| Abstract: |
| A comprehensive and systematic method is introduced for deriving oscillatory $N$-breather solutions for the Camassa-Holm equation, which are a new class of solutions on the oscillatory backgrounds. The process of this method is divided into four distinct but interrelated stages: First, resorting to the B\acklund transformations in Hirota`s bilinear equations, a novel technique is devised to solve the spectral problems of a negative-order KdV equation involving theta-function potentials. Second, using these B\acklund transformations, an $N$-fold Darboux transformation for the Camassa-Holm equation is rigorously formulated. Third, reciprocal and Darboux transformations are applied to construct oscillatory $N$-breather solutions for the Camassa-Holm equation from the spectral function of a negative-order KdV equation. Finally, the reality, boundedness, and smoothness of these novel solutions are rigorously established by expanding the Wronskians into Hirota summations. |
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