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 ${x}_{n}={x}_{0}tanh(\ensuremath{\omega}t\ensuremath{-}kna)$ can propagate in a chain of harmonically coupled particles in the discrete case if the potential $\ensuremath{-}\frac{1}{2}A{x}_{n}^{2}+\frac{1}{4}B{x}_{n}^{4}$ 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 $\ensuremath{\omega}$ and $k$. For large $\ensuremath{\omega}$ the maximum at ${x}_{n}=0$ becomes a minimum, giving a triple-minimum potential. Potential shapes and particle positions are illustrated for various ($\ensuremath{\omega}$,$k$) 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 $E=\frac{{m}_{0}{c}_{s}^{2}}{{(1\ensuremath{-}\frac{{v}^{2}}{{c}_{s}^{2}})}^{\frac{1}{2}}}$, where ${m}_{0}$ is the soliton effective mass, $v$ is the soliton speed, and ${c}_{s}$ is the speed of sound in the mass-spring chain.