Analytical three-periodic solutions of Korteweg–de Vries-type equations

Based on the direct method of calculating the periodic wave solution proposed by Nakamura, we give an approximate analytical three-periodic solutions of Korteweg–de Vries (KdV)-type equations by perturbation method for the first time. Limit methods have been used to establish the asymptotic relationships between the three-periodic solution separately and another three solutions, the soliton solution, the one- and the two-periodic solutions. Furthermore, it is found that the asymptotic three-soliton solution presents the same repulsive phenomenon as the asymptotic three-soliton solution during the interaction.