Mathematics in ultradiscrete integrable systems

1.我们确定了类型为（A_1^<(1)>）的周期盒球系统（pbbs）的基本周期在系统大小N， N→∞的极限下的渐近行为。由于PBBS的可积性，它们的轨道位于相空间的一个小区域，其体积与exp[N]成正比。我们证明了最大基本循环的阶为exp[N ^<1/2>]，但几乎所有的基本循环都小于exp[(log N)^2].2。构造了由Berenstein和Kazhdan提出的仿射李代数的几何晶体，利用带谱参数的矩阵实现，得到了几何晶体张量积相互交织的双分变换（热带R矩阵）。我们也证明了它的唯一性。热带R矩阵可以用几何Kashiwara算子计算，并且满足Yang-Baxter关系。它不保留加法公式的逆运算，也不能产生超离散系统的分段线性方程。构造了B_n^<(1)>、D_n^<(1)>、A_<2n>^<(2)>和D_<n+1>^<(2)>的组合R矩阵。由这些R矩阵给出的具有玻尔兹曼权值的晶格中的孤子散射被表示为Wyle群的作用。我们还建立了一个类似的逆散射方法。

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.