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.