Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic‐Quintic Nonlinearities

Abstract

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.

Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results

“…The modulational instability of continuous waves [17] and many types of optical solitons produced by fractional NLSEs has been theoretically investigated, chiefly by means of numerical methods. These are quasi-linear "accessible solitons" (actually, quasi-linear modes) [20,21], gap solitons supported by spatially periodic (lattice) potentials [26][27][28][29][30], solitary vortices [31,32], multi-pole and multi-peak solitons [33][34][35][36], soliton clusters [37], and solitary states with spontaneously broken symmetry [40][41][42], as well as solitons in optical couplers [43,44]. Dissipative solitons in fractional complex Ginzburg-Landau equation (CGLE) were studied too [39].…”

Families of fundamental and multipole solitons in a cubic-quintic nonlinear lattice in fractional dimension

“…( 1) below. The experimental implementation of the FSE in condensed-matter [56,57] and optical [58] setups, where nonlinearity is a natural feature, has drawn interest to the possibility of existence of solitons in fractional dimensions [59][60][61][62]. In particular, "accessible solitons" [63,64] and self-trapped states of vectorial [65], gap [66], nonlocal [67], vortical [68], and multi-peak types [69] have been predicted in FSE models, as well as soliton clusters [70,71], symmetry breaking of solitons [72,73], coupled solitons [74] and dissipative solitons in a fractional complex-Ginzburg-Landau model [75].…”

Symmetry Breaking of PT-symmetric Solitons in Self-defocusing Saturable Nonlinear Schrödinger Equation

“…And the existence and stability of solitons in PT-symmetric optical lattices in fractional NLSE models were studied [27]. Subsequently, the fork bifurcation of the symmetry breaking for fractional optical solitons were reported [28], and the symmetry breaking behavior of solitons were found in partial PT-symmetric potential [29]. The soliton solutions of the (1+1) dimensional nonhomogeneous cubic-quintic-septimal NLSE with PT-symmetric potential were discussed [30].…”