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

Solitons of the form $x_n=x_{0}tanh(\omega t-kna)$ can propagate in a chain of harmonically coupled particles in the discrete case if the potential $-\frac{1}{2}A{x}_n^2+\frac{1}{4}B{x}_n^4$ 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 $\omega$ and $k$. For large $\omega$ the maximum at $x_n=0\mathrm{}$ becomes a minimum, giving a triple-minimum potential. Potential shapes and particle positions are illustrated for various ($\omega$,$k$) 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 $E=\frac{m_0{c}_s^2}{{(1-\frac{v^2}{c_s^2})}^{\frac{1}{2}}}$, 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.