Numerical methods for computing nonlinear eigenpairs: Part II. Non-iso-homogeneous cases

Standing (solitary)-wave/steady-state solutions in many nonlinear wave motions and Schrodinger flows lead to nonlinear eigenproblems. In [X. Yao and J. Zhou, SIAM J. Sci. Comput., 29 (2007), pp. 1355-1374], a Rayleigh-local minimax method is developed to solve iso-homogeneous eigenproblems. In this subsequent paper, a unified method in Banach spaces is developed for solving non-iso-homogeneous and even nonhomogeneous eigenproblems and applied to solve two models: the Gross-Pitaevskii problem in the Bose-Einstein condensate and the p-Laplacian problem in non-Newtonian flows/materials. First a new active Lagrange functional is formulated to establish a local minimax characterization. A local minimax method is then devised and implemented to solve the model problems. Numerical results are presented. Convergence results of the algorithm and an order of eigensolutions computed by the algorithm are also established.