicit solutions to discrete motion of discrete plane curves which has been introduced by one of the authors recently. Explicit formulas in terms the $ au$ function are presented. Transformation theory of motions of both smooth and discrete curves is developed simultaneously.. |
| 63. | 梶原 健司, 可積分系入門–2次元戸田格子を中心にして–, 離散可積 分系・離散微分幾何チュートリアル 2012, 40, 1-24, 2012.03. |
| 64. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 10.1088/1751-8113/45/4/045206, 45, 4, 045206, 2012.02, We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms of the tau function are presented. Backlund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation.. |
| 65. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves, Journal of Physics A: Mathematical and Theoretical, doi:10.1088/1751-8113/45/4/045206, 45, 045206(16pp), 2012.01, We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms of the τ function are presented. Ba ̈cklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation.. |
| 66. | Bao-Feng Feng, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Yasuhiro Ohta, Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves, Journal of Physics A: Mathematical and Theoretical, doi:10.1088/1751-8113/44/39/395201, 44, 39, 395201, 2011.09. |
| 67. | Bao-Feng Feng, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Yasuhiro Ohta, Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 10.1088/1751-8113/44/39/395201, 44, 39, 395201, 2011.09, We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.. |
| 68. | Hisashi Ando, Mike Hay, Kenji Kajiwara, Tetsu Masuda, An explicit formula for the discrete power function associated with circle patterns of Schramm type, 2011.05, We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric $ au$ functions for the sixth Painlev'e equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface.. |
| 69. | Mike Hay, Kenji Kajiwara and Tetsu Masuda, Bilinearization and special solutions to the discrete Schwarzian KdV equation, Journal of Math-for-Industry, 3, 2011A, 53-62, 2011.04. |
| 70. | 井ノ口順一，梶原健司，松浦望，太田泰広, Semi-discrete modified KdV 方程式と平面離散曲線の時間発展, 九州大学応用力学研究所研究集会報告, 22AO-S8, 2011.04. |
| 71. | Mike Hay, Kenji Kajiwara, Tetsu Masuda, Bilinearization and Special Solutions to the Discrete Schwarzian KdV Equation, Journal of Math-for-Industry, 3, 2011A, 53-62, 2011.04, Various solutions to the discrete Schwarzian KdV equation are discussed. We first derive the bilinear difference equations of Hirota type of the discrete Schwarzian KP equation, which is decomposed into three discrete two-dimensional Toda lattice equations. We then construct two kinds of solutions in terms of the Casorati determinant. We derive the discrete Schwarzian KdV equation on an inhomogeneous lattice and its solutions by a reduction process. We finally discuss the solutions in terms of the $ au$ functions of some Painlev'e systems.. |
| 72. | 井ノ口順一, 梶原健司, 松浦望, 太田泰広, semi-discrete modified KdV方程式と平面離散曲線の時間発展, 九州大学応用力学研究所研究集会報告, 22AO-S8, 75-81, 2011.03. |
| 73. | Kenji Kajiwara, Nobutaka Nakazono and Teruhisa Tsuda, Projective reduction of the discrete Painlevé system of type (A2+A1)(1), International Mathematical Research Notices, 10.1093/imrn/rnq089, Vol. 2010, article ID: rnq089, 2010.05. |
| 74. | Ken-ichi Maruno and Kenji Kajiwara, The discrete potential Boussinesq equation and its multisoliton solutions , Applicable Analysis, 10.1080/00036810903569473 , 89, 4, 593-609, 2010.04. |
