Simple proof of stationary phase method and application to oscillatory bifurcation problems. (January 2020) - Kato, Keiichi; Shibata, Tetsutaro
Abstract: We consider the nonlinear eigenvalue problem − u ′ ′ ( t ) = λ f ( u ( t ) ), u ( t ) > 0, t ∈ I ≔ ( − 1, 1 ), u ( ± 1 ) = 0, where f ( u ) = f 1 ( u ) = u 3 + sin ( u 3 ) ∕ u, f ( u ) = f 2 ( u ) = u + u p sin ( u q ) ( 0 ≤ p < 1, 1 < q ≤ p + 2 ) and λ > 0 is a bifurcation parameter. It is known that, in this case, λ is parameterized by the maximum norm α = ‖ u λ ‖ ∞ of the solution u λ associated with λ and is written as λ = λ ( α ) . We simplify the argument of the stationary phase method and show the asymptotic formulas for λ ( α ) for f 1 ( u ) and f 2 ( u ) as α → ∞ and α → 0 . In particular, the shape of bifurcation diagram of λ ( α ) for f 1 ( u ) seems to be new.