TY - JOUR
T1 - Newton's method and Morse index for semilinear elliptic PDES
AU - Neuberger, John M.
AU - Swift, James W.
N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2001/3
Y1 - 2001/3
N2 - In this paper we primarily consider the family of elliptic PDEs Δu + f(u) = 0 on the square region Ω = (0, 1) × (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an "action" functional, we consider Newton's method on the gradient of that functional. We use a Galerkin expansion, in eigenfunctions of the Laplacian, to find solutions of arbitrary Morse index. Taking f′(0) to be a bifurcation parameter, we analyze the bifurcations from the trivial solution, u ≡ 0, using symmetry arguments and our numerical algorithm. The Morse index of the approximated solutions is provided and support is found concerning several existence and nodal structure conjectures. We discuss the applicability of this method to find critical points of functional in general.
AB - In this paper we primarily consider the family of elliptic PDEs Δu + f(u) = 0 on the square region Ω = (0, 1) × (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an "action" functional, we consider Newton's method on the gradient of that functional. We use a Galerkin expansion, in eigenfunctions of the Laplacian, to find solutions of arbitrary Morse index. Taking f′(0) to be a bifurcation parameter, we analyze the bifurcations from the trivial solution, u ≡ 0, using symmetry arguments and our numerical algorithm. The Morse index of the approximated solutions is provided and support is found concerning several existence and nodal structure conjectures. We discuss the applicability of this method to find critical points of functional in general.
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U2 - 10.1142/S0218127401002444
DO - 10.1142/S0218127401002444
M3 - Article
AN - SCOPUS:0035606761
VL - 11
SP - 801
EP - 820
JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
SN - 0218-1274
IS - 3
ER -