Mathematics in ultradiscrete integrable systems

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。

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.