General high-order localized waves to the Bogoyavlenskii–Kadomtsev–Petviashvili equation

In this work, the Bogoyavlenskii–Kadomtsev–Petviashvili equation is investigated. By means of the Hirota bilinear system and Pfaffian, we demonstrate that the Bogoyavlenskii–Kadomtsev–Petviashvili equation has the Grammian determinant solution. On the basis of the Grammian determinant solution, we derive a class of exponentially localized wave solutions. It is shown that the exponentially localized wave solutions describe the fission and fusion phenomena between kink-type soliton and breather-type soliton. Further, general high-order localized waves consisting of kink-type, lump-type and breather-type solitons are derived by means of the general differential operators. These general high-order localized waves contain more plentiful dynamical behaviors. It is shown that the mixture of multiple wave solutions exhibit the fission and fusion phenomena among kink-type, lump-type and breather-type solitons. In addition, by choosing appropriate parameters, we construct general high-order lump-type soliton solutions. The results show that high-order lump-type solitons propagate with the variable speed on the curves. After collision, the lump-type solitons do not pass through each other, but rather are reflected back. The propagation paths of the lump-type solitons are changed completely.