Exact solution in the discrete case for solitons propagating in a chain of harmonically coupled particles lying in double-minimum potential wells

Abstract

Solitons of the form xn=x0tanh(ωt−kna) can propagate in a chain of harmonically coupled particles in the discrete case if the potential −1/2Axn^2+1/4Bxn^4 giving such solitions in the continuum limit is suitably modified. This modified potential is expressible in closed form, and its shape is a function of ω and k. For large ω the maximum at xn=0 becomes a minimum, giving a triple-minimum potential. Potential shapes and particle positions are illustrated for various (ω,k) combinations. The total energy and its kinetic, potential, and spring energy constituents are also expressible in closed form. In the continuum limit the total energy has the form E=(m0cS^2)/(1−v^2/cS^2)^1/2, where m0 is the soliton effective mass, v is the soliton speed, and cS is the speed of sound in the mass-spring chain.