Nonlinear multiparameter eigenvalue problems: Linearised and nonlinearised solutions

摘要

The paper focuses on a nonlinear multiparameter eigenvalue problem called P. This problem involves n spectral parameters and also depends on n(2) numerical parameters, where n >= 2 is an integer. If these numerical parameters vanish, P degenerates into n linear (one-parameter) eigenvalue problems called P-i(0)(i = (1,n) over bar). In connection with P one can consider another n nonlinear one-parameter eigenvalue problems called P-i. The problems P-i have eigenvalues with as well as without linear counterparts. The paper suggests to consider P-i as 'nonunperturbed' instead of P-i(0). Using properties of eigenvalues of P-i, one manages to prove existence of eigentuples of P. Among the eigentuples found in this way, there are eigetuples with as well as without linear counterparts. Results of the paper are found with a nonclassical approach. Applications of the found results to nonlinear optics are shown. (C) 2019 Elsevier Inc. All rights reserved.