Discrete Modeling of a String and Analysis of a Loop Soliton

A discrete model for an extensible string is proposed and analyzed by a discrete soliton theory and computer simulations. The relation between tension of the string and the size of a loop propagating on the string is obtained analytically by using the soliton theory. We use this relation to investigate dynamics and stability of loops, and it is found that one loop is stable against various kinds of perturbation. It is confirmed numerically that the loop can be formed by moving a boundary along a semicircle. If the moment of the string is introduced, behaviors of the formation are drastically changed and there is a critical value of stiffness of the string beyond which the loop cannot be formed. As for collision of two loops, we, found that two loops do not break after collision if the two are similar. This result of collision can be well explained by our former analysis of a continuous string theory (Nishinari, 1997).