Construction and analysis of multi-lump solutions of dispersive long wave equations via integer partitions

Abstract

In this paper, the relation between the integer partition theory and a kind of rational solution of the dispersion long wave equations is studied. For the integer partition $\lambda =\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\right)$ of positive integer $N$, with the degree vector $m=\left(m_1,m_2,\dots ,m_n\right)$, the corresponding $M$ lump solution can be obtained where $M=N+n{m}_n$. Combined with the generalized Schur polynomial and heat polynomial, the asymptotic positions of peaks are studied, and the arrangement of multi-peak groups in multi-lump solutions are obtained, as well as the relationship between the patterns formed by single-peak groups and the corresponding integer partition.