Bäcklund transform

Summary

A Bäcklund transform is a system of first-order partial differential equations that links two functions $u$ and $v$. The system can be solved for $v$ exactly when $u$ solves one partial differential equation. Then $v$ solves a second equation. When the two equations are the same, it is an auto-Bäcklund transform, and it turns known solutions into new ones. It is one of the main tools of the theory of integrable equations and solitons.