Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1103/PhysRevD.89.065033

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1088/0031-8949/89/03/038006

Language：English   Publishing type：Research paper (scientific journal)  

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and give some exact instanton solutions for various noncommutative settings. We also present a new formula which is crucial to show an origin of the instanton number for U(1) and to prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data.

DOI： 10.1007/s11464-013-0281-2

Language：English   Publishing type：Research paper (scientific journal)  

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data, together with an origin of the instanton number for U(1). We also give a derivation of the ADHM construction from the viewpoint of the Nahm transformation of instantons on four-torus.

DOI： 1088/1742-6596/411/1/012016

Language：Japanese  

Hyper-Kahler 多様体とは, 性質の良い 3 つの複素構造が四元数的に備わった Riemann多様体のことである. 素粒子論においては対称性の高い理論のモジュライ空間として現れ，duality の立証や分配関数の計算などで重要な役割を果たしてきた. この記事では, 主に noncompact な Hyper-Kahler 多様体の性質を，具体的計量の構成・解析により，詳しく議論する. 数学的基礎や関連する話題についても少し触れる.

Language：English   Publishing type：Research paper (scientific journal)  

We discuss the extension of soliton theory and integrable systems to non-commutative spaces, focusing on integrable aspects of non-commutative
anti-self-dual Yang-Mills equations. We present Backlund transformations for the G=U(2) non-commutative anti-self-dual Yang-Mills equations and give a wide class of exact solutions of them (not only instanton-type solutions with nite action). We find that one kind of non-commutative determinants,
quasi-determinants, play a crucial role in the construction of noncommutative solutions. Finally we briefly present some examples of reduction of non-commutative anti-self-dual Yang-Mills equations to non-commutative KdV, NLS and Liouville equations. This is partially based on collaboration with C. Gilson and J. Nimmo (Glasgow).

Language：English   Publishing type：Research paper (scientific journal)  

We present Backlund transformations for the noncommutative anti-self-dual Yang-Mills equation where the gauge group is G=GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasideterminants. We also explain the origins of all of the ingredients of the Backlund transformations within the framework of noncommutative twistor theory. In particular we show that the generated solutions belong to a noncommutative version of the Atiyah-Ward ansatz.

DOI： 10.1098/rspa.2008.0515

Language：English   Publishing type：Research paper (scientific journal)  

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

We study exact multi-soliton solutions of integrable hierarchies on noncommutative space-times which are represented in terms of quasi-determinants of Wronski matrices by Etingof, Gelfand and Retakh. We analyze the asymptotic behavior of the multi-soliton solutions and found that the asymptotic configurations in soliton scattering process can be all the same as commutative ones, that is, the configuration of N-soliton solution has N isolated localized energy densities and the each solitary wave-packet preserves its shape and velocity in the scattering process. The phase shifts are also the same as commutative ones. Furthermore noncommutative toroidal Gelfand-Dickey hierarchy is introduced and the exact multi-soliton solutions are given.

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

Noncommutative Ward's conjecture is a noncommutative version of the original Ward's conjecture which says that almost all integrable equations can be obtained from anti-self-dual Yang-Mills equations by reduction. In this paper, we prove that wide class of noncommutative integrable equations in both (2+1)- and (1+1)-dimensions are actually reductions of noncommutative anti-self-dual Yang-Mills equations with finite gauge groups, which include noncommutative versions of Calogero-Bogoyavlenskii-Schiff eq., Zakharov system, Ward's chiral and topological chiral models, (modified) Korteweg-de Vries, Non-Linear Schroedinger, Boussinesq, N-wave, (affine) Toda, sine-Gordon, Liouville, Tzitzeica, (Ward's) harmonic map eqs., and so on. This would guarantee existence of twistor description of them and the corresponding physical situations in N=2 string theory, and lead to fruitful applications to noncommutative integrable systems and string theories. Some integrable aspects of them are also discussed.

DOI： 10.1016/j.nuclphysb.2006.02.014

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax hierarchies with both space-space and space-time noncommutativities and prove the existence of infinite number of conserved densities. We also give the explicit representations of them in terms of Lax operators. Our results include noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera, modified KdV equations and so on.

DOI： 10.1063/1.1865321

Language：Japanese   Publishing type：Research paper (scientific journal)  

Event date： 2021.3

Language：English   Presentation type：Oral presentation (general)  

Venue：Zoom   Country：Japan  

Event date： 2020.9

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2020.9

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2020.9

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2019.6

Language：English   Presentation type：Oral presentation (general)  

Venue：Sabhal Mor Ostaig, the Gaelic College, the Isle of Skye, UK   Country：United Kingdom  

Event date： 2019.6

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Khazar University, Baku   Country：Azerbaijan  

Event date： 2018.12

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：早稲田本庄セミナーハウス   Country：Japan  

Event date： 2017.12

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Tokyo University of Science   Country：Japan  

Event date： 2017.6

Language：English   Presentation type：Oral presentation (general)  

Country：Italy  

Event date： 2014.12

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Korea, Republic of  

Event date： 2014.11

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Japan  

Event date： 2013.3

Language：English   Presentation type：Oral presentation (invited, special)  

Country：United States  

Event date： 2012.6 - 2012.7

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Greece  

Event date： 2009.11

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Taiwan, Province of China  

Event date： 2009.9

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：早稲田大学   Country：Japan  

Event date： 2009.7

Language：English   Presentation type：Oral presentation (invited, special)  

Country：China  

Event date： 2008.10 - 2008.11

Language：English   Presentation type：Oral presentation (invited, special)  

Country：United Kingdom  

Event date： 2008.7

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Japan  

Event date： 2008.3

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：沼津高専   Country：Japan  

Event date： 2008.2

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Japan  

Event date： 2008.1

Language：English   Presentation type：Oral presentation (invited, special)  

Country：United States  

Event date： 2007.9

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Japan  

Event date： 2006.6

Language：English   Presentation type：Oral presentation (invited, special)  

Country：United Kingdom  

Event date： 2005.11

Language：English   Presentation type：Oral presentation (invited, special)  

Country：United Kingdom  

Event date： 2004.12

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Japan  

Event date： 2004.8

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Germany  

Event date： 2004.2 - 2004.3

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Japan