| 75. | Kenji Kajiwara and Yasuhiro Ohta, Bilinearization and Casorati determinant solutions to non-autonomous 1 + 1 dimensional discrete soliton equations, RIMS Kokyuroku Bessatsu , B13, 53-74, 2009.11. |
| 76. | 太田 泰広, K. Kajiwara, Bilinearization and Casorati determinant solutions to non-autonomous 1+1 dimensional discrete soliton equations, RIMS Kokyuroku Bessatsu, B13, pp. 53-73, 53-73, 2009.10. |
| 77. | Kenji Kajiwara, Masanobu Kaneko, Atsushi Nobe and Teruhisa Tsuda, Ultradiscretization of a solvable two-dimensional chaotic map assciated with the Hesse cubic curve , Kyushu Journal of Mathematics, 63巻2号315-338ページ, 2009.09. |
| 78. | Kenji Kajiwara, Masanobu Kaneko, Atsushi Nobe, Teruhisa Tsuda, ULTRADISCRETIZATION OF A SOLVABLE TWO-DIMENSIONAL CHAOTIC MAP ASSOCIATED WITH THE HESSE CUBIC CURVE, KYUSHU JOURNAL OF MATHEMATICS, 10.2206/kyushujm.63.315, 63, 2, 315-338, 2009.09, We present a solvable two-dimensional piecewise linear chaotic mail that arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the non-trivial ultradiscrete limit of the solution in spite of a problem known as 'the minus-sign problem'. |
| 79. | 梶原健司，中園信孝，津田照久, q-Painlev\'e 方程式の対称化, 九州大学応用力学研究所研究集会報告 20ME-S7 「非線形波動の数理と物理」, 21−28ページ, 2009.06. |
| 80. | Kenji Kajiwara, Atsushi Nobe, Teruhisa Tsuda, Ultradiscretization of solvable one-dimensional chaotic maps, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 10.1088/1751-8113/41/39/395202, 41, 39, 2008.10, We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that the ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed.. |
| 81. | Kenji Kajiwara, Atsushi Nobe and Teruhisa Tsuda, Ultradiscretization of solvable one-dimensional chaotic maps, Journal of Physics A: Mathematical and Theoretical, 41巻，395202, 2008.09. |
| 82. | Kenji Kajiwara and Yasuhiro Ohta, Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation, Journal of the Physical Society of Japan, 77巻, 054004, 2008.05. |
| 83. | Kenji Kajiwara, Yasuhiro Ohta, Bilinearization and casorati determinant solution to the non-autonomous discrete KdV equation, JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN, 10.1143/JPSJ.77.054004, 77, 5, 2008.05, Casorati determinant solution to the non-autonomous discrete KdV equation is constructed by using the bilinear formalism. We present three different bilinear formulations which have different origins.. |
| 84. | Kenji Kajiwara, Marta Mazzocco and Yasuhiro Ohta, A remark on the Hankel determinant formula for solutions of the Toda equation, Journal of Physics A: Mathematical and Theoretical, 40巻，12661-12675, 2007.10, [URL]. |
| 85. | Taro Hamamoto and Kenji Kajiwara, Hypergeometric solutions to the q-Painleve equation of type A_4^{(1)}, Journal of Physics A: Mathematical and Theoretical, 40巻，12509-12524, 2007.10, [URL]. |
| 86. | Kenji Kajiwara, Marta Mazzocco, Yasuhiro Ohta, A remark on the Hankel determinant formula for solutions of the Toda equation, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 10.1088/1751-8113/40/42/S11, 40, 42, 12661-12675, 2007.10, We consider the Hankel determinant formula of the tau functions of the Toda equation. We present a relationship between the determinant formula and the auxiliary linear problem, which is characterized by a compact formula for the tau functions in the framework of the KP theory. Similar phenomena that have been observed for the Painleve II and IV equations are recovered. The case of finite lattice is also discussed.. |
