Localized nonlinear excitation on the doubly periodic backgrounds governed byWeierstrass type elliptic waves of the fifth-order KdV equation

In this work, the linear spectral problems, that is, the Lax pairs of the fifth-order Korteweg-de Vries (KdV) model with traveling and static Weierstrass functions as coefficients, are investigated using the Lamé equation, and the nonlinear localized waves on the Weierstrass elliptic periodic function backgrounds are derived using Darboux transformations, whose propagation and corresponding nonlinear dynamics are also discussed. Furthermore, two types of degenerate nonlinear waves are studied in the limiting cases of the half-periods \omega_1 and \omega_2 of the Weierstrass-\wp(x) function. The findings are beneficial for understanding and forecasting some nonlinear phenomena in related physical fields, including oceanography and hydrodynamics.