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On the existence of optical vortex solitons propagating in saturable nonlinear media

In this paper, an existence theory is established for ring-profiled optical vortex solitons. We consider such solitons in the context of an electromagnetic light wave propagating in a self-focusing nonlinear media and governed by a nonlinear Schr\"odinger type equation. A variational principle and constrained minimization approach is used to prove the existence of positive solutions for an undetermined wave propagation constant. We provide a series of explicit estimates related to the wave propagation constant, a prescribed energy flux, and vortex winding number. Further, on a Nehari manifold, the existence of positive solutions for a wide range of parameter values is proved. We also provide numerical analysis to illustrate the behavior of the soliton's amplitude and wave propagation constant with respect to a prescribed energy flux and vortex winding number.


INTRODUCTION
An exciting area of research in modern optics is the study of optical vortices. In a light wave, optical vortices are formed by wave dislocations or defects [28]. Its applications are found across numerous branches of nonlinear science, such as quantum information processing, wireless communications, and some not directly related to wave propagation, for instance condensed matter physics, particle interactions, and cosmology [5-7, 12, 19, 31, 34, 38, 41]. An interesting class of optical vortices are the ring-profiled optical vortices. Such vortices can be considered as a ring of light with a black spot at its center. In terms of a light beam, such black spots represent a zero light intensity.
Of particular interest, is the theoretical description of a complex-valued light wave propagating in a nonlinear media and governed by a nonlinear Schrödinger equation [1,11,20,21,25,27,30,36]. Rigorous mathematical treatments of such nonlinear problems present mathematical challenges and have been considered by mathematical analysts [2-4, 8-10, 24, 33, 39, 40]. Our interest is motivated by the work of Skryabin and Firth [36] and the mathematical analysis of Liu and Ren [24], and Zhang and Yang [40].
Consider the propagation of an electromagnetic wave in the longitudinal z-direction over the transverse plane of coordinates (x, y) perpendicular to the z-axis. In dimensionless form, the evolution of the slowly varying electric field envelope, E, is modelled by the nonlinear Schrödinger equation [36], where ∇ 2 ⊥ is the Laplace operator over the transverse plane of coordinates. The function F depends on the total field intensity, I, i.e., I = |E| 2 , and encapsulates the nonlinear properties of an optical medium. Examples for F (I) include 1. F (I) = I (pure Kerr nonlinearity), 2. F (I) = I − αI 2 (cubic-quintic model), 3. F (I) = I(1 + αI) −1 (saturable nonlinearity), where α ∈ R is a parameter describing the nonlinearity saturation [12]. The saturation constant may be defined as α = l nl /l d , with l nl and l d denoting the nonlinear and diffraction lengths, respectively. Note that the saturable nonlinearity and cubic-quintic models reduce to the pure-Kerr nonlinearity when the saturation constant is zero.
We focus on spatial optical solitons [12,23,35,37]. Spatially localized solutions of (1.1), which do not change their intensity profile during propagation, can be described under the spatial soliton ansatz E(x, y, z) = u(x, y) exp(iκz + iφ(x, y)), (1.2) where u and φ are real valued functions representing the soliton amplitude and phase, respectively, and κ ∈ R is the wave propagation constant. In view of (1.2), the nonlinear Schrödinger equation (1.1) transforms into the coupled system Under an appropriate balance between the nonlinear and diffraction lengths, the electromagnetic radiation may become self-trapped and form a self-induced wave-guide. Solutions to (1.3) are referred to as self-trapped nondiffracting solutions in a self-focusing saturable nonlinearity.
The existence theory in this paper seeks self-trapped positive radially symmetric solutions of (1.3) with a phase singularity at its center. Such solutions describe ring-profiled vortex solitons and can be found under the n-vortex ansatz u = u(r), φ = nθ, r = x 2 + y 2 , θ = arctan(y/x), (1.4) where r, θ are polar coordinates over R 2 and n ∈ Z is the vortex winding number.
Due to the presence of the vortex core or, equivalently, the regularity of u at r = 0, we impose the condition u(0) = 0. Moreover, such ring-like beams remain localized. Thus allowing us to mathematically impose the "boundary" condition u(R) = 0 for R > 0 sufficiently large, where R represents the distance from the vortex core.
Using (1.4), and in a saturable nonlinear media, the system (1.3) reduces to the n-vortex (1.5) An important parameter characterization of spatial solitons is its energy flux. Using the n-vortex ansatz, the soliton energy flux is defined as The rest of the paper is summarized as follows. In section 2, we give a necessary condition for the existence of nontrivial solutions. In section 3, we treat (1.5) as a nonlinear eigenvalue problem and use a constrained minimization approach, subject to a prescribed energy flux constraint, to prove the existence of positive solution pairs (u, κ). In section 4, we prove the existence of positive solutions for a wide range of parameter values over a Nehari manifold.
In section 5, we supplement our results by using a finite element formalism to compute the soliton's amplitude and wave propagation constant for a prescribed energy flux. A summary is provided in section 6.

