N-breather solutions of the Camassa–Holm equation on oscillatory backgrounds

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 Bäcklund 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 Bäcklund 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.