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Ba¨cklund transformations Transformations between solutions of differential equations, in particular soliton equations. They can be used to construct nontrivial solutions from the trivial solution.

Formally: Two evolution equations ut = K(x, u, u1, ..., um) and vt = G(y, v, v1, ..., vn) are said to be equivalent under a Ba¨cklund transformation if there exists a transformation of the form y = ψ(x, u, u1, ..., un), v = φ(x, u, u1, ..., un).