NECESSARY CONDITION FOR NONTRIVIAL SOLUTIONS
The n-vortex equation (1.5) may be viewed as the Euler-Lagrange equations of a corresponding action functional. For a sufficiently large distance R, we prove that the action functional is indefinite, and, therefore, a direct minimization approach is not possible.
Consider the action functional I κ : H → R defined as with |n| ≥ 1 and α > 0. H is the completion of (the space of differentiable functions over [0, R] which vanish at the two endpoints of the interval) and is equipped with the inner product We may treat H as an embedded subspace of W 1,2 0 (B R ), composed of radially symmetric functions enjoying the property u(0) = 0 for any u ∈ H, where B R := {(x, y) ∈ R 2 : x 2 + y 2 ≤ R 2 }.
We are interested in finite energy solutions of (1.5). From the inequality where r 0 (≈ 2.404825) is the first positive zero of the Bessel function J 0 [39].
As a consequence of the following lemma, when the distance from the vortex core R is sufficiently large, we prove that the functional (2.1) is indefinite.
Lemma 2.2 Let κ < α −1 and β > 0. If the distance from the vortex core satisfies then there exists an element u 0 ∈ H such that Proof. Set R = 2a and define (2.17) By direct calculation we obtain, with a = R/2. For any ǫ > 0, choose b sufficiently large such that Hence, In order for the right hand term of (2.22) to be negative, let ǫ = 1 3 (α −1 − κ) and R be such (2.23) For any β > 0, choose b satisfying (2.21) and, such that, ||u 0 || 2 H > β. With these values of b and R, we have I κ (u 0 ) < 0.
From inequality (2.22), if the distance from the vortex core is sufficiently large, then Therefore, for a sufficiently large distance R and κ < α −1 , the functional I κ is indefinite and as such, a direct minimization is not possible.

EXISTENCE VIA CONSTRAINED MINIMIZATION
Using a variational principle and constrained minimization problem, we prove the existence of positive solutions of the n-vortex equation (1.5). In this scenario, the wave propagation constant κ is undetermined and appears as a Lagrange multiplier. We provide a series of explicit estimates for the wave propagation constant, vortex winding number, and a prescribed energy flux.
We view (1.5) as a nonlinear eigenvalue problem Define the action functional I as, and the soliton energy flux constraint functional Q as

Consider the nonempty admissible class
where E(u) is a defined by (2.6). In order to prove the existence of a solution pair (u, κ), it suffices to show that a solution to the following exist: where Q 0 is a prescribed value for the energy flux and κ is the Lagrange multiplier. (i) There exists a solution pair (u, κ) satisfying u(r) > 0 for r ∈ (0, R).
As a result, the minimization problem (3.5) is well-defined. Let {u j } be a minimizing sequence of (3.5), i.e., choose a sequence of functions {u j } in A such that Since {u j } minimizes (3.5) and using (3.6), there exist C > 0 independent of j such that where u j,r := d dr u j . The distributional derivative of u satisfies ||u| r | ≤ |u r |, and the functionals I and Q are both even, i.e., I(u) = I(|u|) and Q(u) = Q(|u|). As a consequence, we assume that the sequence {u j } consists of nonnegative valued functions. Moreover, we take these functions to be radially symmetric over the disk B R and vanishing on its boundary. From (3.7) and (2.5) it can be seen that the functions u j belong in W 1,2 0 (B R ) under the radially symmetric reduced norm, Using (3.7) and (2.5), we show the sequence {u j } is bounded in W 1,2 0 (B R ), Without loss of generality, since we are in a reflexive space, we may assume the weak convergence of {u j } to an element u ∈ W 1,2 0 (B R ). As a result, it now suffices to show that u j converges to a minimizer of (3.5) and belongs in A.

