arXiv subjects

Atsushi Nobe

Publications and source records attributed to Atsushi Nobe.

16 recordsLinked to original sources

arXiv2024

The Volterra lattice, Abel's equation of the first kind, and the SIR epidemic models

The Volterra lattice, when imposing non-zero constant boundary values, admits the structure of a completely integrable Hamiltonian system if the system size is sufficiently small. Such a Volterra lattice can be regarded as an epidemic model known as the SIR model with vaccination, which extends the celebrated SIR model to account for vaccination. Upon the introduction of an appropriate variable transformation, the SIR model with vaccination reduces to an Abel equation of the first kind, which corresponds to an exact differential equation. The equipotential curve of the exact differential equation is the Lambert curve. Thus, the general solution to the initial value problem of the SIR model with vaccination, or the Volterra lattice with constant boundary values, is implicitly provided by using the Lambert W function.

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arXiv2023

Exact solutions to SIR epidemic models via integrable discretization

An integrable discretization of the SIR model with vaccination is proposed. The conserved quantities of the continuous model are inherited to the discrete model through the discretization, since the discretization is based on the intersection structure of the non-algebraic invariant curve defined by the conserved quantities. Uniqueness of the forward/backward evolution of the discrete model is demonstrated in terms of the single-valuedness of the Lambert W function on the positive real axis. Furthermore, the exact solution to the continuous SIR model with vaccination is constructed via the integrable discretization. The discretization procedure similarly applied to the original SIR model leads to two kinds of integrable discretization, and the exact solution to the continuous SIR model is also deduced. It is moreover shown that the discrete SIR model geometrically linearizes the time evolution by using the non-autonomous parallel translation of the line intersecting the invariant curve.

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arXiv2021

Periodicity, linearizability and integrability in seed mutations of type $A^{(1)}_N$

In the network of seed mutations arising from a certain initial seed, an appropriate path emanating from the initial seed is intendedly chosen, noticing periodicity of the exchange matrices in the path each of which is assigned to the generalized Cartan matrix of type $A^{(1)}_N$. Then dynamical property of the seed mutations along the path, which is referred to as of type $A^{(1)}_N$, is intensively investigated. The coefficients assigned to the path form certain $N$ monomials that posses periodicity with period $N$ under the seed mutations and enable to obtain the general terms of the coefficients. The cluster variables assigned to the path of type $A^{(1)}_N$ also form certain $N$ Laurent polynomials possessing the same periodicity as the monomials generated by the coefficients. These Laurent polynomials lead to sufficiently number of conserved quantities of the dynamical system derived from the cluster mutations along the path. Furthermore, by virtue of the Laurent polynomials with periodicity, the dynamical system is non-autonomously linearized and its general solution is concretely constructed. Thus the seed mutations along the path of type $A^{(1)}_N$ exhibit discrete integrability.

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arXiv2019

A family of integrable and non-integrable difference equations arising from cluster algebras

The one-parameter family of second order nonlinear difference equations each of which is given by $$ x_{n-1}x_nx_{n+1}=x_{n-1}+(x_n)^{β-1}+x_{n+1} \qquad(β\in\mathbb{N}) $$ is explored. Since the equation above is arising from seed mutations of a rank 2 cluster algebra, its solution is periodic only when $β\leq3$. In order to evaluate the dynamics with $β\geq4$, algebraic entropy of the birational map equivalent to the difference equation is investigated; it vanishes when $β=4$ but is positive when $β\geq5$. This fact suggests that the difference equation with $β\leq4$ is integrable but that with $β\geq5$ is not. It is moreover shown that the difference equation with $β\geq4$ fails the singularity confinement test. This fact is consistent with linearizability of the equation with $β=4$ and reinforces non-integrability of the equation with $β\geq5$.

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arXiv2019

Generators of rank 2 cluster algebras of affine types via linearization of seed mutations

From the viewpoint of integrable systems on algebraic curves, we discuss linearization of birational maps arising from the seed mutations of types $A^{(1)}_1$ and $A^{(2)}_2$, which enables us to construct the set of all cluster variables generating the corresponding cluster algebras. These birational maps respectively induce discrete integrable systems on algebraic curves referred to as the types of the seed mutations from which they are arising. The invariant curve of type $A^{(1)}_1$ is a conic, while the one of type $A^{(2)}_2$ is a singular quartic curve. By applying the blowing-up of the singular quartic curve, the discrete integrable system of type $A^{(2)}_2$ on the singular curve is transformed into the one on the conic, the invariant curve of type $A^{(1)}_1$. We show that the both discrete integrable systems of types $A^{(1)}_1$ and $A^{(2)}_2$ commute with each other on the conic, the common invariant curve. We moreover show that these integrable systems are simultaneously linearized by means of the conserved quantities and their general solutions are respectively obtained. By using the general solutions, we construct the sets of all cluster variables generating the cluster algebras of types $A^{(1)}_1$ and $A^{(2)}_2$, respectively.

