Analytical three-periodic solution and interaction for nonlocal Boussinesq equation

Based on a multidimensional Riemann ϑ function and the coupled Hirota bilinear system, we use the Perturbation method to construct the asymptotic analytical three-periodic solution of the nonlocal Boussinesq equation. The analytical three-periodic solution is a direct generalization of the one- and two-periodic solutions and it has 6 fundamental periods. To prove that this analytical solution is correct, we verify it with numerical results obtained by the Gauss-Newton method. In addition, the asymptotic relationship between the periodic solution and soliton solution is established by a small-amplitude limit method. Interestingly, we find that when the three-periodic solution has a local maximum, its asymptotic three-soliton solution has a global maximum. During the nonlinear interaction, both the three-periodic solution and its asymptotic soliton can exhibit repulsive phenomenon, respectively.