Ba¨cklund transformations (BT) refer to transformations between the solutions of either the same or two different differential equations. Their origin may be traced back to the 1875 studies of A. V. Ba¨cklund on pseudospherical surfaces [134], i.e., surfaces of constant negative curvature. On such surfaces the line element in terms of suitable coordinates u and v may be expressed as

ds2 = α2(du2 + 2 cosωdudv + dv2), (8.1.1)

where −1/α2 is the constant total curvature of the surface and ω is the angle between the asymptotic lines. It may be shown that the angle ω satisfies [135]


∂u∂v = sinω, (8.1.2)

which is known as the Sine-Gordon equation. A solution of this equation corresponds to a surface of constant negative curvature. In trying to generate such surfaces Ba¨cklund discovered that a new solution ω1 could be obtained from a given solution ω0 by means of the following transformation:

∂u ( ω1 − ω0

2 ) = a sin(

ω1 + ω0 2

), (8.1.3)

∂v ( ω1 + ω0

2 ) = a−1 sin(

ω1 − ω0 2

), (8.1.4)

where a is an arbitrary constant. It is obvious that for such a transformation to be of any practical use one must be able to find the solution ω0. However, the efficacy of such transformations rests on the fact that in many cases, the initial solution ω0 can often be obtained by inspection. In the case of (8.1.2), for example, it is seen that ω0 = 0 is a solution and can therefore be used for generating new solutions.