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Linear stability of both sign-definite (positive) and sign-indefinitesolitary waves near pitchfork bifurcations is analyzed for the generalizednonlinear Schroedinger equations with arbitrary forms of nonlinearity andexternal potentials in arbitrary spatial dimensions. Bifurcations oflinear-stability eigenvalues associated with pitchfork bifurcations areanalytically calculated. It is shown that the smooth solution branch switchesstability at the bifurcation point. In addition, the two bifurcated solutionbranches and the smooth branch have the opposite (same) stability when theirpower slopes have the same (opposite) sign. One unusual feature on thestability of these pitchfork bifurcations is that the smooth and bifurcatedsolution branches can be both stable or both unstable, which contrasts suchbifurcations in finite-dimensional dynamical systems where the smooth andbifurcated branches generally have opposite stability. For the special case ofpositive solitary waves, stronger and more explicit stability results are alsoobtained. It is shown that for positive solitary waves, their linear stabilitynear a bifurcation point can be read off directly from their power diagram.Lastly, various numerical examples are presented, and the numerical resultsconfirm the analytical predictions both qualitatively and quantitatively.

Stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear Schroedinger equations