In recent years, considerable effort has been devoted to the investigation of optical solitary waves [1] due to their fundamental impact on nonlinear wave propagation, spawning exciting applications along the way such as supercontinuum light sources [2], soliton lasers [3], and an improved understanding of the development and control of rogue waves (RWs) [4]. A special type of solitary waves, the RW phenomenon, which was first observed in the ocean, is a rare, short-lived, and high-energy event with amplitude much higher than the average wave crests around it [5]. The typical feature of RWs is that they suddenly appear and increase up to a very high and abnormal amplitude to finally disappear without a trace [6,7]. The experimental observation and theoretical analysis of RWs have ranged from Bose–Einstein condensates (BECs) [8,9] to optical systems [10–12], oceans [13], superfluids [14], and plasmas [15,16]; see more details in two recent review articles [17,18]. One possible mathematical model to describe such RWs is the rational solution of one-dimensional nonlinear Schrödinger (NLS) type equations. Moreover, research has diversified, also addressing optical solitary waves in higher-dimensional media, which display a more complex phenomenology due to the increased degrees of freedom [19].

In this context, the formation of self-trapped wave packets or light bullets, is one of the most exciting yet experimentally unsolved problems in optics [20,21]. Light bullets are spatiotemporal solitons that form when a suitable nonlinearity arrests both spatial diffraction and temporal group-velocity dispersion. Despite considerable theoretical work, experimental research on light bullets is rare, and one of the main reasons is that in nonlinear propagation, three-dimensional light bullets tend to disintegrate due to inherent instabilities. However, different situations are found in BECs and nonlinear optics with temporally or spatially modulated parameters. In particular, it was shown in Ref. [22] that complete stabilization of a cylindrical (2+1)-dimensional [(2+1)D] spatial soliton can be secured in a layered medium with nonlinearity management. A scheme for stabilizing spatiotemporal solitons in media with cubic self-focusing nonlinearity and dispersion management was proposed in Ref. [23]. The formation of tandem structures, which are composed of periodically alternating linear dispersive and nonlinear layers, was studied in Refs. [24,25]. Moreover, alteration of atomic scattering length achieved by Feshbach resonance has been used to dynamically stabilize higher-dimensional bright solitons [26]. Thus, the study of the (2+1)D and (3+1)D variable coefficients NLS equations (NLSEs) has recently been one of the central issues in the field of nonlinear optics. One of the interesting challenges concerns how to characterize the nonlinear light bullets on analytical level [27–29]. In general, the analytical study of the multidimensional light bullets is impeded by the lack of the corresponding integrable systems. Therefore, several approaches have been recently developed to overcome this limitation. The traveling wave and light bullet soliton solutions to the generalized NLSE in (3+1)D for a cubic nonlinearity were first developed in Ref. [30] for anomalous dispersion and were generalized in Ref. [31] for normal dispersion by using the F-expansion technique. Exact solutions for varying potential and nonlinearity were found in Ref. [32] by similarity transformations. Nonautonomous rogue wave solutions have also been found for the generalized NLSEs with variable coefficients in three-dimensional spaces [33] based on the similarity analysis idea.

Very recently, the spatiotemporal dynamics of RW solutions in a composite (2+1)D were investigated in Ref. [34]. A novel type of light bullets, which take the shape of RWs and travel on a finite (2+1)D space-time background, has been obtained. It was shown that both the fundamental and second-order RWs have a directional preference or a bullet nature that can propagate in a certain direction with transverse double localization. Such special (2+1)D RW behavior has been called rogue wave bullets. We shall in this paper proceed along this direction to get rogue wave bullets of a new inhomogeneous (3+1)D integrable system where coefficients depend on time and transverse radial coordinates. The main result of the present work is the possibility to obtain a single optical sphere and homocentric optical spheres for an inhomogeneous (3+1)D NLSE with spherical symmetry.

