Ralph WILLOX

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Title
 Professor
Field
 Applied Mathematics
Research interests
 Integrable systems, discrete (nonlinear) dynamical systems, cellular automata.
Current research

A major part of my research activity focuses on the study of nonlinear integrable systems and their algebraic properties, and especially on the construction of particular classes of solutions for integrable systems and on the algebraic methods required for this purpose. I am particularly interested in so-called discrete integrable systems (discrete in space and time) as these systems often encompass a multitude of integrable systems in certain limits. A special limiting procedure that can be performed on such discrete systems, the ultradiscrete limit, allows one to obtain cellular automata that exhibit soliton-like behaviour. As it turns out, these solitonic cellular automata are related to (quantum) solvable lattice models, at their zero temperature limit. The role so-called Yang-Baxter maps play in this connection is a particularly important research topic.

I also study the possible applications of discretization and ultradiscretization techniques, originally developed in the context of integrable systems, to non-integrable dynamical systems. The aim is to obtain discretizations that exhibit the same dynamics as the original continuous systems, over a vast range of the discretization parameters. In many cases, going to the ultradiscrete limit allows one to obtain cellular automata that preserve the essential dynamics of the original continuous systems.
Selected publications
  1. "Generalised QRT mappings with periodic coefficients''， R. Ramani，B. Grammaticos and R. Willox，Nonlinearity 24 (2011) 113-126.
  2. "Solving the ultradiscrete KdV equation''，R. Willox， Y. Nakata，J. Satsuma， R. Ramani and B. Grammaticos， J. Phys. A: Math. Theor. 43 (2010) FT 482003.
  3. "Yang-Baxter maps from the discrete BKP equation'''，S. Kakei， J.J.C. Nimmo and R. Willox， SIGMA 6 (2010) 028, 11 pages.
  4. "A discrete-time model for cryptic oscillations in predator-prey systems'''，R. Willox， A. Ramani and B. Grammaticos， Physica D 238 (2009), 2238--2245.
  5. "On two (not so) new integrable partial difference equations'''，A. Ramani，B. Grammaticos， J. Satsuma and R. Willox， J. Phys. A: Math. Theor. 42 (2009) FT 282002 (6pp).
Memberships and activities

Mathematical Society of Japan

Japan Society of Industrial and Applied Mathematics (JSIAM)

Member of the General Assembly of the "Instituts Internationaux de Chimie et Physique, fondés par E. Solvay"

Journal of Physics A : Mathematical and Theoretical, Advisory Board member