| 87. | Taro Hamamoto, Kenji Kajiwara, Hypergeometric solutions to the q-Painlevé equation of type $A_4^{(1)}$, Journal of Physics A: Theoretical and Matematical, 10.1088/1751-8113/40/42/S01, 40, 42, 12509-12524, 2007.10, We consider the q-Painlevé equation of type $A_4^{(1)}$ (a version of q-Painlevé V equation) and construct a family of solutions expressible in terms of certain basic hypergeometric series. We also present the determinant formula for the solutions.. |
| 88. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Point configurations, Cremona transformations and the elliptic difference Painleve equation, Séminaires et Congrès , 14巻，175-204., 2007.08. |
| 89. | Nalini Joshi, Kenji Kajiwara and Marta Mazzocco, Generating function associated with the Hankel determinant formula for solutions of the Painleve IV equation, Funkcialaj Ekvacioj, 49(3), 451-468, 2006.12. |
| 90. | Taro Hamamoto, Kenji Kajiwara and Nicholas S. Witte, Hypergeometric solutions to the q-Painleve equation of type $(A_1+A_1')^{(1)}$, Interenational Mathematics Research Notices, 2006, Article ID 84619, 2006.10. |
| 91. | Hiromichi Goto and Kenji Kajiwara, Generating Function Related to the Okamoto Polynomials for the Painlev\'e IV Equation, Bulletin of the Australian Mathematical Society, 71, 3, 517-526, Vol.71(3)(2005) 517-526, 2005.08. |
| 92. | K Kajiwara, A Mukaihira, Soliton solutions for the non-autonomous discrete-time Toda lattice equation, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 10.1088/0305-4470/38/28/008, 38, 28, 6363-6370, 2005.07, We construct N-soliton solution for the non-autonomous discrete-time Toda lattice equation, which is a generalization of the discrete-time Toda equation such that the lattice interval with respect to time is an arbitrary function in time.. |
| 93. | Kenji Kajiwara and Atsushi Mukaihira, Soliton Solutions for the Non-autonomous Discrete-time Toda Lattice Equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/38/28/008, 38, 28, 6363-6370, Vol.38(28) (2005) 6363-6370, 2005.06. |
| 94. | Nalini Joshi, Kenji Kajiwara and Marta Mazzocco, Generating Function Associated with the Determinant Formula for the Solutions of the Painlev\'e II Equation, Ast\'erisque, 297, 67-78, Vol.274(2004) 67-78, 2005.06. |
| 95. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Cubic Pencils and Painlev\'e Hamiltonians, Funkcialaj Ekvacioj, Vol.48(1) (2005) 147-160, 2005.04. |
| 96. | 梶原 健司, 増田 哲, 野海 正俊, 太田 泰広, 山田 泰彦, $q$-Painlevé方程式の超幾何解 (可積分系数理の展望と応用), 数理解析研究所講究録, 1422, 77-98, 2005.04. |
| 97. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Construction of Hypergeometric Solutions to the q-Painlev\'e Equations, International Mathematical Research Notices, 24, 1439-1463, Vol.2005(24) (2005) 1439-1463, 2005.01. |
| 98. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta, Yasuhiko Yamada, Construction of Hypergeometric Solutions to the q-Painlevé Equations, International Mathematics Research Notices, 10.1155/IMRN.2005.1439, 2005, 24, 1439-1463, 2005.01, Hypergeometric solutions to the q-Painlevé equations are constructed by direct linearization of disrcrete Riccati equations. The decoupling factors are explicitly determined so that the linear systems give rise to q-hypergeometric equations.. |
| 99. | 梶原 健司, 増田 哲, 野海 正俊, 太田 泰広, 山田 泰彦, Cremona変換と楕円差分Painleve方程式 : 高次元的な枠組みへの試論 (可積分系理論とその周辺 : 課題と展望を探る), 数理解析研究所講究録, 1400, 197-263, 2004.10. |
| 100. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Hypergeometric solutions to the q-Painleve equations, International Mathematical Research Notices, 47, 2497-2521, 2004:47 (2004) 2497-2521, 2004.08. |
