Accessible soliton solutions with initial phase-front curvature in strongly nonlocal nonlinear media

Based on the extended fractional dimensional nonlinear Schrödinger equation and the variable separation method, a fractional accessible soliton solution with initial phase curvature is proposed for the first time. The soliton solution of the model is composed of hypergeometric functions and generalized Laguerre polynomials in fractional dimensional space, namely, Hypergeometric–Laguerre–Gaussian soliton. The theoretical results indicate that a series of different types of solitons are generated with the change of the beam parameters, forming a fractious family of solitons. At the same time, solitons produce a splitting phenomenon similar to that of the Hermitian beams. Additionally, the initial phase curvature also affects the stability of beam propagation, suppressing the formation of soliton.


Introduction
Over the past few decades, optical solitons have been attracting attention from various fields of physics.The optical soliton formed by the relationship between equilibrium dispersion and Kerr nonlinear effect is called time optical soliton (Wang et al. 2022;Fang et al. 2021), and is usually applied to optical fiber communication; Spatial optical solitons (Li et al. 2023;Guo et al. 2020) are formed by the nonlinear autofocusing effect of balanced beams, which are usually found in non-localized media.The propagation dynamics of two kinds of optical solitons are usually described by the nonlinear Schrödinger equation.Based on the nonlinear Schrödinger equation, various solitons or breathing wave forms of beam propagation are found (Geng et al. 2023;Wen et al. 2023;Shen et al. 2021).
1155 Page 2 of 13 In nature, the most common strongly nonlocal nonlinear media materials are nematic liquid crystals (Conti et al. 2004) and lead glasses (Rotschild et al. 2005), whose nonlocality can prevent wave collapse (Bang et al. 2009;Parola et al.1998) and play an important role in the formation and propagation of solitons.By setting strong non-local physical conditions, the nonlinear Schrödinger equation will be reduced to the famous Snyder-Mitchell model (SMM) (Snyder er al. 1997).Theoretically, based on SMM, various soliton forms are obtained.For example, Laguerre-Gaussian and Hermite-Gaussian solitons (Song et al. 2020), hyperbolic-sine-Gaussian solitons (Yang et al. 2018), twisted Gaussian Schell-model solitons (Zhang et al. 2022b).In addition, an exact soliton solution with initial phase front curvature is given by SMM.Studies have shown that the initial phase front curvature causes the pulsation behavior of the beam propagating in a strong non-local medium, causing periodic interference mode between the two beams of light, and as an application, it is expected to determine the initial phase front curvature by measuring the spacing between interference fringes (Zhang et al. 2007).
However, a growing body of research shows that the behavior and properties of physical phenomena may differ significantly from the integer dimension in the fractional dimension.For example, Longhi introduced a laser implementation method for fractional quantum harmonic oscillators based on the spatial fractional Schrödinger equation (Longhi 2015).Dai et al. predicted multipolar solitons in NLFSE (Bo et al. 2023).On the other hand, fractional calculus provides a variety of methods for dealing with Laplace operators, including the quantum Riesz fractional derivative involving Levi flights (Zhong et al. 2016b), as well as the multidimensional Stillinger formulation (Stillinger 1977) in non-integer dimensional spaces (Baleanu et al. 2010;Sandev et al. 2014).
Calculus in fractional dimensional spaces has been widely used to simulate various phenomena in different physics-related settings.For example, it can be used to describe excitons in amorphous solids (Lenzi et al. 2009) and excitons in quantum wells (Matos-Abiague et al. 1998).Considering the strong nonlocal nonlinearity, Zhong et al. used the nonlinear fractional dimension Schrödinger equation to model the optical soliton (Zhong et al. 2016a).Since this optical soliton has a harmonic oscillator potential, this model is suitable for many physical applications (Lenzi et al. 2009;Matos-Abiague et al. 1998).He obtained the exciton wave functions by solving the nonlinear Schrödinger equation in fractional dimensional space (He 1990(He , 1991)).Zhong and his team solved the accessible solitons in the fractional dimensional Schrödinger equation (Zhong et al. 2016a, b), which further enriched the variety of optical solitons.
In this paper, a novel type of accessible soliton with an initial phase pre-curvature, based on the fractional dimensional Schrödinger equation, is constructed for the first time.Traditionally, the behavior of solitons has been typically described in the integer dimension.but our research shows that accessible solitons exhibit entirely new characteristics and behaviors in the fractional dimension space.In addition, the evolution properties of such solitons are studied in detail.By separating variables in cylindrical coordinates, a hypergeometric Laguerre Gaussian (LG) soliton solution is obtained.The effect of fractional dimension on accessible solitons modulated by different parameters is demonstrated, revealing various profiles and properties of the solitons.The beam pulsation behavior caused by initial curvature is discussed.The introduction of this initial phase curvature leads to periodic changes and fluctuation behavior of the beam, which provides a new idea for beam regulation and modulation in the field of optical information processing.In addition, the presence of fractional dimension can lead to changes in the spatial distribution and interaction modes of the medium, bringing new possibilities for fields such as optical information processing.

