From breather solutions to lump solutions: A construction method for the Zakharov equation

Abstract

Periodic solutions of the Zakharov equation are investigated. By performing the limit operation λ 2l–1 → λ 1 on the eigenvalues of the Lax pair obtained from the n-fold Darboux transformation, an order-n breather-positon solution is first obtained from a plane wave seed. It is then proven that an order-n lump solution can be further constructed by taking the limit λ 1 → λ 0 on the breather-positon solution, because the unique eigenvalue λ 0 associated with the Lax pair eigenfunction Ψ(λ 0) = 0 corresponds to the limit of the infinite-periodic solutions. A convenient procedure of generating higher-order lump solutions of the Zakharov equation is also investigated based on the idea of the degeneration of double eigenvalues in multi-breather solutions.