| 101. | 後藤弘道，梶原健司, Okamoto多項式のHankel行列式表示にに付随する母函数, 九州大学応用力学研究所研究集会報告 15ME-S3 「非線形波動および非線形力学系の数理とその応用」, 220-226, 2004.04. |
| 102. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta, Yasuhiko Yamada, Cubic Pencils and Painlev'e Hamiltonians, 2004.03, We present a simple heuristic method to derive the Painlev'e differential equations from the corresponding geometry of rational surafces. We also give a direct relationship between the cubic pencils and Seiberg-Witten curves.. |
| 103. | Kenji Kajiwara, On a q-Painleve III equation.II: rational solutions, Journal of Nonlinear Mathematical Physics, Vol.22 282-303, 2003.08. |
| 104. | K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, 10E9 solution to elliptic Painleve equation, Journal of Physics A: Mathematica and General, Vol.36 L263-L272, 2003.05. |
| 105. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta, Yasuhiko Yamada, 10E9) solution to the elliptic Painlevé equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/36/17/102, 36, 17, L263-L272, 2003.05, A τ function formalism for Sakai's elliptic Painlevé equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the elliptic Painlevé equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of die elliptic Painlevé equation expressed in terms of the elliptic hypergeometric function 10E9.. |
| 106. | 梶原健司, 離散パンルヴェ方程式の理論：現状と展望, 九州大学応用力学研究所研究集会報告 14ME-S7 「非線形波動および非線形力学系に関する最近の話題」, 144-154, 2003.04. |
| 107. | Kenji Kajiwara and Kinji Kimura, On a $q$-Painlev\'e III Equation. I: Derivations, Symmetry and Riccati Type Soutions, Journal of Nonlinear Mathematical Physics, Vol.10, 86-102, 2003.02. |
| 108. | Kenji Kajiwara, Masatoshi Noumi and Yasuhiko Yamada, q-Painlev\'e Systems Arising from q-KP Hierarchy, Letters in Mathematical Physics, 10.1023/A:1022216308475, 62, 3, 259-268, Vol.62, 259-268., 2002.12. |
| 109. | Kenji Kajiwara, Masatoshi Noumi and Yasuhiko Yamada, Discrete Dynamical Systems with W(A{(1)}{m-1} x A{(1)}_{n-1}) Symmetry, Letters in Mathematical Physics, 10.1023/A:1016298925276, 60, 3, 211-219, Vol.60, 211-219, 2002.06. |
| 110. | K Kajiwara, M Noumi, A Yamada, Discrete dynamical systems with W(A(m-1)((1)) x A(n-1)((1))) symmetry, LETTERS IN MATHEMATICAL PHYSICS, 60, 3, 211-219, 2002.06, We give a birational realization of affine Weyl group of type Ad(m-1)((1)) x Ad(n-1)((1)). We apply this representation to construct some discrete integrable systems and discrete Painleve' equations. Our construction has a combinatorial counterpart through the ultra-discretization procedure.. |
| 111. | 梶原健司，木村欣司, q-Painlev\'e III 方程式： 導出・対称性・特殊解, 九州大学応用力学研究所研究集会報告 13ME-S4 「非線形波動現象の理論と応用」, 53-60, 2002.04. |
| 112. | Katsunori Iwasaki, Kenji Kajiwara and Toshiya Nakamura, Generating Function Associated with the Rational Solutions of the Painlev\'e II Equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/35/16/101, 35, 16, L207-L211, Vol.35, L207-L211, 2002.04. |
| 113. | Katsunori Iwasaki, Kenji Kajiwara, Toshiya Nakamura, Generating function associated with the rational solutions of the Painlevé II equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/35/16/101, 35, 16, L207-L211, 2002.04, We consider the Hankel determinant representation for the rational solutions of the Painlevé II equation. We give an explicit formula for the generating function of the entries in terms of logarithmic derivative of the Airy function.. |
