Analytical derivation of kink breather solutions in a traveling periodic wave background, highlighting their relationship to Riemann's theta functions.
We characterize a general traveling periodic wave of the defocusing mKdV (modified Korteweg--de Vries) equation by using a quotient of products of Jacobi's elliptic theta functions. Compared to the standing periodic wave of the defocusing NLS (nonlinear Schr\"{o}dinger) equation, these solutions are special cases of Riemann's theta function of genus two. Based on our characterization, we derive a new two-parameter solution form which defines a general three-parameter solution form with the scaling transformation. Eigenfunctions of the Lax system for the general traveling periodic wave are also characterized as quotients of products of Jacobi's theta functions. As the main outcome of our analytical computations, we derive a new solution of the defocusing mKdV equation which describes the kink breather propagating on a general traveling wave background.
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Arruda et al. (2025) studied this question.
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