Existence of dnoidal-type standing waves on the loop coupled to soliton tails on half-lines is shown via the Implicit Function Theorem, with orbital (in)stability analyzed using perturbation and Krein-von Neumann theory.
Airy and Schr\"odinger-type equations on looping-edge graphs and applications
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abstract
The aim of this work is to study the Airy and Schr\"odinger operators on looping-edge graphs, a class of metric graphs consisting of a circle and a finite number $N$ of infinite half-lines attached to a common vertex. For the Airy operator, we characterize all extensions generating unitary and contractive dynamics in terms of self-orthogonal subspaces and linear operators acting on indefinite inner product spaces (Krein spaces) associated to the boundary values at the vertex. Employing similar abstract techniques, we then describe a systematic way to produce self-adjoint extensions of the Schr\"odinger operator that are compatible with prescribed boundary relations on looping-edge and $\mathcal{T}$-shaped graphs.
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math.AP 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Existence and (in)stability of standing waves for the nonlinear Schr\"odinger Equations on looping-edge graphs with $\delta'$-type interactions
Existence of dnoidal-type standing waves on the loop coupled to soliton tails on half-lines is shown via the Implicit Function Theorem, with orbital (in)stability analyzed using perturbation and Krein-von Neumann theory.