Multiple lump solutions and dynamics of the generalized (3+1)-dimensional KP equation

Abstract

In this paper, the bilinear method is employed to investigate the multiple lump solutions of the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional KP equation. With the aid of the variable transformation, this equation is reduced to a dimensionally reduced equation. The reduced equation can be transformed into a bilinear equation. On the basis of the bilinear forms and a special quadratic function, the 1-lump solution is constructed, which has a positive peak and two negative peaks. Three polynomial functions are introduced to derive the 3-lump solutions. The 3-lump wave has a “triangular” structure. As the parameters tend to zero, the 3-lump wave becomes the lump wave with two adjacent cusps. The 6-lump solutions are constructed. Four kinds of lump waves appear as the parameters increase. In addition, the high-order 8-lump solutions are also obtained. All the peaks of the 6-lump and the 8-lump wave tend to the same height when the parameters are sufficiently large. The dynamical behaviors of the multiple lump solutions are discussed. All the results are useful in explaining the dynamical phenomena of the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional KP equation.