| 114. | Tetsu Masuda, Yasuhiro Ohta and Kenji Kajiwara, A Determinant Formula for a Class of Rational Solutions of Painlev\'e V Equation, Nagoya Mathematical Journal, 168, 1-25, Vol. 168, 1-25., 2002.01. |
| 115. | Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada, q-Painlev'e systems arising from q-KP hierarchy, 2001.12, A system of q-Painlev'e type equations with multi-time variables t_1,...,t_M is obtained as a similarity reduction of the N-reduced q-KP hierarchy. This system has affine Weyl group symmetry of type A^{(1)}_{M-1} imes A^{(1)}_{N-1}. Its rational solutions are constructed in terms of q-Schur functions.. |
| 116. | Kenji Kajiwara, Masatoshi Noumi and Yasuhiko Yamada, A Study on the Fourth q-Painlev\'e Equation, Journal of Physics A: Mathematical and General, Vol.34,8563-8581, 2001.10. |
| 117. | Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada, A study on the fourth q-Painlevé equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/34/41/312, 34, 41, 8563-8581, 2001.10, A q-difference analogue of the fourth Painlevé equation is proposed. Its symmetry structure and some particular solutions are investigated.. |
| 118. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Determinant Formulas for the Toda and Discrete Toda Equations, Funkcialaj Ekvacioj, Vol. 44,291-307, 2001.08. |
| 119. | Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada, Discrete dynamical systems with $W(A^{(1)}_{m-1} imes A^{(1)}_{n-1})$ symmetry, 2001.06, We give a birational realization of affine Weyl group of type $A^{(1)}_{m-1} imes A^{(1)}_{n-1}$. We apply this representation to construct some discrete integrable systems and discrete Painlev'e equations. Our construction has a combinatorial counterpart through the ultra-discretization procedure.. |
| 120. | 増田哲，太田泰広，梶原健司, Painlev\'e V 方程式の有理解と universal character, 京都大学数理解析研究所講究録1203「パンルヴェ方程式の解析」, 97-108, 2001.04. |
| 121. | 増田哲, 太田泰広, 梶原健司, Painleve V 方程式の有理解とuniversal character (パンルヴェ方程式の解析), 数理解析研究所講究録, 1203, 97-108, 2001.04. |
| 122. | 増田哲，梶原健司, Painlev\'e 方程式の有理解に対する Schur 関数型表示, 京都大学数理解析研究所講究録 1170 「離散可積分系に関する最近の話題」, 99--110, 2000.09. |
| 123. | 増田哲, 梶原健司, Painleve方程式の有理解に対するSchur関数型表示 (離散可積分系に関する最近の話題), 数理解析研究所講究録, 1170, 99-110, 2000.09. |
| 124. | 梶原健司，高橋大輔，松木平淳太，西成活裕, 超離散系に対する特異点閉じ込めテスト, 九州大学応用力学研究所研究集会報告11ME-S4「非線形波動のメカニズム--現象とモデルの数理構造--」, 115-122, 2000.03. |
| 125. | Kenji Kajiwara, Ken-ichi Maruno and Masayuki Oikawa, Biliearization of Discrete Soliton Equations through the Singularity Confinement Test, Chaos, Solitons & Fractals, 10.1016/S0960-0779(98)00265-3, 11, 1-3, 33-39, Vol.11, 33-40., 2000.03. |
| 126. | Ken-ichi Maruno, Kenji Kajiwara and Masayuki Oikawa, A Note on Integrable Systems Related to Discrete Time Toda Lattice, CRM Proceedings and Lecture Notes, Vol.25, 303-314, 2000.01. |
| 127. | K Kajiwara, K Maruno, M Oikawa, Bilinearization of discrete soliton equations through the singularity confinement test, CHAOS SOLITONS & FRACTALS, 11, 1-3, 33-39, 2000.01, An elementary and systematic method to construct exact solutions for discrete soliton equations is presented. In this method, the singularity patterns, obtained through the singularity confinement test, give us critical information for bilinearization. (C) 1999 Elsevier Science Ltd. All rights reserved.. |
| 128. | Kenji Kajiwara and Tetsu Masuda, On the Umemura Polynomials for the Painlev\'e III Equation, Physics Letters A, Vol.260, 462-467, 1999.09. |
