Spectral Truncation in a Class of Nonlinear Compactified Variational Systems

Abstract

We analyze a class of quasilinear variational functionals defined on the compactified manifold M₄ × S¹ characterized by saturating gradient nonlinearities. In this setting, the discrete winding index enters the nonlinear energy density through the invariant

  s = |∇ψ|² + (n² / R²) |ψ|².

We prove that this internal coupling induces a structural loss of virial balance under an L²-constraint. Using L²-preserving dilations, Pohozaev-type identities, and concentration–compactness arguments, we establish existence of constrained minimizers for admissible indices and prove that localized stationary states cease to exist beyond a computable critical index n₍crit₎. Strict monotonicity of the corresponding energy levels is also demonstrated. These results characterize a truncation phenomenon in a class of nonlinear compactified variational systems.