Backlund Transformations and Nonlinear Superposition Principle

In the last century a Sweden mathematician Backlund, considering the geometry of surfaces with constant negative curvature, showed a way to obtain the hierarchy of sine-Gordon solutions when a new solution can be build on the bases of known solutions. The transformation, as applied to the sine-Gordon equation, has the form

where , - transformation parameter, - solutions of the equation .

Assuming that the diagram for the construction of the solutions

is commutative, one can eliminate the partial derivatives and find the analytical expression of the Backlund transformations

where , - are the parameters of transformation, is the n-parametric solution.

This formula gives a way to build the multi-soliton solutions.