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

[Maple Math]
[Maple Math]

where [Maple Math] , [Maple Math] - transformation parameter, [Maple Math] - solutions of the equation [Maple Math] .

Assuming that the diagram for the construction of the solutions

[Maple OLE 2.0 Object]

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

[Maple Math]

where [Maple Math] , [Maple Math] - are the parameters of transformation, [Maple Math] is the n-parametric solution.

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