The model
The introduction of the concept of fractional dimension has sparked the interest of many researchers and has been widely applied following Mandelbrot's (Mandelbrot 1982) pioneering work.In the framework of the fractional dimension model, the Laplace operator in the a-dimensional space proposed by Stellinger (Stillinger 1977;Matos-Abiague 2001) is employed to solve the nonlinear Schrödinger equation, aiming to investigate the anisotropic excitations dynamics (Eid et al. 2009).Recent research progress has introduced the fractional-dimensional Laplace operator into nonlinear optics, and the optical soliton solution is obtained by solving the nonlinear fractional-dimensional Schrödinger equation (Zhong et al. 2016a, b).Considering the beams propagate along the z direction in the strongly nonlocal nonlinear medium, the complex amplitude field Φ(r, , z) can be expressed by the nonlinear Schrödinger equation: where (r, ) is the set of polar coordinates obtained from the transformation of rectangular coordinates (x, y) , 0 ⩽ r < ∞, 0 ⩽ ⩽ 2 , ∇ 2 is the fraction-dimensional Laplacian opera- tor defined as (Eid et al. 2009;Sandev et al. 2014): 1 <  ⩽ 2.  > 0 is the parameter related to the beam power.In particular, Eq. ( 1) is transformed into the well-known standard nonlinear Schrödinger equation when = 2.
The solution of a separated variable is allowed by Eq. ( 1), so we set Φ(r, , z) = u(r, , z)G (r, z) .Substituting this quasi-solution into Eq.( 1), one obtains Combining with Ref. (Deng et al. 2011;Shen et al. 2006), we can obtain that Eq. (3) has a Gauss solution: where P 0 is the initial power, and the value is 1. w(z), c(z) and (z) are the beam width, wavefront curvature, and phase offset of Gauss beam, respectively. ( 1155 Page 4 of 13 Taking w(0) = w 0 , c(0) = c 0 and (0) = 0 as the boundary conditions and plugging this ansatz into Eq.( 6), the analytic expressions of these parametric equations can be obtained where = 2 w 4 0 , c 0 and 0 are real numbers.In particular, if c 0 = 0 , it can be seen from Eq. (6a) that the self-focusing effect overcomes the diffraction effect when  > 1 , and the beam initially compresses.When  < 1 , the opposite is true, and the beam initially expands.The self-focusing balances the diffraction completely when = 1 , the beam width does not change with the propagation distance, namely, accessible soliton.

