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In this paper, we derive seed and 1-soliton solutions of the Q2 equation in the Adler–Bobenko–Suris list. The seed solutions of Q2 are obtained using those of \(\mbox{Q}1(\delta)\) and an non-auto Bäcklund transformation connecting them. Then using an auto Bäcklund transformation, two types of Q2 one-soliton solutions are obtained based on its seed solutions. These obtained solutions are new and cannot be derived as degenerations from any known soliton solutions.

On one-soliton solutions of the Q2 equation in the ABS list