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The multi-soliton solution of the sine-Gordon equation in the presence ofelliptic-function background is derived by the inverse scattering method. Thekey tool in our formulation is the Lax pair written by $4\times4$ matrixdifferential operators given by Takhtadzhyan and Faddeev in 1974, which enablesus to use the conventional form of the integral representation of the Jostsolutions and Krichever's theory of commuting differential operators. As aby-product we also provide generalized orthogonality and completeness relationsfor eigenfunctions associated with indefinite inner product. The multi-solitonsolution is expressed by a determinant of theta functions and the shift of thebackground lattice due to solitons is also determined using addition formula.One kink and one breather solutions are presented by animated gifs.

Multi-soliton solutions of the sine-Gordon equation with elliptic-function background