From the compact embedding
Hence, u is radially symmetric and satisfies the boundary condition In view of (3.7) and using Fatou's lemma, we get where the finiteness of the right hand side of (3.8c) follows from (2.4) and (2.5). Therefore, from (2.5) and (3.8), we get the weak lower semi-continuity of the functional I, i.e., Using (3.9), together with (3.5), gives Moreover, the finite-energy condition also holds from (3.7) and (3.8a)-(3.8c). In particular, ru 2 , u 2 r , ru 2 r , and ln(1 + αu 2 ) are all in L(0, R). To show that u(0) = 0, we follow as in [39]. Let {u j } be a sequence in W 1,2 (ǫ, R) where ǫ ∈ (0, R). For any ǫ ∈ (0, R), {u j } is bounded in W 1,2 (ǫ, R). The compact embedding Thus, for any pair r 1 , r 2 ∈ (0, R) such that r 1 < r 2 and using C from (3.7), we get Since u j → u uniformly as j → ∞, we take j → ∞ above to get The right hand side of (3.11) goes to zero as r 1 , r 2 → 0; since u 2 r is in L(0, R). Hence, the following limit exists, As a consequence, the boundary condition u(0) = 0 is achieved.
Therefore, the function u, obtained as the limit of the minimizing sequence {u j }, is a solution to the constrained minimization problem (3.5), and there is a real number κ such that (u, κ) satisfies (3.1).
Further, we may suppose that there is a point r 0 ∈ (0, R) such that u(r 0 ) = 0. Since r 0 would be a minimum point for u(r), we have u r (r 0 ) = 0. However, by the uniqueness theorem of the initial value problem of ordinary differential equations, u(r) = 0 for all r ∈ (0, R), thus contradicting the energy flux constraint Q(u) = Q 0 > 0. Hence, u(r) > 0 for all r ∈ (0, R). A standard bootstrap method may then be used to conclude that u is a classical solution of (1.5).
(ii) Let (u, κ) be the solution pair obtained in part (i). Using u 4 1+αu 2 ≤ u 4 in (2.10), we get We treat u as a radially symmetric function defined over R 2 with its support contained in the disk B R . From the classical Gagliardo-Nirenberg inequality over R 2 , we write As a result, we get Rearranging the terms in (3.15), and using the prescribed energy flux constraint gives as claimed. (i) The wave propagation constant satisfies (ii) If the vortex winding number satisfies |n| ≥ Q 0 /π, then κ < 0.
(iii) For κ > 0, the solution pair (u, κ) satisfies for r sufficiently large and C κ > 0 is a constant depending on κ only.
Remarks. Beam confinement requires the exponential decay of the soliton amplitude u at infinity [36]. From the exponential decay estimate given in Theorem 3.2 (iii), we see that this occurs for κ > 0. Theorem 3.2 (ii) states that the vortex winding number must satisfy |n| < Q 0 /π. On the other hand, from Theorem 3.1 (ii), if the wave propagation constant is positive, then the prescribed energy flux must satisfy Q 0 > 1 4 . As will be seen in Section 6, the condition on the prescribed energy flux, Q 0 > 1 4 , is not sufficient to conclude that the propagation constant is positive.
When the prescribed energy flux Q 0 is fixed, and the distance from the vortex core R goes to infinity, the necessary condition given in Theorem 2.1, together with the lower bound for the wave propagation constant, gives the inequality which is in agreement with the results of Skryabin and Firth [36], for any self-trapped solutions of their model.

