Nonlinear Evolution Equations, Inverse Scattering and Cellular Automata
There is a class of nonlinear evolution equations which have solitons as special solutions. Significantly many of these equations fall into a class for which the Inverse Scattering Transform method applies. A discussion of this method of solution for certain prototype nonlinear evolution equations in continuous media in 1 + 1 and 2 + 1 dimensions is given. It also turns out that there are cellular automata which admit soliton solutions and have many similar properties, but there are significant differences as well. One of these cellular automata is discussed in some detail.
KeywordsSolitary Wave Cellular Automaton Soliton Solution Cellular Automaton Nonlinear Evolution Equation
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- Ablowitz, M.J. and Segue, H., Solitons and the Inverse Scattering Transform, SIAM Studies 4 (1981).Google Scholar
- Ablowitz, M.J. and Fokas, A.S., Comments on the Inverse Scattering Transform and Related Nonlinear Evolution Equations, monograph: Nonlinear Phenomena. ed. K.B. Wolf, Springer, (1983) 3–24.Google Scholar
- [2b]Fokas, A.S. and Ablowitz, M.J., monograph: Nonlinear Phenomena. ed. K.B. Wolf, Springer (1983), 137–183, Lectures on the Inverse Scattering Transform for Multidimensional (2+1) Problems.Google Scholar
- Park, J. Stieglitz, K. and Thurston, W., Soliton-like Behavior in Automata, Phys. D 19D (1986).Google Scholar
- Ablowitz, M.J. and Segue, H., Fluid Mech. 92, 691–715.Google Scholar
- Wolfram, S., Cellular Automaton Supercomputing, in Proceedings of the Scientific Applications and Algorithms Design for High Speed Computing Workshop, Urbana, IL, April (1986).Google Scholar
- Gardner, M., Sci. Am. 223 Oct. 120, Nov. 116 (1970); 224 Jan. 104, Feb. 112, Apr. 114 (1971).Google Scholar
- Papatbeodorou, T. and Fokas, A.S., Evolution Theory, Periodic Particles and Solitons,INS#85, to appear Stud. Appl. Math..Google Scholar
- Fokas, A.S. Papadopoulaou, E. Saridakis, Y. and Ablowitz, M.J., Interaction of Simple Particles in Soliton Cellular Automata,INS#97, to appear Stud. Appl. Math..Google Scholar
- Keiser, J., Masters Thesis, On the Computation of Periodic Particles for Soliton Cellular Automata, Clarkson University (May 1989).Google Scholar
- Ablowitz, M.J., Keiser, J. and Herbst, B.M., Nonlinear Evolution Equations, Solitons, Chaos and Cellular Automata, INS # 123, to appear proceedings conference on Nonlinear Physics held in Shanghai, China April 23–30, 1989.Google Scholar