Conditions and Stability Analysis for Saddle-Node Bifurcations of Solitary Waves in Generalized Nonlinear Schrödinger Equations

Abstract

Saddle-node bifurcations of solitary waves in generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions are analyzed. First, general conditions for these bifurcations are derived. Second, it is shown analytically that the linear stability of these solitary waves does not switch at saddle-node bifurcations, which is in stark contrast with finite-dimensional dynamical systems where stability switching takes place. Third, it is shown that this absence of stability switching does not contradict the Vakhitov-Kolokolov stability criterion or the results in finite-dimensional dynamical systems. Fourth, it is shown that this absence of stability switching holds not only for real potentials but also for complex potentials. Lastly, various numerical examples will be given to confirm these analytical findings. In particular, saddle-node bifurcations with both branches of solitary waves being stable will be presented.

Stability switching at transcritical bifurcations of solitary waves in generalized nonlinear Schrödinger equations

“…Indeed, it has been shown that for saddle-node and pitchfork bifurcations, stability properties in the GNLS equations differ significantly from those in finitedimensional dynamical systems [8,9,17]. For instance, at a saddle-node bifurcation point, there is no stability switching in the GNLS equations (2.1) [8,9], but any dynamical-system textbook would say that such stability switching takes place [16]. Thus it may be more appropriate to view this similar stability on transcritical bifurcations in the GNLS equations and finite-dimensional dynamical systems as a happy surprise rather than a trivial expectation.…”

“…Indeed, various solitary wave bifurcations in these equations have been reported. Examples include saddle-node bifurcations (also called fold bifurcations) [2,4,5,6,7,8,9], pitchfork bifurcations (sometimes called symmetry-breaking bifurcations) [7,10,11,12,13,14], and transcritical bifurcations [15]. These three types of bifurcations have also been classified in [15], where analytical conditions for their occurrence were derived.…”

“…However, as was explained in [9,17], those stability results from finite-dimensional dynamical systems cannot be extrapolated to the generalized nonlinear Schrödinger equations. Thus this stability in the generalized nonlinear Schrödinger equations has to be studied separately.…”

“…Thus this stability in the generalized nonlinear Schrödinger equations has to be studied separately. For saddle-node bifurcations of solitary waves, this stability question has been analyzed in [8,9]. It was shown that no stability switching takes place at a saddle-node bifurcation, which is in stark contrast with finite-dimensional dynamical systems where stability switching generally takes place [16].…”

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

“…Three major types of bifurcations have been classified [11]. Of these bifurcations, stability of solitary waves near saddle-node bifurcations has been analyzed in [19,20]. It was shown that no stability switching takes place at a saddle-node bifurcation, which dispels a pervasive misconception that such stability switching should occur.…”