When can solitons compute?

We explore the possibility of using soliton interactions in a one-dimensional
bulk medium as a basis for a new kind of computer. Such a structure
is gateless” { all computations are determined by an input stream
of solitons. Intuitively, the key requirement for accomplishing this is that
soliton collisions be nonoblivious; that is, solitons should transfer state
information during collisions. All the well known systems described by
integrable partial di erential equations (PDEs) { the Korteweg-de Vries,
sine-Gordon, cubic nonlinear Schrodinger, and perhaps all integrable systems
{ are oblivious when displacement or phase is used as state. We
present a cellular automaton (CA) model, the oblivious soliton machine
(OSM), which captures the interaction of solitons in systems described by
such integrable PDEs. We then prove that OSMs with either quiescent
or periodic backgrounds can do only computation that requires time at
most cubic in the input size, and thus are far from being computationuniversal.
Next, we de ne a more general class of CA, soliton machines
(SMs), which describe systems with more complex interactions. We show
that an SM with a quiescent background can have at least the computational
power of a nite-tape Turing machine, whereas an SM with a
periodic background can be universal. The search for useful nonintegrable
(and nonoblivious) systems is challenging: We must rely on numerical
solution, collisions may be at best only near-elastic, and collision elasticity
and nonobliviousness may be antagonistic qualities. As a step in this
direction, we show that the logarithmically nonlinear Schrodinger equation
(log-NLS) supports quasi-solitons (gaussons) whose collisions are, in
fact, very near-elastic and strongly nonoblivious. It is an open question
whether there is a physical system that realizes a computation-universal
soliton machine.