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 differential equations (PDEs) – the Korteweg-de Vries, sine-Gordon, cubic nonlinear Schrödinger, 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 define 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 finite-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 Schrödinger 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.