Degenerate lump chain solutions and rouge wave solutions of the (4+1)-dimensional nonlinear evolution equation

In this article, we take the (4+ 1)-dimen sional nonlinear evolution equation as an example and construct the degenerate solutions of the lump chains and the rogue wave solutions, respectively. In the process of degradation, we can observe the coupling phenomenon between lump chains. In the event of complete solution degradation, the period tends to infinity, resulting in the centralization of multi-lump solutions. Then, we apply the long-wave limit approach and perturb phase to derive the rogue wave solutions from the N-soliton solution. Also, we obtain images of the rogue wave solutions. We find that higher-order rogue wave solutions generatemultiple stable structures. And in the limit, the rogue wave solutions have a similar structure to the central region of the degenerate solutions, i.e., multi-lump structures. This finding links the three types of solutions and provides a simple way to find multilump solutions. In addition, the dynamical behaviors of these solutions help solve issues in the fluctuation theory, marine science, and other related domains.