The three-dimensional inhomogeneous NLSE with variable coefficients can be written in a dimensionless form: i(∂∂tψ)+β(r,t)∇2ψ−v(r,t)ψ−g(r,t)|ψ|2ψ=0,(1)where ψ(r,t) is the complex envelope of the optical field, r=x2+y2+z2 is the distance from a point (x,y,z) to the origin of the coordinates, and ∇2ψ=r−2∂/∂r(r2∂ψ/∂r) is the 3D Laplacian describing beam diffraction or group velocity dispersion in a 1D time-domain configuration. The external potential v(r,t) and nonlinear coefficient g(r,t) are real-valued functions of time and spatial coordinates. This equation arises in many fields such as nonlinear optics [1,21] and BECs [32,35–37]. The 1D version of Eq. (1) was considered in Refs. [38–40], where the soliton, together with first- and second-order RW solutions, was obtained. We present here 3D RW solutions to the NLSE in (3+1)D. In order to investigate the dynamic properties of the optical rogue wave solution for Eq. (1), we perform a specific reduction, namely, ψ(r,t)=ρ(r)exp[iϕ(r,t)]Φ[X(t),T(r,t)],(2)where the functions ρ(r) and ϕ(r,t) represent the amplitude and the phase, respectively. The complex function Φ satisfies the following NLSE with constant coefficients: i[∂∂XΦ(X,T)]+∂2∂T2Φ(X,T)+2ϵ|Φ(X,T)|2Φ=0,(3)which is obtained by substituting Eq. (2) into Eq. (1) with the following specific transformation: T=αr+t,X=t,ρ=1rα,ϕ=−αr2−t2,(4)β=1α2,g=−2ϵαr2,v=14.(5)Here ϵ=±1andα is a positive constant.

According to the above transformation defined by Eqs. (2), (4), and (5), we set α=1,ϵ=1, and then Eq. (1) leads to a solvable three-dimensional inhomogeneous NLSE with spatial nonlinearities: i∂∂tψ+(∂2∂r2ψ+2r∂∂rψ)−14ψ+2r2|ψ|2ψ=0.(6)This equation is a solvable model due to the established transformation and the solvability of the NLSE, which is the main result of this paper. We shall focus on rational-like solutions of Eq. (6), which provide novel localized optical spheres and thus generate new kinds of light bullets. These optical spheres oscillate along the radius direction and do not travel like the usual light bullets.

In optics, spatially inhomogeneous nonlinearities can be realized in various ways [41]. In a BEC, Eq. (6) describes the evolution of matter waves, where the spatially modulated nonlinearity landscape can be generated by the Feshbach resonance in nonuniform external fields [42,43]. Nonlinearity can also be modulated in optical structures, e.g., in photonic crystal fibers with the holes infiltrated with a highly nonlinear material, for example, index-matching nonlinear liquids [44,45].

We next investigate the features of the amplitude Urw1=|ψrw| of the 3D RW solution from Eq. (7). Indeed, there exists a critical point t0 such that Urw1 oscillates with respect to r when t<t0, but it is a monotonic function of r when t>t0.

The radius of these spheres increases to a certain value and then reaches an upper limit, which corresponds to localized feature of the rogue wave. In this process, the radius r of the sphere may have oscillation. Equivalently, r is not a monotonic function of time t, although it is bounded. In other words, the isosurface of Urw1 forms size-bounded sphere, which is a strong reflection of the localized nature of rogue waves in three dimensions. Moreover, the polynomial form of the rogue wave in the one-dimensional NLSE is reflected by the oscillation of the radius r of the isosurface. Therefore, the behavior of the sphere of the isosurface represents the nature of the first-order rogue wave of the NLSE: polynomial and having a doubly localized property. The asymptotical radius of the isosurface valued at Urw1=k is given by ras=1k.

In conclusion, we have shown that in the (3+1)-dimensional NLSE with varying coefficients, localized solutions in the form of rational formulas can exist owing to a specific transformation that allows us to reduce the dimensionality of the equation from (3+1) dimensions to (1+1) dimensions. These solutions are localized both in space and in time, and thus their corresponding isosurfaces are single spheres or homocentric spheres, which oscillate along the radius direction and are completely different from the well-known standard traveling light bullets. They can be interpreted as prototypes of RW light bullet solutions in the (3+1)-dimensional time-space. The other properties of the new nonautonomous RW light bullets have been studied analytically. Our analytical findings are confirmed by numerical plots of these solutions. A linear stability analysis in terms of spherical harmonic modes has been investigated. We have found that the RW light bullet solutions are stable for higher modes and transversely unstable for lower modes of the perturbation. Experimental advances have recently provided a strong incentive in the area of RWs in complex media [54]. Note that a demonstration of the direct observation of RWs in self-excited 3D longitudinal plasma density waves was reported in Ref. [55] by using self-excited dust acoustic waves as a platform. We believe that the results obtained here can stimulate further research on the experiments and help to understand the behavior of 3D RWs in a wide range of nonlinear physical areas.

Acknowledgment. J. He gratefully acknowledges support from the University of Lille for his visit. C. G. L. Tiofack acknowledges support from the Ministry of Higher Education and Research, Hauts-de-France Council, and ERDF through the Contract de Projets Etat-Region (CPER Photonics for Society P4S). C. G. L. Tiofack and M. Taki appreciate helpful discussions with S. Coulibaly.