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New Solutions to the Ultradiscrete Soliton Equations
Author(s) -
Hirota Ryogo
Publication year - 2009
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2009.00438.x
Subject(s) - korteweg–de vries equation , soliton , toda lattice , traveling wave , collision , mathematics , ball (mathematics) , mathematical physics , mathematical analysis , physics , quantum mechanics , nonlinear system , integrable system , computer science , computer security
New solutions to the ultradiscrete soliton equations, such as the Box–Ball system, the Toda equation, etc. are obtained. One of the new solutions which we call a “negative‐soliton” satisfies the ultradiscrete KdV equation (Box–Ball system) but there is not a corresponding traveling wave solution for the discrete KdV equation. The other one which we call a “static‐soliton” satisfies the ultradiscrete Toda equation but there is not a corresponding traveling wave solution for the discrete Toda equation. A collision of a soliton with a negative‐soliton generates many balls in a box over the capacity of the box in the Box–Ball system, while a collision of a soliton with the static‐soliton describes, in the ultradiscrete limit, transmission of a soliton through junctions of a “nonuniform Toda equation.” We have obtained exact solutions describing these phenomena.