This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of predefining candidate basis pools, our novel method adaptively computes bases, considering the equation's nature and structural characteristics of the solution. It further leverages companion matrix techniques to generate initial guesses for subsequent computations. Thus this approach not only yields numerous initial guesses for solving such equations but also adapts orthogonal basis functions to effectively address discretized nonlinear systems. Through a series of numerical experiments, this paper demonstrates the method's effectiveness and robustness. By reducing computational costs in various applications, this novel approach opens new avenues for uncovering multiple solutions to differential equations with polynomial nonlinearities.

翻译：本文提出了一种创新方法——自适应正交基方法，专门用于计算具有多项式非线性的微分方程的多解。与预定义候选基池的传统做法不同，我们的新方法根据方程的性质和解的结构特征自适应地计算基函数。该方法进一步利用伴随矩阵技术为后续计算生成初始猜测。因此，这种方法不仅为求解此类方程提供了大量初始猜测，还能自适应地优化正交基函数以有效处理离散化后的非线性系统。通过一系列数值实验，本文证明了该方法的有效性和鲁棒性。通过降低各种应用场景中的计算成本，这种新方法为揭示多项式非线性微分方程的多解开辟了新的途径。