改进的PNC方法及其在非线性偏微分方程多解问题中的应用

2017年，李昭祥等提出了一种偏牛顿-校正法（Partial Newton-Correction Method，简记为PNC方法），并利用它成功地计算出了三类非线性偏微分方程的多重不稳定解.本文在PNC方法的基础上，提出并发展了一种改进的PNC方法.首先，利用Nehari流形$\mathcal{N}$与零平凡解的可分离性，建立并证明了$\mathcal{N}$的某特殊子流形$\mathcal{M}$上的全局分离定理及其推广（即局部分离定理）.全局分离定理只跟非线性偏微分算子或相应的非线性泛函本身有关，而与具体的计算方法无关.对一些典型的非线性偏微分方程多解问题（比如，Henon方程问题），该全局分离定理的分离条件，经验证是成立的.另一个方面，通过修改或补充原辅助变换的定义，去掉了原辅助变换的奇异性；接着建立并证明了某些非线性偏微分方程问题的新未知解与该非线性偏微分算子零核空间的密切关系；在证明中，去掉了在原奇异变换下所需的标准收敛（standard convergence）假设.最后，计算实例与数值结果验证了改进的PNC方法的可行性和有效性；同时表明子流形$\mathcal{M}$与已知解的可分离性是PNC方法和本文新方法能成功找到多解的关键.

IMPROVED PARTIAL NEWTON-CORRECTION METHOD AND ITS APPLICATIONS IN FINDING MULTIPLE SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

In 2017,Li et al.developed a partial Newton-correction(PNC)method and applied it to successfully solve three classes of nonlinear PDEs with multiple unsteady solutions.Based on PNC method,we propose a new numerical method here.Firstly,by virtue of the separbility of Nehari manifold N and zero solution,we establish and prove a global separation theorem and its generalization(i.e,a local separation result)on somne sub-manifold M of N.The global separation result is only related to the nonlinear partial diferential operator or the corresponding nonlinear functional itself,but not to a specific numerical method.For some typical nonlinear PDEs with multiple solutions(e.g,Henon equation problem),the separa-tion conditions of the global separation theorem are proved to be valid.On the other hand,by revising or supplementing the definition of the original augmented singular transform(AST)1used in PNC method,we manage to remove the singularity in PNC method.Then,we show that some unknown solutions of certain nonlinear PDEs are closely related to their zero kernel space.In the proof,we remove the so-called standard convergence assumption.Finally,several numerical examples not only demonstrate the feasibility and effectiveness of the proposed method,but also illustrate that the key to the success of PNC method and of our new method is the separability of sub-manifold M and known solutions.