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E.W. Kirr, P.G. Kevrekidis, and D.E. Pelinovsky

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Symmetry-breaking bifurcation in the nonlinear
Schr¨odinger equation with symmetric potentials

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Commun. Math. Phys. 308, 795–844 (2011)

**Abstract:**

We consider the focusing (attractive) nonlinear Schrodinger (NLS) equation with an external,
symmetric potential which vanishes at infinity and supports a linear bound state. We prove that
the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the
potential has a non-degenerate local maxima at zero. Under a generic assumption we show that
the bifurcation is either subcritical or supercritical pitchfork. In the particular case of doublewell
potentials with large separation, the power of nonlinearity determines the subcritical or
supercritical character of the bifurcation. The results are obtained from a careful analysis of
the spectral properties of the ground states at both small and large values for the corresponding
eigenvalue parameter.

We employ a novel technique combining concentration–compactness and spectral properties
of linearized Schr¨odinger type operators to show that the symmetric ground states can either
be uniquely continued for the entire interval of the eigenvalue parameter or they undergo a
symmetry–breaking pitchfork bifurcation due to the second eigenvalue of the linearized operator
crossing zero. In addition we prove the appropriate scaling for the L^{q},
q >= 2 and H^{1} norms of any stationary states in the limit of large values
of the eigenvalue parameter. The scaling and our novel technique imply that all ground states
at large eigenvalues must be localized near a critical point of the potential and bifurcate
from the soliton of the focusing NLS equation without potential localized at the same point.

The theoretical results are illustrated numerically for a double-well potential obtained after
the splitting of a single-well potential. We compare the cases before and after the splitting,
and numerically investigate bifurcation and stability properties of the ground states which are
beyond the reach of our theoretical tools.

**Keywords:**

nonlinear Schrodinger equation, double-well potentials,
symmetric and asymmetric stationary states,
pitchfork bifurcations, stability, Lyapunov-Schmidt reductions