| Abstract: |
| For the Gerdjikov-Ivanov (GI) equation, we rigorously prove the equivalence between the Wronskian- and Grammian-type solutions derived from the elementary and binary Darboux transformations, respectively. The proof is finished by making complete Wronskian expansions and establishing the relations between the corresponding numerators and denominators of two determinant solutions. Meanwhile, some determinant identities are obtained as a byproduct upon comparing the coefficients of the same terms in the expansions. Furthermore, we conduct asymptotic analysis for N-soliton solutions on the zero and plane-wave backgrounds. Explicit asymptotic expressions are obtained as t goes to infinity, yielding the physical information of interacting solitons, such as amplitudes, velocities, and phase shifts before and after collisions. In particular, we derive the general parametric conditions for synchronous N-soliton collisions at arbitrary space-time points on both backgrounds. This scenario may be useful for understanding complex behavior for a large number of solitons, e.g., the generation of rogue waves via soliton collisions. |
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