| 129. | K Kajiwara, T Masuda, On the Umemura polynomials for the Painleve III equation, PHYSICS LETTERS A, 260, 6, 462-467, 1999.09, A determinant expression of Jacobi-Trudi type for the rational solutions of the Painleve III (P-III) equation is presented. Entries of determinant are given by the Laguerre polynomials. Degeneration of this determinant expression to that for the rational solutions of P-II is discussed by applying the coalescence procedure. (C) 1999 Published by Elsevier Science B.V. All rights reserved.. |
| 130. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta, Yasuhiko Yamada, Determinant Formulas for the Toda and Discrete Toda Equations, 1999.08, Determinant formulas for the general solutions of the Toda and discrete Toda equations are presented. Application to the $ au$ functions for the Painlev'e equations is also discussed.. |
| 131. | Kenji Kajiwara and Tetsu Masuda, A Generalization of Determinant Formulae for the Solutions of Painlev\'e II and XXXIV Equations, Journal of Physics A: Mathematical and General, Vol. 32, 3763-3778, 1999.05. |
| 132. | K Kajiwara, T Masuda, A generalization of determinant formulae for the solutions of Painleve II and XXXIV equations, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 32, 20, 3763-3778, 1999.05, A generalization of determinant formulae for the classical solutions of Painleve XXXIV and Painleve II equations are constructed using the technique of Darboux transformation and Hirota's bilinear formalism. It is shown that the solutions admit determinant formulae even for the transcendental case.. |
| 133. | Daisuke Takahashi and Kenji Kajiwara, On the Integrability Test for Ultradiscrtete Equations, 京都大学数理解析研究所講究録 1098「離散可積分系の応用数理」, 1-13, 1999.04. |
| 134. | 丸野健一，梶原健司，及川正行, Discrete potential soliton方程式と特異点閉じ込め, 九州大学応用力学研究所研究集会報告10ME-S1「ソリトン理論の新展開」, 123-128, 1999.04. |
| 135. | 増田哲，梶原健司, Darboux Transformation for the Painlev\'e XXXIV and Painlev\'e II Equations, 九州大学応用力学研究所研究集会報告10ME-S1「ソリトン理論の新展開」, 135-142, 1999.04. |
| 136. | 高橋 大輔, 梶原 健司, On integrability test for ultradiscrete equations (Applied Mathematics of Discrete Integrable Systems), 数理解析研究所講究録, 1098, 1-13, 1999.04. |
| 137. | Yoshimasa Nakamura, Kenji Kajiwara and Hironori Shiotani, On an Integrable Discretization of the Rayleigh Quatient Gradient System and the Power Method with the Optimal Shift, Journal of Computational and Applied Mathematics, Vol.96, 77-90, 1998.09. |
| 138. | K. M. Tamizhmani, A. Ramani, B. Grammaticos and K. Kajiwara, Coalescense Cascades and Special Function Solutions for the Continuous and Discrete Painlev\'e Equation, Journal of Physics A: Mathematical and General, Vol.31, 5799--5810, 1998.07. |
| 139. | 丸野健一，梶原健司, 離散ソリトン方程式と singularity confinement, 九州大学応用力学研究所研究集会報告9ME-S2「ソリトン理論の新展開」, 96-102, 1998.05. |
| 140. | 梶原健司，太田泰広, Painlev\'e IV 方程式の代数解と Schur 関数, 九州大学応用力学研究所研究集会報告 9ME-S2「ソリトン理論の新展開」, 117-124, 1998.05. |
| 141. | Ken-ichi Maruno, Kenji Kajiwara and Masayuki Oikawa, Casorati Determinant Solutions for the Discrete Relativistic Toda Lattice Equation, Physics Letters A, 10.1016/S0375-9601(98)00150-9, 241, 6, 335-343, Vol.241, 335-343., 1998.05. |
| 142. | 中尾真一郎，梶原健司，高橋大輔, Multiplicative dPII とその超離散極限について, 九州大学応用力学研究所研究集会報告 9ME-S2「ソリトン理論の新展開」, 125-130, 1998.05. |
| 143. | K Maruno, K Kajiwara, M Oikawa, Casorati determinant solution for the discrete-time relativistic Toda lattice equation, PHYSICS LETTERS A, 241, 6, 335-343, 1998.05, The discrete-time relativistic Toda lattice (dRTL) equation is investigated by using the bilinear formalism. Bilinear equations are systematically constructed with the aid of the singularity confinement method. It is shown that the dRTL equation is decomposed into the Backlund transformations of the discrete-time Toda lattice equation. The N-soliton solution is explicitly constructed in the form of the Casorati determinant. (C) 1998 Published by Elsevier Science B.V.. |
