Asymptotic analysis of multi-lumps solutions in the Kadomtsev-Petviashvili-(I) equation

Inspired by the works of Y. Ohta and J. Yang, one constructs the lumps solutions in the Kadomtsev-Petviashvili-(I) equation using the Grammian determinants. It is shown that the locations of peaks will depend on the real roots of Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. Also, one can prove that all the locations of peaks are on a vertical line when time approaches - $\infty$, and then they will be on a horizontal line when time approaches $\infty$, i.e., there is a rotation $\frac{\pi}{2}$ after interaction.