Superposition behaviour between lump solutions and different forms of N-solitons $$(N\rightarrow \infty )$$ for the fifth-order Korteweg–de Vries equation

Abstract

A lump-type solution of the $$(2+1)$$ -dimensional generalised fifth-order Korteweg–de Vries (KdV) equation is obtained from the two-soliton solution by applying the parametric limit method. Some theorems and corollaries about the superposition behaviour between lump solutions and different forms of N-soliton $$(N\rightarrow \infty )$$ solutions are constructed, and detailed proofs are given. Besides,we give a large number of examples and spatial evolution graphics to illustrate the effectiveness of the described theorems and corollaries. Some new nonlinear phenomena and superposition behaviour, such as rational-exponential type, rational-cosh-cos type, rational-sin type, rational-logarithmic type etc., are simulated and shown for the first time. Finally, we also illustrate the superposition between high-order lump-type solutions and N-soliton solutions.