State-transition mechanisms for (2+1)-dimensional Sawada-Kotera equation with the variable coefficients in plasma physics and fluid dynamics

In this paper, we investigate the linear and tortuous transformed nonlinear waves in the (2+1)-dimensional Sawada-Kotera equation with variable coefficients, which may provide valuable theoretical insights for experimental design in plasma physics and fluid dynamics. Through characteristic line analysis and state transition conditions, the general transformed waves under various inhomogeneous parameters are modulated. The results demonstrate that the selected inhomogeneous parameters have minimal impact on the isolation and oscillatory characteristics of the transformed waves. Furthermore, we present a series of S-type tortuous transformed waves with identical inhomogeneous coefficients corresponding to different wave number ratios in the ($x$-$t$) plane, highlighting that the wave number ratio significantly influences the locality and oscillatory nature. Additionally, a novel type of tortuous M-shaped transformed wave, characterized by an initial intersection followed by a parallel configuration, is also reported. Our findings indicate that the wave number ratio plays a crucial role in shaping the dynamic characteristics of tortuous transformed waves, including their shape, velocity, and other properties.