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

Periodic solutions of the Zakharov equation are investigated. By performing the limit operation λ_2l–1 → λ₁ 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 λ₁ → λ₀ on the breather-positon solution, because the unique eigenvalue λ₀ associated with the Lax pair eigenfunction Ψ(λ₀) = 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.