Solitons, nonlinear wave transitions and characteristics of quasi-periodic waves for a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation in fluid mechanics and plasma physics

A (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation describing many nonlinear phenomena in fluid dynamics and plasma physics is considered. The $N$-solitons and breathers are obtained by basing on its Hirota’s bilinear form and taking the complex conjugate condition on parameters of $N$-solitons. What is more, breathers can be transformed into a series of nonlinear localized waves by the mechanism of breather transformation. Then through the multi-dimensional Riemann-theta function and the bilinear method, the high-dimensional complex three-periodic wave solutions are constructed systematically, which are the generalization of one-periodic wave and two-periodic wave solutions. By a limiting procedure, the asymptotic relations between the quasi-periodic waves and solitons are strictly established. Additionally, a novel analytical method of characteristic line is introduced to analyze statistically the dynamical characteristics of the quasi-periodic waves. The analytical method employed in this paper can be further extended to investigate the other complex high-dimensional nonlinear integrable equations.