| 144. | Kenji Kajiwara and Yasuhiro Ohta, Determinant Structure of the Rational Solutions for the Painlev\'e IV Equation, Journal of Physics A: Mathematical and General, Vol. 31, 2431--2446, 1998.03. |
| 145. | Kenji Kajiwara, Yasuhiro Ohta, Determinant structure of the rational solutions for the Painlevé IV equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/31/10/017, 31, 10, 2431-2446, 1998.03, Rational solutions for the Painlevé IV equation are investigated by Hirota bilinear formalism. It is shown that the solutions in one hierarchy are expressed by 3-reduced Schur functions, and those in another two hierarchies by Casorati determinants of the Hermite polynomials, or by a special case of the Schur polynomials.. |
| 146. | A Nagai, T Tokihiro, J Satsuma, R Willox, K Kajiwara, Two-dimensional soliton cellular automaton of deautonomized Toda-type, PHYSICS LETTERS A, 10.1016/S0375-9601(97)00591-4, 234, 4, 301-309, 1997.09, A deautonomized version of the two-dimensional Toda lattice equation is presented. Its ultra-discrete analogue and soliton solutions are also discussed. (C) 1997 Published by Elsevier Science B.V.. |
| 147. | Kenji Kajiwara, Kazushi Yamamoto, Yasuhiro Ohta, Rational solutions for the discrete Painlevé II equation, Physics Letters, Section A: General, Atomic and Solid State Physics, 10.1016/S0375-9601(97)00397-6, 232, 3-4, 189-199, 1997.07, The rational solutions for the discrete Painlevé II equation are constructed based on the bilinear formalism. It is shown that they are expressed by a determinant whose entries are given by the Laguerre polynomials. The continuous limit to the Devisme polynomial representation of the rational solutions for the Painlevé II equation is also discussed. © 1997 Elsevier Science B.V.. |
| 148. | K Maruno, K Kajiwara, S Nakao, M Oikawa, Bilinearization of discrete soliton equations and singularity confinement, PHYSICS LETTERS A, 229, 3, 173-182, 1997.05, The singularity confinement method is applied to the systematic derivation of the bilinear equations for discrete soliton equations, Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed. (C) 1997 Published by Elsevier Science B.V.. |
| 149. | Jarmo Hietarinta, Kenji Kajiwara, Rational solutions to d-PIV, 1997.05, We study the rational solutions of the discrete version of Painleve's fourth equation d-PIV. The solutions are generated by applying Schlesinger transformations on the seed solutions -2z and -1/z. After studying the structure of these solutions we are able to write them in a determinantal form that includes an interesting parameter shift that vanishes in the continuous limit.. |
| 150. | F Nijhoff, J Satsuma, K Kajiwara, B Grammaticos, A Ramani, A study of the alternate discrete Painleve II equation, INVERSE PROBLEMS, 12, 5, 697-716, 1996.10, We study a three-point mapping that has the Painleve-II Pn equation as continuous limit. The qualifier 'alternate' is due to the fact that there already exists another discrete form of Pn We present the Lax pair for this discrete equation, as well as its auto-Backlund transform. Special rational solutions as well as solutions in terms of the (discrete) Airy function are obtained for particular values of the parameter of the equation.. |
