arXiv subjects

Thamizharasi Tamizhmani

Publications and source records attributed to Thamizharasi Tamizhmani.

3 recordsLinked to original sources

arXiv2026

Singularity interactions for lattice equations: introducing the taishi

We study the singularities of some selected integrable lattice equations and show they all admit three types of singularities: one of finite and two of infinite extent. In particular, we show that all the equations we study possess a recently discovered, strip-like, singularity which is known under the moniker of ''taishi''. We study in detail the interaction of these taishi with the remaining two types of singularities and show that the rich behaviour first obtained in the case of the Korteweg-deVries equation is also present for other equations. Moreover, we find that in all cases this behaviour can be encoded in terms of simple symbolic dynamics which are just the dynamics governing a Box & Ball cellular automaton.

nlin.SI↗

arXiv2026

On unconventional discretisations

We present various discretisation techniques that mark a departure from the classical algorithms of numerical analysis, in the aim of preserving the physical behaviour of the solutions to the differential systems we are discretising, even for discretisation steps that are not very small. These techniques are inspired by the principles laid down by Mickens and are close to those proposed by Hirota and Kahan. Detailed applications of these methods are presented for the logistic equation and for the Lotka-Volterra system. We discuss the importance of preserving positivity whenever the latter is an expected property of the solution and give a practical rule for achieving this. Possible problems arising from the discretisations we propose are also discussed. A section is devoted to discretisations of integrable systems aimed at preserving integrability. Finally we conclude this review with a discussion of the possible relevance of a discrete space-time.

math.NA↗

arXiv2022

On the singularity structure of a discrete modified-Korteweg-deVries equation

We study the singularities of a modified lattice Korteweg-deVries (KdV) equation and show that it admits three families of singularities, with analogous properties to those found in the lattice KdV equation. The first family consists of localised singularities which can occupy an arbitrarily large domain but which are, nevertheless, always confined. The second family consists of one or more lines extending all the way from the south-west to the north-east on the plane, involving a single finite value that depends on the parameter that appears in the equation. We argue that the infinite extent of this singularity is not incompatible with the confinement property or with the integrability of the equation. The third family consists of horizontal strips in which the product of values on vertically adjacent lattice sites is equal to 1. In the case of the lattice KdV equation this type of singularity was dubbed `taishi'. The taishi for the modified lattice KdV equation can interact with singularities of the other two families, giving rise to very rich {and quite intricate} singularity structures. Nonetheless, these interactions can be described in a compact way through the formulation of a symbolic representation of the dynamics that is similar to, but in a sense, simpler than that for the KdV case. We give an interpretation of this symbolic representation in terms of a Box&Ball system related to the ultradiscrete mKdV equation. These results show that taishi-type singularities are not limited to the lattice KdV equation, but might very well be a general feature of integrable lattice equations with deep connections to other facets of their integrability.

nlin.SI↗