# Laurent phenomenon algebras and the discrete BKP equation

@article{Okubo2016LaurentPA, title={Laurent phenomenon algebras and the discrete BKP equation}, author={Naoto Okubo}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2016}, volume={49} }

We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation, the discrete Sawada–Kotera equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.

## 3 Citations

### Laurent phenomenon algebras arising from surfaces

- Mathematics
- 2016

It was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces. We link…

### Laurent phenomenon algebras arising from surfaces II: Laminated surfaces

- MathematicsSelecta Mathematica
- 2020

It was shown by Fock and Goncharov (Dual Teichmüller and lamination spaces. Handbook of Teichmüller Theory, 2007), and Fomin et al. (Acta Math 201(1):83–146, 2008) that some cluster algebras arise…

### The solution to the initial value problem for the ultradiscrete Somos-4 and 5 equations

- Mathematics
- 2017

We propose a method to solve the initial value problem for the ultradiscrete Somos-4 and Somos-5 equations by expressing terms in the equations as convex polygons and regarding max-plus algebras as…

## References

SHOWING 1-10 OF 19 REFERENCES

### Bilinear equations and q-discrete Painlevé equations satisfied by variables and coefficients in cluster algebras

- Mathematics
- 2015

We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to…

### Discrete Integrable Systems and Cluster Algebras

- Mathematics
- 2018

We construct the directed graph (quiver) for which the associated cluster algebra gives the Hirota‐Miwa equation, and prove that the difference equations obtained from its reductions have the Laurent…

### The Laurent Phenomenon

- MathematicsAdv. Appl. Math.
- 2002

A unified treatment of the phenomenon of birational maps given by Laurent polynomials is suggested, which covers a large class of applications and settles in the affirmative a conjecture of D. Gale on integrality of generalized Somos sequences.

### Laurent phenomenon algebras

- Mathematics
- 2012

We generalize Fomin and Zelevinsky's cluster algebras by allowing exchange polynomials to be arbitrary irreducible polynomials, rather than binomials.

### Cluster mutation-periodic quivers and associated Laurent sequences

- Mathematics
- 2011

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the…

### Cluster algebras IV: Coefficients

- MathematicsCompositio Mathematica
- 2007

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these…

### Hirota’s difference equations

- Mathematics
- 1997

A review of selected topics for Hirota’s bilinear difference equation (HBDE) is given. This famous three-dimensional difference equation is known to provide a canonical integrable discretization for…

### Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

- Mathematics
- 2001

Abstract: We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonical…

### Laurent phenomenon sequences

- Mathematics
- 2013

In this paper, we undertake a systematic study of sequences generated by recurrences $$x_{m+n}x_m = P(x_{m+1}, \ldots , x_{m+n-1})$$xm+nxm=P(xm+1,…,xm+n-1) which exhibit the Laurent phenomenon. Some…

### Nonlinear Partial Difference Equations. : I. A Difference Analogue of the Korteweg-de Vries Equation

- Mathematics, Physics
- 1977

A systematic method for isolating certain nonlinear partial difference equations that exhibit solitons is proposed. A nonlinear partial difference equation which approximates the Korteweg-de Vries…