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arXiv2018

Mutations of the cluster algebra of type $\boldsymbol{A^{(1)}_1}$ and the periodic discrete Toda lattice

A direct connection between two sequences of points, one of which is generated by seed mutations in the cluster algebra of type $A^{(1)}_1$ and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations in the cluster algebra of type $A^{(1)}_1$ as addition of points on an elliptic curve arising as the spectral curve of the Toda lattice.

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arXiv2018

A geometric realization of the periodic discrete Toda lattice and its tropicalization

An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the periodic discrete Toda lattice, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the periodic discrete Toda lattice. Finally, the formulation is lifted to the framework of tropical geometry, and a tropical geometric realization of the periodic box-ball system is constructed via tropical curve intersections.

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arXiv2018

Birational maps conjugate to the rank 2 cluster mutations of affine types and their geometry

Mutations of the cluster variables generating the cluster algebra of type $A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and its resolution of the singularity induces a discrete integrable system on a conic governed by a cubic birational map conjugate to the cluster mutations of type $A^{(2)}_2$. Moreover, it is shown that the conic is also the invariant curve of the quadratic birational map arising from the cluster mutations of type $A^{(1)}_1$ and the two birational maps on the conic are commutative. Finally, the commutative birational maps are reduced as singular limits of additions of points on an elliptic curve arising as the spectral curve of the discrete Toda lattice of type $A^{(1)}_1$.

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arXiv2015

On the addition formula for the tropical Hesse pencil

We give the addition formula for the tropical Hesse pencil, which is the tropicalization of the Hesse pencil parametrized by the level-three theta functions, via those for the ultradiscrete theta functions. The ultradiscrete theta functions are reduced from the level-three theta functions through the procedure of ultradiscretization by choosing their parameters appropriately. The parametrization of the level-three theta functions firstly introduced in \cite{KKNT09} gives an explicit correspondence between the amoeba of the real part of the Hesse cubic curve and the tropical Hesse curve. Moreover, through the parametrization, we obtain the subtraction-free forms of the addition formulae for the level-three theta functions, which lead to the addition formula for the tropical Hesse pencil in terms of the ultradiscretization. Using the parametrization, the tropical counterpart of the Hesse configuration is also given.

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arXiv2013

A geometric realization of the ultradiscrete periodic Toda lattice via tropical plane curves

This is a review article on a tropical geometric realization of the ultradiscrete periodic Toda lattice (UD-pTL). Time evolution of the UD-pTL is translated into an addition on the Picard group of its spectral curve, which is a tropical hyperelliptic curve of arbitrary genus depending on the system size. The addition on the Picard group can be realized by using intersection of several tropical plane curves, one of which is the spectral curve. In addition, the tropical eigenvector map, which maps a point in the phase space of the UD-pTL into a set of points on the spectral curve, can also be realized by using intersection of tropical curves. Thus, if the initial values are given then the time evolution of the UD-pTL is completely translated into a motion of intersection points of tropical plane curves. Moreover, all tropical plane curves appearing in the curve intersection are explicitly given in terms of the conserved quantities of the UD-pTL.

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arXiv2011

Addition in Jacobians of tropical hyperelliptic curves

We show that there exists a surjection from the set of effective divisors of degree $g$ on a tropical curve of genus $g$ to its Jacobian by using a tropical version of the Riemann-Roch theorem. We then show that the restriction of the surjection is reduced to the bijection on an appropriate subset of the set of effective divisors of degree $g$ on the curve. Thus the subset of effective divisors has the additive group structure induced from the Jacobian. We finally realize the addition in Jacobian of a tropical hyperelliptic curve of genus $g$ via the intersection with a tropical curve of degree $3g/2$ or $3(g-1)/2$.

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arXiv2011

The group law on the tropical Hesse pencil

We show that the addition of points on the tropical Hesse curve can be realized via the intersection with a tropical line. Then the addition formula for the tropical Hesse curve is reduced from those for the level-three theta functions through the ultradiscretization procedure. A tropical analogue of the Hessian group, the group of linear automorphisms acting on the Hesse pencil, is also investigated; it is shown that the dihedral group of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.

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arXiv2011

A tropical analogue of the Hessian group

We investigate a tropical analogue of the Hessian group $G_{216}$, the group of linear automorphisms acting on the Hesse pencil. Through the procedure of ultradiscretization, the group law on the Hesse pencil reduces to that on the tropical Hesse pencil. We then show that the dihedral group $\mathcal{D}_3$ of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.

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arXiv2010

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.

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arXiv2009

Ultradiscretization of a solvable two-dimensional chaotic map associated with the Hesse cubic curve

We present a solvable two-dimensional piecewise linear chaotic map which arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the nontrivial ultradiscrete limit of the solution in spite of a problem known as "the minus-sign problem."

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arXiv2008

Ultradiscretization of solvable one-dimensional chaotic maps

We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the $m$-th multiplication formula of the elliptic functions is also discussed.

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