It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions; (ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS; and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions; (ii) it guarantees high-order accuracy; and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature.

翻译：非线性偏微分方程（PDE）常存在多个不同解，且每个解可能具有独特的物理意义。求解多解的典型方法是采用牛顿法结合不同初始猜测，这些猜测需理想地落入各自解的吸引域。本文提出一种快速精确的多解数值方法，包含三个核心组成部分：（i）对底层PDE进行精心设计的谱伽辽金离散，生成具有多解的非线性代数系统（NLAS）；（ii）采用有效的收缩技术从其他未知解中剔除已知（已发现）解，构建收缩型NLAS；（iii）采用可行的非线性最小二乘与信赖域方法（LSTR）依次求解NLAS及收缩型NLAS，逐一获取多个解。通过大量微分方程实例及与现有方法的对比，我们证明谱LSTR-收缩法具有以下优势：（i）初始值选择高度灵活，甚至可从相同初始猜测出发求解所有多解；（ii）保证高阶精度；（iii）快速定位多个不同解，并探索文献中尚未报道的新解。