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Mikio Murata

Publications and source records attributed to Mikio Murata.

6 recordsLinked to original sources
arXiv2013

Spatial pattern of discrete and ultradiscrete Gray-Scott model

Ultradiscretization is a limiting procedure transforming a given difference equation into a cellular automaton. In addition the cellular automaton constructed by this procedure preserves the essential properties of the original equation, such as the structure of exact solutions for integrable equations. In this article, we propose a discretization and an ultradiscretization of Gray-Scott model which is not an integrable system and which gives various spatial patterns with appropriate initial data and parameters. The resulting systems give a travelling pulse and a self-replication pattern with appropriate initial data and parameters. The ultradiscrete system is directly related to the elementary cellular automaton Rule 90 which gives a Sierpinski gasket pattern. A $(2+1)$D ultradiscrete Gray-Scott model that gives a ring pattern, a self-replication pattern and a chaotic pattern, is also constructed.

nlin.PS↗
arXiv2008

Lax forms of the $q$-Painlevé equations

All $q$-Painlevé equations which are obtained from the $q$-analog of the sixth Painlevé equation are expressed in a Lax formalism. They are characterized by the data of the associated linear $q$-difference equations. The degeneration pattern from the $q$-Painlevé equation of type $A_2$ is also presented.

nlin.SI↗
arXiv2007

Two-component soliton systems and the Painlevé equations

We give an extension of the two-component KP hierarchy by considering additional time variables. We obtain the linear $2\times 2$ system by taking into consideration the hierarchy through a reduction procedure. The Lax pair of the Schlesinger system and the sixth Painlevé equation is given from this linear system. A unified approach to treat the other Painlevé equations from the usual two-component KP hierarchy is also considered.

nlin.SI↗
arXiv2003

New Expressions for Discrete Painlevé Equations

It is known that discrete Painlevé equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlevé equations in which the symmetries become clearly visible. We know how to obtain discrete Painlevé equations from certain rational surfaces in connection with the extended affine Weyl groups. By means of this representation, we clarify the relation between the equation and the surface.

nlin.SI↗
arXiv2002

Riccati Solutions of Discrete Painlevé Equations with Weyl Group Symmetry of Type $E_8^{(1)}$

We present a special solutions of the discrete Painlevé equations associated with $A_0^{(1)}$, $A_0^{(1)*}$ and $A_0^{(1)**}$-surface. These solutions can be expressed by solutions of linear difference equations. Here the $A_0^{(1)}$-surface discrete Painlevé equation is the most generic difference equation, as all discrete Painlevé equations can be obtained by its degeneration limit. These special solutions exist when the parameters of the discrete Painlevé equation satisfy a particular constraint. We consider that these special functions belong to the hypergeometric family although they seems to go beyond the known discrete and $q$-discrete hypergeometric functions. We also discuss the degeneration scheme of these solutions.

nlin.SI↗