Stability and the Geometry of Unstable Modes

This note extends the Chen–Moutinho framework on the asymptotic stability of multi-solitons by identifying the correct geometric form of the unstable mode projection. The central observation is that the projection cannot be defined through the standard Hilbert inner product, since the linearized NLS operator is not self-adjoint. Instead, the system’s canonical symplectic form defines the true dual pairing, establishing a coherent projection onto the unstable and stable subspaces. Under this symplectic geometry, each soliton contributes a two-dimensional hyperbolic plane in phase space: one expanding (ionization-like) and one contracting (relaxation-like) direction. The resulting picture unifies spectral instability in supercritical NLS with the canonical phase-space structure of quantum systems, where the unstable nodes correspond to electron-like escape directions from a bound state. This interpretation reframes asymptotic stability not as a purely analytic bound, but as a geometric classification of all perturbative trajectories in the Hamiltonian field. It converts the analytic cancellation of unstable modes into a geometric constraint: motion must remain symplectically orthogonal to every ionization vector.

Significance:
This contribution clarifies the geometric foundation of asymptotic stability. Prior work constructed projections that eliminated unstable growth but did not specify the underlying geometry of those projections. By reinterpreting the unstable mode decomposition through the symplectic form rather than the (L^2) inner product, this note establishes the correct Hamiltonian pairing that governs all stability mechanisms in the nonlinear Schrödinger hierarchy. The framework reveals that each soliton’s instability is not an analytical artifact but a geometric saddle in canonical phase space—an electron-like escape axis whose cancellation defines the codimension-m manifold of bounded multi-soliton states. This geometric reformulation bridges spectral theory, symplectic geometry, and quantum interpretation, and lays groundwork for generalizing stability analysis to nonlinear plasma and field systems with identical canonical structure.

Category:
Nonlinear Stability & Spectral Geometry

Hashtags:
#asymptoticstability #symplecticgeometry #nls #nonlinearpde #hamiltonianstructure #spectraltheory #unstablemodes #jtpmath