This is just Newton's second law for the j -th particle. The first factor k ( x j + 1 + x j − − 1 − − 2 x j ) is just the usual Hooke's law form for the force. The factor with α α is the nonlinear force. We can rewrite this in terms of continuum quantities by defining c = κ κ / ρ ρ to be the wave speed, where κ κ = k / h is the Young's modulus for the string, and ρ ρ = m / h 3 is the density: