Nonlinear spectral problems arise across a range of fields, compreso
mechanical vibrations, fluid-solid interactions, and photonic crystals.
Discretizing infinite-dimensional nonlinear spectral problems often introduces
significant computational challenges, particularly spectral pollution and
invisibility, which can distort or obscure the true underlying spectrum. Noi
present the first general, convergent computational method for computing the
spectra and pseudospectra of nonlinear spectral problems. Our approach uses new
results on nonlinear injection moduli and requires only minimal continuity
assumptions: specifically, continuity with respect to the gap metric on
operator graphs, making it applicable to a broad class of problems. We use the
Solvability Complexity Index (SCI) hierarchy, which has recently been used to
resolve the classical linear problem, to systematically classify the
computational complexity of nonlinear spectral problems. Our results establish
the optimality of the method and reveal that Hermiticity does not necessarily
simplify the computational complexity of these nonlinear problems.
Comprehensive examples — including nonlinear shifts, Klein–Gordon equations,
wave equations with acoustic boundary conditions, time-fractional beam
equations, and biologically inspired delay differential equations —
demonstrate the robustness, accuracy, and broad applicability of our
methodology.

Questo articolo esplora i giri e le loro implicazioni.

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