Bright solitons from the nonpolynomial Schrödinger equation with inhomogeneous defocusing nonlinearities

Abstract

Extending the recent work on models with spatially nonuniform nonlinearities, we study bright solitons generated by the nonpolynomial self-defocusing (SDF) nonlinearity in the framework of the one-dimensional (1D) Muñoz-Mateo -Delgado (MM-D) equation (the 1D reduction of the Gross-Pitaevskii equation with the SDF nonlinearity), with the local strength of the nonlinearity growing at |x| → ∞ faster than |x|. We produce numerical solutions and analytical ones, obtained by means of the Thomas-Fermi approximation (TFA), for nodeless ground states, and for excited modes with 1, 2, 3, and 4 nodes, in two versions of the model, with steep (exponential) and mild (algebraic) nonlinear-modulation profiles. In both cases, the ground states and the single-node ones are completely stable, while the stability of the higher-order modes depends on their norm (in the case of the algebraic modulation, they are fully unstable). Unstable states spontaneously evolve into their stable lower-order counterparts. [22], etc. This subject has been greatly upheld by the use of the Feshbachresonance (FR) technique, i.e., the control of the strength of the inter-atomic interactions by externally applied fields [23][24][25], which opens the possibility to implement sophisticated nonlinear patterns. In particular, the management of localized solutions of the Gross-Pitaevskii equation (GPE) [26] by means of the spatially inhomogeneous nonlinearity, which may be created by external nonuniform fields that induce the corresponding FR landscape, has attracted a great deal of interest in theoretical studies [27][28][29][30][31][32][33][34][35][36].In this vein, the existence of bright solitons in systems with purely repulsive, alias self-defocusing (SDF) nonlinearity, in the absence of external linear potentials, was recently predicted [37]. This result is intriguing because the existence of such solutions, supported by SDF-only nonlinearities, without the help of a linear potential, was commonly considered impossible. In the setting introduced in Ref.[37], the system is described by a nonlinear Schrödinger (NLS) equation with the SDF cubic term, whose strength increases in space rapidly enough towards the periphery. The discovery of bright solitons in this setting has ushered studies of solitary modes in other models with spatially growing repulsive nonlinearities, both local [38][39][40][41][42][43] and nonlocal [44]. More specifically, in Ref. [38] it was demonstrated that spatially inhomogeneous

Asymmetric localized states in periodic potentials with a domain-wall-like Kerr nonlinearity

“…As such, the study of soliton phenomena in truly periodic nonlinear lattice (the nonlinearity embedded into a linear uniform medium) is particularly intriguing, since it departs from the case in conventional linear lattices mentioned above with modulated refractive index. The stabilization of solitons and vortical ones in nonlinear lattices [34][35][36][37][38][39][40][41][42][43], combined linear and nonlinear lattices [44][45][46][47] is increasingly being studied in past years, and recently the interest is also on the scenarios with inhomogeneous modulations of nonlinearity [48][49][50][51][52][53][54][55][56]. In particular, spatially inhomogeneous nonlinear media with a defocusing nonlinearity, whose local strength grows fast enough from the pivot to the periphery, can uphold a vast variety of localized states, both the fundamental and higher-order solitons, which are in the forms of solitary vortices (with arbitrarily vortex charges) [48], vortex rings [48], soliton gyroscopes [49] and skyrmions [50], hopfions, complex hybrid modes, localized dark solitons and vortices [56].…”

Exact solutions for bright and dark solitons in spatially inhomogeneous nonlinearity

“…The media with uniform SDF cubic nonlinearity in the presence of anti-Gaussian-type losses allow stable dissipative solitons [16]. Bright solitons have also been found in SDF quintic [17] and nonpolynomial [18,19] nonlinearities. Moreover, stable bright solitons can occur in a situation where the homogeneous SDF nonlinearity is modulated by a singular or localized SF nonlinearity with algebraic profiles [20].…”

“…Subsequently, several other types of the SDF cubic nonlinearities have been shown to support stable bright solitons [9][10][11][12][13][14]. In addition, this interesting finding has been extended to diverse settings [15][16][17][18][19]. For example, it was shown that bimodal systems with an anti-Gaussian-type SDF cubic nonlinearity can support stable bright vector solitons [15].…”

Bright solitons in nonlinear media with a self-defocusing double-well nonlinearity

“…The formation of stable bright solitons requires diverging nonlinearities. Several different types of such nonlinearities have been discussed [10][11][12][13][14][15][16][17][18][19][20][21][22]. In certain situations, exact analytical results for bright soliton solutions have been obtained.…”

“…In certain situations, exact analytical results for bright soliton solutions have been obtained. In most of these previous works, spatially inhomogeneous SDF nonlinearities display a single-well structure, including antiGaussian [10,12,15,17,18], exponential [10,14,[20][21][22], and algebraic [11,16,22] nonlinearities.…”