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D. Noja, D. Pelinovsky, and G. Shaikhova

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Bifurcations and stability of standing waves
in the nonlinear Schrodinger equation
on the tadpole graph

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Nonlinearity 28 (2015) 2343-2378

**Abstract:**

We develop a detailed rigorous analysis of edge bifurcations of standing waves in
the nonlinear Schrodinger (NLS) equation on a tadpole graph
(a ring attached to a semi-infinite line subject to the Kirchhoff boundary conditions at the junction).
It is shown in the recent work by using explicit Jacobi elliptic functions that
the cubic NLS equation on a tadpole graph admits a rich structure of standing waves. Among these,
there are different branches of localized waves bifurcating from the edge of the essential spectrum
of an associated Schrodinger operator.

We show by using a modified Lyapunov-Schmidt reduction method that the bifurcation of localized standing waves occurs
for every positive power nonlinearity. We distinguish a primary branch of never vanishing standing waves
bifurcating from the trivial solution and an infinite sequence of higher branches with
oscillating behavior in the ring. The higher branches bifurcate from
the branches of degenerate standing waves with vanishing tail outside the ring.

Moreover, we analyze stability of bifurcating standing waves. Namely, we show that the primary branch
is composed by orbitally stable standing waves for subcritical power nonlinearities, while all
nontrivial higher branches are linearly unstable near the bifurcation point. The stability character of
the degenerate branches remains inconclusive at the analytical level,
whereas heuristic arguments based on analysis of embedded eigenvalues of negative Krein signatures
support the conjecture of their linear instability at least near the bifurcation point.
Numerical results for the cubic NLS equation show that this conjecture is valid and that the degenerate
branches become spectrally stable far away from the bifurcation point.

**Keywords**:

Nonlinear Schrodinger equation; quantum graphs, standing wave solutions,
existence and stability, edge bifurcations.