The Peierls–Nabarro and Benjamin–Ono Equations

An intimate connection between the Peierls–Nabarro equation in crystal-dislocation theory and the travelling-wave form of the Benjamin–Ono equation in hydro-dynamics is uncovered. It is used to prove the essential uniqueness of Peierlsʹ solution of the Peierls–Nabarro equation and to give, in closed form, all solutions of the analogous periodic problem. The latter problem is shown to be an example of global bifurcation with no secondary, symmetry-breaking, bifurcations for a nonlinear Neumann boundary-value problem or, equivalently, for an equation involving the conjugate operator, which is the Hilbert transform of functions on the unit circle.