A spectral Levenberg–Marquardt-Deflation method for multiple solutions of semilinear elliptic systems

Numerous practical problems give rise to nonlinear differential equations that may exhibit multiple nontrivial solutions relevant to applications. Efficiently computing these solutions is crucial for a profound understanding of these problems and enhancing various applications. Therefore, the development of a numerical method capable of finding multiple solutions efficiently is imperative. Additionally, the provision of an efficient iteration process is vital for promptly obtaining multiple solutions. In the current paper, we introduce a novel algorithm for identifying multiple solutions of semilinear elliptic systems, where the trust region Levenberg–Marquardt method, combined with the deflation technique, is designed to compute multiple solutions for the first time. Based on several numerical experiments, our algorithm demonstrates efficacy in efficiently identifying multiple solutions, even when the nonlinear term appearing in these equations involves solely the first derivative. Moreover, we validate the efficiency of our algorithm and unveil previously undiscovered solutions in the existing literature