Exact solutions of the elliptic sine equation in two space dimensions with application to the Josephson effect

Exact time-independent solutions of the elliptic sine equation ∂²ψ∂x²+∂²ψ∂y²=sinψ are derived with the help of a new B\"acklund transformation and the associated Bianchi diagrams. A formula is developed which enables us to generate without additional quadratures an infinite number of real two-space dimensional solutions α. We apply these solutions to the propagation of magnetic flux through a large two-dimensional Josephson tunneling junction and discuss briefly the experimental implications. We call the α solutions soliton-like solutions, since they can be shown to carry through the Josephson junction an integral number n of positive flux quanta: Φₙ⁽⁺⁾=nΦ₀, where Φ₀ denotes a single quantum of superconducting magnetic flux. The multiple soliton-like solutions possess infinite total energy and can be labeled by a topological quantum number. The analysis is strictly classical.