Mathematics in ultradiscrete integrable systems

超离散可积系统中的数学

• 批准号: 12440046
• 负责人: TOKIHIRO Tetsuji
• 金额: $ 8.7万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440046/
• 关键词: ultradiscrete systems; integrable systems; cellular automaton; integrable lattice models; crystal theory; tropical; 可解格子模型; 箱玉系; 超散系

1.We have determined asymptotic behaviour of the fundamental cycle of periodic box-ball systems (PBBSs) of type (A_1^<(1)>)in the limit of the system size N, N → ∞. Due to the integrability of the PBBS, their orbits are located in a small region of the phase space the volume of which is proportional to exp[N]. We have proved that the maximum fundamental cycle is of order of exp[N ^<1/2>], but that almost all the fundamental cycle is less than exp[(log N)^2].2.We constructed the geometric crystal, which was proposed by Berenstein and Kazhdan, for the affine Lie algebra Using the realization by matrices with spectral parameters, the birational transformation (tropical R matrix), which intertwines the tensor product of the geometric crystals, is obtained. We also proved its uniqueness.The tropical R matrix is comutable with the geometric Kashiwara operators and satisfies the Yang-Baxter relation.It does not preserve the inverse operation to addition formulae and produce piecewise linear equations for ultradiscrete systems.3.We have constructed the combinatorial R matrices for B_n^<(1)>, D_n^<(1)>, A_<2n>^<(2)> and D_<n+1>^<(2)>. The soliton scattering in the lattices with Boltzmann weight given by these R matrices are expressed as the action by Wyle group. We also constructed an analogue to the inverse scattering methods.

1.在系统规模N，N→∞的极限下，我们确定了(A_1^&lt；(1)&gt；)型周期箱球系统(PBBS)基本循环的渐近行为。由于PBB的可积性，它们的轨道位于相空间的一个小区域内，该区域的体积与exp[N]成正比。我们证明了最大基本循环是exp[N^&lt；1/2&gt；]的量级，但几乎所有的基本循环都小于exp[(Log N)^2]。2.我们构造了Berenstein和Kazhdan提出的几何晶体，对于仿射李代数，利用带谱参数的矩阵实现，得到了将几何晶体的张量积缠绕在一起的双子变换(热带R矩阵)。3.构造了B_n^&lt；(1)&lt；(1)&gt；(1)&gt；，A_n&lt；2n&gt；^&lt；(2)&gt；以及D_n^&lt；n+1&gt；(2)&gt；由这些R矩阵给出的玻尔兹曼权晶格中的孤子散射被表示为Wyle群的作用。我们还构造了一个类似于逆散射方法的方法。

项目成果

A.Nagai: "Conserved quantities of box ball system"Glasgow Math. J.. 43A. 91-97 (2001)

A.Nagai：“盒子球系统的守恒量”格拉斯哥数学。

T.Kimijima, T.Tokihiro: "Initial value problem of discrete periodic Toda equation and its ultradiscretization"Inverse Problems. 18. 1705-1732 (2002)

T.Kimijima、T.Tokihiro：“离散周期户田方程的初值问题及其超离散化”反问题。

G.Hatayama, A.Kuniba, T.Takagi: "Scattering rules in soliton cellular automata associated with crystal bases"Contemporary Mathematics. 297. 151-182 (2002)

G.Hatayama、A.Kuniba、T.Takagi：“与晶体基相关的孤子元胞自动机中的散射规则”当代数学。

F.Yura: "On a periodic soliton cellular automaton"J.Phys. A : Math. Gen.. 35. 3787-3801 (2002)

F.Yura：“关于周期性孤子元胞自动机”J.Phys。

A.D.Lorenzo: "Quasi-classical descendeants of disordered vertex models with boundaries"Nucl.Phys. B. 644. 409-432 (2002)

A.D.Lorenzo：“具有边界的无序顶点模型的准经典后代”Nucl.Phys。