A Class of Special Solutions for the Ultradiscrete Painlev\'e II Equation

A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter

Max-Plus Algebra for Complex Variables and Its Application to Discrete Fourier Transformation

A generalization of the max-plus transformation, which is known as a method to derive cellular automata from integrable equations, is proposed for complex numbers. Operation rules for this transformation is also studied for general number of complex variables. As an application, the max-plus transformation is applied to the discrete Fourier transformation. Stretched coordinates are introduced to obtain the max-plus transformation whose imaginary part coinsides with a phase of the discrete Fourier transformation

Bilinear Equations and B\"acklund Transformation for Generalized Ultradiscrete Soliton Solution

Ultradiscrete soliton equations and B\"acklund transformation for a generalized soliton solution are presented. The equations include the ultradiscrete KdV equation or the ultradiscrete Toda equation in a special case. We also express the solution by the ultradiscrete permanent, which is defined by ultradiscretizing the signature-free determinant, that is, the permanent. Moreover, we discuss a relation between B\"acklund transformations for discrete and ultradiscrete KdV equations.Comment: 11 page

N-soliton solutions to the DKP equation and Weyl group actions

We study soliton solutions to the DKP equation which is defined by the Hirota bilinear form, {\begin{array}{llll} (-4D_xD_t+D_x^4+3D_y^2) \tau_n\cdot\tau_n=24\tau_{n-1}\tau_{n+1}, (2D_t+D_x^3\mp 3D_xD_y) \tau_{n\pm 1}\cdot\tau_n=0 \end{array} \quad n=1,2,.... where $\tau_0=1$. The $\tau$-functions $\tau_n$ are given by the pfaffians of certain skew-symmetric matrix. We identify one-soliton solution as an element of the Weyl group of D-type, and discuss a general structure of the interaction patterns among the solitons. Soliton solutions are characterized by $4N\times 4N$ skew-symmetric constant matrix which we call the $B$-matrices. We then find that one can have $M$-soliton solutions with $M$ being any number from $N$ to $2N-1$ for some of the $4N\times 4N$ $B$-matrices having only $2N$ nonzero entries in the upper triangular part (the number of solitons obtained from those $B$-matrices was previously expected to be just $N$).Comment: 22 pages, 12 figure

Effects of time and duration of rearing with bottom sand on the occurrence and expansion of staining-type hypermelanosis in the Japanese flounder Paralichthys olivaceus

We previously reported that the progression of staining-type hypermelanosis spontaneously ceased at a specific time and area in Japanese flounder Paralichthys olivaceus. To examine whether time is a limiting factor in the spontaneous cessation of staining, we experimentally controlled the initiation and duration of staining by manipulating the bottom substrate condition in the fish tanks. At 151 days post hatching (DPH; 11 weeks), spontaneous cessation of staining was observed in fish reared in tanks without a sandy substrate. However, staining resumed (or was initiated) in tanks where sand was removed from 11 weeks, indicating a strong but temporary effect of bottom sand and the absence of time limitation in the staining progression by 151 DPH. Extended duration of the inhibitory period of hypermelanosis expansion (9 weeks or more) aided in only a 20 % reduction of the final staining area because of the increased rate of staining expansion. The bottom sandy substrate decreased the visibility of the staining area in individuals, but this was observed only before the completion of the staining expansion. These findings are discussed in relation to possible presence of area limitation of future staining, as well as the fundamental nature of staining

An ultradiscrete integrable map arising from a pair of tropical elliptic pencils

We present a tropical geometric description of a piecewise linear map whose invariant curve is a concave polygon. In contrast to convex polygons, a concave one is not directly related to tropical geometry; nevertheless the description is given in terms of the addition formula of a tropical elliptic curve. We show that the map is arising from a pair of tropical elliptic pencils each member of which is the invariant curve of the ultradiscrete QRT map.Comment: 14 pages, 4 figure

Ultra-discrete Optimal Velocity Model: a Cellular-Automaton Model for Traffic Flow and Linear Instability of High-Flux Traffic

In this paper, we propose the ultra-discrete optimal velocity model, a cellular-automaton model for traffic flow, by applying the ultra-discrete method for the optimal velocity model. The optimal velocity model, defined by a differential equation, is one of the most important models; in particular, it successfully reproduces the instability of high-flux traffic. It is often pointed out that there is a close relation between the optimal velocity model and the mKdV equation, a soliton equation. Meanwhile, the ultra-discrete method enables one to reduce soliton equations to cellular automata which inherit the solitonic nature, such as an infinite number of conservation laws, and soliton solutions. We find that the theory of soliton equations is available for generic differential equations, and the simulation results reveal that the model obtained reproduces both absolutely unstable and convectively unstable flows as well as the optimal velocity model.Comment: 9 pages, 6 figure