TY - JOUR

T1 - Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

AU - Kirr, E.

AU - Kevrekidis, P. G.

AU - Pelinovsky, D. E.

PY - 2011/12

Y1 - 2011/12

N2 - We consider the focusing (attractive) nonlinear Schrödinger (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 a subcritical or supercritical pitchfork. In the particular case of double-well 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.

AB - We consider the focusing (attractive) nonlinear Schrödinger (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 a subcritical or supercritical pitchfork. In the particular case of double-well 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.

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U2 - 10.1007/s00220-011-1361-3

DO - 10.1007/s00220-011-1361-3

M3 - Article

AN - SCOPUS:80955151651

SN - 0010-3616

VL - 308

SP - 795

EP - 844

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -