A New Approach for Numerically Solving Nonlinear Eigensolution Problems

By considering a constraint on the energy profile, a new implicit approach is developed to solve nonlinear eigensolution problems. A corresponding minimax method is modified to numerically find eigensolutions in the order of their eigenvalues to a class of semilinear elliptic eigensolution problems from nonlinear optics and other nonlinear dispersive/diffusion systems. It turns out that the constraint is equivalent to a constraint on the wave intensity in L-(p+1) norm. The new approach enables people to establish some interesting new properties, such as wave intensity preserving/control, bifurcation identification, etc., and to explore their applications. Numerical results are presented to illustrate the method.