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

Periodic solutions of the Zakharov equation are investigated in this paper. Through performing the limit operation $\lambda_{2l-1}\to\lambda_1$ 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 through taking the limit $\lambda_1\to\lambda_0$ on the breather-positon solution, because the unique eigenvalue $\lambda_0$ associated with the Lax pair eigenfunction $\Psi(\lambda_0)=0$ 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.