Construction and application of degenerate solutions of nonlinear wave equations

Abstract Nonlinear wave equations (NWEs) enjoy widespread applications in many fields, and their solutions are usually comparatively complicated to obtain. Nevertheless, degenerate solutions could reflect the inherent laws as special solutions with degenerate properties only under certain conditions. Consequently, this paper discusses the problems in constructing degenerate solutions to nonlinear wave equations, along with their applications. In this paper, the problem of constructing degenerate solutions to the NWE was numerically studied using the finite difference method. The KdV model was adopted, along with the establishment of appropriate initial and boundary conditions. Space and time in the equation can be discretized into a calculative difference equation by applying the finite difference method. Error analysis and verification of Von Neumann stability were performed to analyze the stability and propagation characteristics of the degenerate solution. The results of error analysis show that with the reduction of the time step from 0.01 s to 0.001 s and the space step from 0.1 m to 0.01 m, the root mean square error is 0.0005. Through in-depth research on the construction of degenerate solutions, this paper opens new vistas for theoretical development regarding the NWE. It provides theoretical support for numerical simulations and experimental design in related application fields.