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Energy dissipating flows for solving nonlinear eigenpair problems
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DOI：10.1016/j.jcp.2018.09.012.png)
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Abstract
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This work is concerned with computing nonlinear eigenpairs, which model solitary waves and various other physical phenomena. We aim at solving nonlinear eigenvalue problems of the general form T (u) = lambda Q (u). In our setting T is a variational derivative of a convex functional (such as the Laplacian operator with respect to the Dirichlet energy), Q is an arbitrary bounded nonlinear operator and lambda is an unknown (real) eigenvalue. We introduce a flow that numerically generates an eigenpair solution by its steady state. Analysis for the general case is performed, showing a monotone decrease in the convex functional throughout the flow. When T is the Laplacian operator, a complete discretized version is presented and anlalyzed. We implement our algorithm on Korteweg and de Vries (KdV) and nonlinear Schrodinger (NLS) equations in one and two dimensions. The proposed approach is very general and can be applied to a large variety of models. Moreover, it is highly robust to noise and to perturbations in the initial conditions, compared to classical Petiashvili-based methods. (C) 2018 Elsevier Inc. All rights reserved.
Keywords:
Eigenpair
Fixed point solutions
Solitons
Variational calculus
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