The transitions and mechanisms of nonlinear waves in the (2+1)-dimensional Sawada–Kotera (2DSK) equation are studied by means of characteristic line and phase shift analysis, and the dynamic behavior of various nonlinear transformed waves is analyzed. Firstly, we obtain the N-soliton solution based on the Hirota bilinear method, from which the breath wave solution is constructed by changing the parameters into a complex form in pairs, and lump solution is obtained via the long wave limit method. Then the mechanism of the breath wave solution transformation is studied by characteristic line analysis, we present the types of transformed nonlinear waves, including quasi-anti-dark soliton, M-shaped soliton, W-shaped soliton, multi-peak soliton, and quasi-periodic wave soliton, and the distribution diagrams of these nonlinear waves on the $(\alpha ,\beta )$ plane is rendered. We further reveal the gradient properties of the transformed wave. In addition, the transformed wave is decomposed into a solitary wave and a periodic wave component, and the formation mechanism, locality, and oscillation properties of the nonlinear transformed wave are explained through the nonlinear superposition. Furthermore, we demonstrate that the geometric properties of the characteristic lines vary with time essentially resulting in the time-varying properties of nonlinear waves, which have never been found in (1+1)-dimensional systems. Based on high-order nonlinear waves, the state transitions of the mixed solution and the second-order breath wave solution are investigated. We show several collision models of nonlinear waves, and reveal that the phase shift difference between the solitary and the periodic wave component leads to the deformable collision of the transformed wave. Such phase shift is due to time evolution and wave interaction. Finally, the dynamic process of nonlinear wave collision under the combined action of time and collision is presented.

通过特征线和相移分析，研究了(2+1)维Sawada-Kotera(2DSK)方程中非线性波的转变和机制，分析了各种非线性变换波的动力学行为。首先，我们基于Hirota双线性方法得到N-孤子解，从中通过将参数成对改变为复数形式构造呼吸波解，并通过长波极限方法获得集总解。然后通过特征线分析研究了呼吸波解变换的机理，给出了变换后的非线性波的类型，包括准反暗孤子、M型孤子、W型孤子、多峰孤子和拟-周期波孤子，以及这些非线性波在$(\alpha ,\beta )$飞机被渲染。我们进一步揭示了变换波的梯度特性。此外，将变换波分解为孤立波和周期波分量，通过非线性叠加解释非线性变换波的形成机理、局部性和振荡特性。此外，我们证明了特征线的几何特性随时间变化，本质上导致非线性波的时变特性，这在 (1+1) 维系统中从未发现过。基于高阶非线性波，研究了混合解和二阶呼吸波解的状态转移。我们展示了几种非线性波的碰撞模型，并揭示了孤立波和周期波分量之间的相移差异导致变换波的可变形碰撞。这种相移是由于时间演化和波相互作用。最后给出了时间和碰撞共同作用下非线性波浪碰撞的动力学过程。