| 151. | Kenji Kajiwara, Yasuhiro Ohta, Determinant structure of the rational solutions for the Painlevé II equation, Journal of Mathematical Physics, 10.1063/1.531648, 37, 9, 4693-4704, 1996.09, Two types of determinant representations of the rational solutions for the Painlevé II equation are discussed by using the bilinear formalism. One of them is a representation by the Devisme polynomials, and another one is Hankel determinant representation. They are derived from the determinant solutions of the KP hierarchy and Toda lattice, respectively. © 1996 American Institute of Physics.. |
| 152. | J SATSUMA, K KAJIWARA, B GRAMMATICOS, J HIETARINTA, A RAMANI, BILINEAR DISCRETE PAINLEVE-II AND ITS PARTICULAR SOLUTIONS, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 10.1088/0305-4470/28/12/025, 28, 12, 3541-3548, 1995.06, By analogy to the continuous Painleve-II equation, we present particular solutions of the discrete Painleve II (d-P-II) equation. These solutions are of a rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the tau-function for d-P-II. Two different forms of bilinear d-P-II are obtained and we show that they can be related by a simple gauge transformation.. |
| 153. | K KAJIWARA, Y OHTA, J SATSUMA, Q-DISCRETIZATION OF THE 2-DIMENSIONAL TODA EQUATIONS, THEORETICAL AND MATHEMATICAL PHYSICS, 99, 3, 668-674, 1994.06, q-Discrete versions of the two-dimensional Toda molecule equation and the two-dimensional Toda lattice equation are proposed through the direct method. The Backlund transformation and the Lax pair of the former are obtained. Moreover, the reduction to the q-discrete cylindrical Toda equations is also discussed.. |
| 154. | Kenji Kajiwara, Yasuhiro Ohta, Junkichi Satsuma, q-discrete Toda molecule equation, Physics Letters A, 10.1016/0375-9601(93)90705-5, 180, 3, 249-256, 1993.09, A q-discrete version of the two-dimensional Toda molecule equation is proposed through the direct method. Its solution, Bäcklund transformation and Lax pair are discussed. The reduction to the q-discrete cylindrical Toda molecule equation is also discussed. © 1993.. |
| 155. | 梶原 健司, 薩摩 順吉, ソリトン方程式のq-離散化(非線型可積分系の研究の現状と展望), 数理解析研究所講究録, 822, 163-175, 1993.03. |
| 156. | 梶原 健司, 松木平 淳太, 薩摩 順吉, 三次形式で書かれる非線形波動方程式の解(流体中の非線形波動の数理的側面), 数理解析研究所講究録, 782, 782, 183-194, 1992.05. |
| 157. | K KAJIWARA, J SATSUMA, Q-DIFFERENCE VERSION OF THE 2-DIMENSIONAL TODA LATTICE EQUATION, JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN, 10.1143/JPSJ.60.3986, 60, 12, 3986-3989, 1991.12, A q-difference version of the two-dimensional Toda lattice equation is proposed. Through a suitable reduction, it reduces to the q-difference version of the cylindrical Toda lattice equation. It is shown that the reduced equation admits solutions expressed by the q-Bessel function.. |
| 158. | K KAJIWARA, J SATSUMA, THE CONSERVED QUANTITIES AND SYMMETRIES OF THE 2-DIMENSIONAL TODA LATTICE HIERARCHY, JOURNAL OF MATHEMATICAL PHYSICS, 10.1063/1.529387, 32, 2, 506-514, 1991.02, The conserved quantities and symmetries of the soliton equations included in the 2DTL (two-dimensional Toda lattice) hierarchy are considered by means of the theory of the tau function. A typical example is presented for the 2DTL equation that is the simplest nontrivial equation in the hierarchy. Then, by applying suitable reductions, the cases of the Toda lattice equation and the sine-Gordon equation are discussed.. |
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