Numerical calculation and characteristics of N-periodic waves of a (4+1)-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff equation in fluid physics and plasma physics

In this paper, an effective method for deriving the N-periodic wave solutions of the (4+1)-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff equation from fluid physics and plasma physics is presented by means of the bilinear Bäcklund transformation and the Riemann-theta function. The problem of constructing of the N-periodic wave solutions can be transformed into a system of algebraic equations. The difficulty of solving these algebraic equations is essentially a nonlinear least square problem, which can be solved by the Gauss–Newton numerical algorithm. It is proved that the N-periodic waves converge to the corresponding N-solitons under the limit of small amplitude. Furthermore, an analytical method related to the characteristic lines for the N-periodic waves is employed to analyze quantitatively the dynamical characteristics. The numerical simulations of all the quasiperiodic wave solutions are consistent with the theoretical results. The method presented can be further extended to other high-dimensional integrable systems to explore complex wave phenomena.