In this paper, in order to find more solutions to a nonvariational quasilinear PDE, a new augmented singular transform (AST) is developed to form a barrier surrounding previously found solutions so that an algorithm search from outside cannot pass the barrier and penetrate into the inside to reach a previously found solution. Thus a solution found by the algorithm must be new. Mathematical justifications of AST formulation are established. A partial Newton-correction method is designed accordingly to solve the augmented problem and to satisfy a constraint in AST. The new method is applied to numerically investigate bifurcation, symmetry-breaking phenomena to a non-variational quasilinear elliptic equation through finding multiple solutions. Such phenomena are numerically captured and visualized for the first time, and still open for theoretical verification. Since the formulation is general and simple, it opens a door to solve other multiple solution problems.

在本文中，为了找到更多关于非变分拟线性PDE的解决方案，开发了一种新的增强奇异变换（AST）来形成围绕先前找到的解决方案的障碍，以便从外部进行搜索的算法无法通过障碍并渗透到内部以获得先前找到的解决方案。因此，该算法找到的解决方案必须是新的。建立了AST公式的数学依据。相应地设计了局部牛顿校正方法以解决增广问题并满足AST中的约束。通过寻找多个解，该新方法被用于数值研究非变分拟线性椭圆方程的分叉，对称破坏现象。首次以数字方式捕获并可视化了此类现象，并仍在进行理论验证。由于该公式通用且简单，因此为解决其他多种解决方案问题打开了一扇门。