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.
Kato,Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, Second ed., 1980
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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.