Soliton molecules for combined mKdV-type bilinear equation

Starting from the multi-soliton solutions obtained by the Hirota bilinear method, the soliton molecule structures for the combined mKdV-type bilinear equation ( D t + ∑ n = 1 N α n D x 2 n + 1 ) f * · f = 0 are investigated using the velocity resonance mechanism. The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper. With particular selections of the involved arbitrary parameters, especially the wave numbers, it is confirmed that, besides the usual multi-bright soliton molecules, the multi-dark soliton molecules and the mixed multi-bright-dark soliton molecules can also be obtained. In addition, we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions. The results demonstrate that when N ≥ 4, the combined mKdV-type bilinear equation no longer admits soliton molecules comprising more than four solitons.