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.