Universal Critical Power for Nonlinear Schrödinger Equations with a Symmetric Double Well Potential

Abstract

Abstract. Here we consider stationary states for nonlinear Schrödinger equations with symmetric double well potentials. These stationary states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a simple pitch-fork bifurcation, and a couple of saddle points which unstable branches collapse in an inverse pitch-fork bifurcation. In this paper we show that in the semiclassical limit, or when the barrier between the two wells is large enough, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value (2); in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is an universal constant in the sense that it does not depends on the shape of the double well potential.Spontaneous symmetry breaking phenomenon is a rather important effect that arises in a wide range of physical systems modeled by nonlinear equations. In classical physics spontaneous symmetry breaking occurs in optics, and it has been experimentally observed for laser beams in Kerr media and focusing nonlinearity [1,2]. Another natural setting in which spontaneous symmetry breaking phenomenon may arise is for Bose Einstein condensates with an effective double well formed by the combined effect of a parabolic-like trap and a periodical-like optical lattice [3,4,5]. Also, the study of gases of pyramidal molecules, like the ammonia N H 3 , it is a topic where spontaneous symmetry breaking phenomenon actually plays a crucial role. In [6,7] has been introduced a nonlinear mean field model of a gas of pyramidal molecules; in this model spontaneous symmetry breaking explaining the presence of two asymmetrical degenerate ground states, corresponding to the different localization of the molecules, has been predicted with a full agreement with experimental data [7,8].The n-dimensional linear Schrödinger equation with a symmetric potential with double well shape has stationary states of a definite even and odd-parity. However, the introduction of a nonlinear term (which usually models, in quantum mechanics, an interacting many-particle system) may give rise to asymmetrical states related to a spontaneous symmetry breaking effect. The governing equations are nonlinear Schrödinger equations of Gross-Pitaevskii typewhere ǫ is the strength of the nonlinear term, µ > 0 is the nonlinearity power, and H 0 is the linear Hamiltonian with a symmetric double well potential. When µ = 1 we have a cubic nonlinearity and the resulting equation has been largely studied [9, Date: December 24, 2017.

Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

“…As μ → μ 0 , these two new solution branches merge with the smooth branch. Examples of pitchfork bifurcations reported so far are all symmetry-breaking bifurcations [11,[13][14][15][16][17][18][19], where a smooth branch of symmetric or antisymmetric solitary waves exists on both sides of the bifurcation point, but two new branches of asymmetric solutions appear on only one side of the bifurcation point. A transcritical bifurcation is where there are two smooth branches of solitary waves which exist on both sides of the bifurcation point μ 0 , and these solutions on both branches approach each other as μ → μ 0 .…”

“…Indeed, various solitary wave bifurcations in miscellaneous nonlinear wave models have been reported. Examples include saddle-node bifurcations (also called fold bifurcations) [2][3][4][5][6][7][8][9][10][11], pitchfork bifurcations (sometimes called symmetry-breaking bifurcations) [11][12][13][14][15][16][17][18][19], transcritical bifurcations [7], and so on. Most of these reports on bifurcations are numerical.…”

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

“…Under certain conditions, these solitary waves undergo saddle-node bifurcations at special values of l [6][7][8][9]. A signature of these bifurcations is that on one side of the bifurcation point l 0 , there are no solitary wave solutions; but on the other side of l 0 , there are two distinct solitary-wave branches which merge with each other at l ¼ l 0 : To derive conditions for these bifurcations, we introduce the linearization operator of Eq.…”

“…Examples include the Boussinesq equations and the fifth-order Korteweg-de Vries equation in water waves [2][3][4], the Swift-Hohenberg equation in pattern formation [5], the nonlinear Schrödinger (NLS) equations with localized or periodic potentials in nonlinear optics and Bose-Einstein condensates [6][7][8][9], and many others. Motivated by stability switching of saddle-node bifurcations in finite-dimensional dynamical systems, it is widely believed that in nonlinear partial differential equations, stability of solitary waves also always switches at saddle-node bifurcations (see [5][6][7][8] for examples). Even though it was claimed on numerical evidence that both branches of saddle-node bifurcations were stable for various solitons in a Kronig-Penney model with cubic-quintic nonlinearity [10], that numerical evidence was not reliable since many solitons which the authors claimed to be stable are actually unstable.…”

Lasing on nonlinear localized waves in curved geometry

“…In general, self-localized nonlinear waves require an extra formation power to bifurcate from the linear modes [18], but geometrical constraints may reduce or cancel this threshold. In this work, we use the vertical cavity surface emitting lasers (VCSELs) as a platform and fabricate a series of curved surfaces, including circular-ring and elliptical-ring cavities, as well as a reference "cold" device with no trapping potential.…”