Multi-rogue waves and triangular numbers

Multi-rogue wave solutions of integrable equations have a very specific number of elementary components within their structures. These numbers are given by the “triangular numbers” for the n^th -order solution. This contrasts with the case of multi-soliton solutions, where the number of solitons is n. This fact reveals a significant difference between the higher-order rogue waves and the higher-order solitons. Each n^th step of generation of multi-rogue wave solutions adds n elementary rogue waves to the solution, in contrast to n-soliton solutions, where each step adds only one soliton to the existing n−1 solitons in the composition. We provide the mathematical analysis for the number of ‘elementary particles’ in the composite rogue wave structures.