The nth-order degenerate breather solution for the Kundu-Eckhaus equation

Abstract In this paper, we formulate the compact determinant representation of the formula of n th-order breather solution for the Kundu–Eckhaus (KE) equation. Then, we obtain the formula of the n th-order degenerate breather solution (breather-positon, b-positon for short) for the KE equation by using the Taylor expansion with respect to degenerate eigenvalues λ 2 k − 1 → λ 1 ( k = 1 , 2 , … , n + 1 ) . B-positon, which is a special kind of breather solution, is recently recognized as a key role being responsible for generating rogue wave. According to the related formula, the exact expression of first-order b-positon is constructed. Furthermore, the dynamics of the first-, second- and third-order b-positons of the KE equation are discussed in detail, and the approximate trajectories and space-dependent ‘phase shift’ of the collision of b-positons are depicted by explicit expressions, respectively, which may be used to predict where rogue wave occurs.

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