Hypergeometric laguerre-gaussian soliton family
From Eq. ( 12), one can find that the beam is affected by the beam width, the loss, the generalized Laguerre polynomials, the angular distribution, and the fractional dimension, where the angular distribution is represented by hypergeometric functions.Furthermore, the results of Eq. ( 12) are discussed and demonstrated by selecting different parameter values.Without special circumstances, some initial parameters are selected as = 1, w 0 = 1 , c 0 = 0, a 0 = 0, 0 = 0.
To understand the characteristics of Hypergeometric LG solitons, the effects of fractional dimension α and radial quantum number n on soliton morphology are discussed firstly.As shown in Fig. 1, the normalized intensity distribution of the rings soliton at the origin is unchanged and around the origin varies with different parameter values.For example, as the radial quantum number n increases, the number of rings increases, and 1155 Page 6 of 13 the increase in fractional dimension α causes the energy around the center to gradually weaken.Furthermore, we find that when α approaches one, the intensity at the outer ring and origin is always stronger than that at the inner ring and attenuates to the conventional LG solitons at α = 2. Next, we discuss the influence of fractional dimension α and exponent m on the properties of wave packets.As shown in Fig. 2, contrary to the case when m = 0, the intensity at the center of the propagation axis is zero.Figure 2a shows the image of soliton intensity under different parameters.The number of solitons sidelobes increases the increase of the index m, and the quantitative relationship is 4 m, m set to the finite positive integer.On the other hand, as α increase to 2, the soliton intensity is transformed from non-uniform distribution to uniform distribution.As we can see from Fig. 2b, the soliton intensity in the x-axis direction does not change with α.On the  The decrease of the soliton intensity in the y-axis direction may be the change of the effective refractive index in the medium, which is related to the nonlinear effect and spatial distribution of the beam.When α = 2, the medium exhibits isotropy, meaning that the refractive indices in both x and y directions are equal, and therefore the soliton intensities in these directions are consistent.
We further study multi-polar soliton, Fig. 3 gives the properties of the soliton for some non-integers m.One can find from Fig. 3a that the soliton appears self-splitting phenomenon like Twisted Hermite-Gaussian Schell-model beams (Zhang et al. 2022a).Interestingly, for non-integer m in Fig. 3b, α acts on both x and y directions.It is not difficult to find that the increase of α causes the axial intensity to weaken.The possible physical reason is that the change of fractional dimension α may affect the nonlinear effects and spatial distribution of the light beam in the medium, leading to the weakening of axial intensity.
For η ≠ 1, one obtains breathers.The cross-sectional intensity of the breathers propagating to the distance Z = √ 2 z is shown in Fig. 4, with the same α, n, m but different .When η > 1, the nonlinearity effect dominates, leading to self-focusing of the beam and a decrease in spot size with propagation distance.On the contrary, when η < 1, the diffraction effect is dominant, resulting in the beam width expanding as the breathers propagate.The physical reason for the change in beam width can be attributed to the balance between diffraction and nonlinearity effects in the medium.It is worth noting that when η = 1, the spot size no longer changes with the propagation distance.The evolution of beam width can be confirmed by Eq. (7a).
Figure 5 shows the evolution curves of beam width under different initial parameters.When η = 1, the beam width oscillates periodically with the variation of the initial curvature c 0 .In addition, the increase of |c 0 | causes the increase of amplitude, as shown in Fig. 5a.This means that in this case, beam broadening and convergence occur periodically.When c 0 = 0.1, for different η in Fig. 5b, the maximum or minimum position of the beam spot size produces displacement.This means that the initial curvature has an effect on the position or shape of the beam.We can see the stability of the solitons is destroyed by the initial curvature, forming periodic oscillation.The results indicate that when the input beam is a spherical wave, the nonlinear and dispersive effects between any positions in the medium cannot balance, which eventually leads to the propagation of the beam becoming unstable and undergoing periodic oscillations.These observations are consistent with the findings in the references (Zhang et al. 2007) and further validate the phenomenon described herein.
Usually, these phenomena can be simply understood.Under special circumstances, the nonlinear refractive index is induced by the intensity distribution in different regions, which causes the refractive index distribution of the medium to be uneven, thereby forming multipole solitons.For spherical waves, the equiphase surface is not in the same plane, which makes the nonlinear medium unable to form an effective spatial waveguide, so the beam width exhibits pulsating behavior.

Conclusion
In summary, fractional-dimension accessible solitons in strongly nonlocal nonlinear media with initial curvature are solved and studied by the method of separating variables.The soliton solution consists of two special functions, namely the Hypergeometric function and the generalized Laguerre polynomial.Through the modulation of three parameters α, m, and n, the profile of solitons take on different shapes, including ring, crescent, and necklace.The findings indicate that the x-axis intensity remains unchanged under the change of α when m is a positive integer.The change of beam width follows the same pattern as that of conventionally accessible solitons, and it is independent of the fractional dimension α.The stability of solitons is disrupted by the initial phase-front curvature.This research is beneficial is beneficial to further understand the characteristics of fractional dimension wave packets in strongly nonlocal nonlinear media.And the feasibility of finding the selfsimilar solution of the fractional dimensional nonlinear Schrödinger equation is verified.Furthermore, the existence of fractional dimensionality can cause changes in the spatial distribution and interaction modes of the medium, leading to novel opportunities in fields such as optical information processing and communication.

Fig. 1
Fig. 1 For m = 0, the influence of the different α on the variation of accessible Hypergeometric LG solitons with parameter n. a The three-dimensional distribution of initial intensity, that is, at z = 0.The parameters are n = 1, 2, α = 1.1, 2, from left to right.b Initial intensity distribution for different parameters n and α. n = 1 (top row), n = 2 (below row), α = 1.1, 1.4, 1.7, 2 from left to right

Fig. 2
Fig. 2 Hypergeometric LG solitons family with initial positions modulated by different parameters.a The parameters are m = 1, 2, 3 from top to bottom, α = 1.1, 1.4, 1.7, 2 from left to right, n = 0. b Corresponding to the third row in (a), the non-normalized beam intensity curves in the x and y directions, respectively

Fig. 3
Fig. 3 An intensity image of the solitons for different parameters.a The values of parameters m from left to right are m = 1, 1.5, 1.7, using parameters α = 1.1 and n = 0. b The values of parameters α from left to right are α = 1.1, 1.5, 1.9, using parameters m = 0.575 and n = 1

Fig. 5
Fig. 5 The evolution plots of the beam's width with the propagation distance Z = √ 2 z for different parameters.a For the different initial curvature parameters c 0 at η = 1.b For the η < 1 and η > 1, from left to right, c 0 = 0.1