SOLUTIONS ON THE NEHARI MANIFOLD
Recall the action functional I κ : H → R defined as, where |n| ≥ 1 and α > 0.
Standard arguments show that I κ ∈ C 3 (H, R). Also, and ·,· denotes the usual duality between H and its dual space H −1 . Let γ κ : H → R be defined by If u ∈ H\{0} is a critical point of I κ , then for every v ∈ H, I κ (u), v = 0. Setting v = u, it follows that γ κ (u) = 1 2 I κ (u), u = 0. Hence, u ∈ M, and the Nehari manifold contains all nontrivial critical points of I κ on H. Proof. The forward implication follows directly from the definition of the Nehari manifold.
We now justify the other direction. For every u ∈ M, (4.5) By definition of γ κ , for every u ∈ M, we get For any critical point u o ∈ M of I κ | M , there exists a Lagrange multiplier, ξ ∈ R, such that I ′ κ (u 0 ),ũ = ξ γ ′ κ (u 0 ),ũ , for everyũ ∈ H. Using the definition of M, gives As a result, using (4.6), it follows that ξ = 0. Therefore, the critical points of I κ | M are also the critical points of I κ .
Lemma 4.1 indicates that M is a natural constraint for I κ . From the necessary condition obtained in Theorem 2.1, the Nehari manifold contains no critical points of I κ when κ ≥ Lemma 4.2 If the distance from the vortex core satisfies then the Nehari manifold is not empty.
Note that, for every κ > −   (4.17) which is a contradiction. Hence, {u j } ∞ j=1 must be a bounded sequence in H. If necessary, passing to a subsequence, we have u j ⇀ u in H, u j → u in L 2 (B R ), and u j (r) → u(r) a.e. r ∈ [0, R] as j → ∞. Using the dominated convergence theorem, we get It follows for any |n| ≥ 1, Hence, u j → 0 in H, which contradicts (4.14). Therefore, there exists a constant C 3 > 0 such that R 0 ru 2 1 + αu 2 dr ≥ C 3 for every u ∈ M. (4.21) Using (4.6), and Holder's inequality, we get The set M is a paracompact and complete topological space.
Proof. The paracompactness of the set M follows identically to the proof in Lemma 3.6, [24]. To show that M is complete, we let {u j } ∞ j=1 be a sequence in M such that u j → u in H as j → ∞. From the compact embedding, W 1,2 (B R ) ⊂⊂ L p (B R ) for p ≥ 1, we note that Since ||u|| H ≥ C 1 > 0, by (4.14), we conclude that u ∈ M.
Similarly to [24], using Lemma 4.3, we may deduce that M is a regular C 2 -Banach manifold and, moreover, using Lemma 4.4 that M is a Finsler manifold. We now look for nontrivial solutions of the n-vortex equation (1.5), as critical points of I κ restricted to the manifold M.
The boundedness of {u j } ∞ j=1 and (4.32), gives then applying (4.34), we have Using the definition of I κ and γ κ , we obtain Applying the dominated convergence theorem, we get We use the evenness of the functional I κ to get a positive solution. Moreover, u(r) > 0 for all r ∈ (0, R). Suppose there is a point r 0 ∈ (0, R) such that u(r 0 ) = 0. Then r 0 would be a minimum point for u(r) and u r (r 0 ) = 0. By the uniqueness theorem of the initial value problem of ordinary differential equations, u(r) = 0 for all r ∈ (0, R), thus contradicting the fact that u ∈ M.

FINITE ELEMENT FORMALISM
We utilize the variational principle used in Section 3 and a finite element formalism to compute the solution pair (u, κ) to the problem (1.5), for a prescribed energy flux (1.6). We use MATLAB's Optimization Toolbox [26] and the Chebfun package [13] to solve (5.3). In particular, we obtain a minimum using the objective function as In order to compute the wave propagation constant, we use a Lagrange multiplier λ ∈ R, such that I(u),ũ = λ (u),ũ . More explicitly, there exists a λ ∈ R such that As in [36], we consider the case when the saturation constant, vortex winding number, and distance from the vortex core are: α = 0.1, n = 1, and R = 8, respectively. With this particular choice of parameter values and using Theorem 2.1 and Theorem 3.2(iii), we get 0 < κ < 9.9470. (5.6) Equivalently, using (5.5), we get that the prescribed energy flux satisfies 13.6 < Q 0 < ∞.
The inequalities (5.6) and (5.7) are the necessary conditions for postive exponentially decaying solutions. This in turn, numerically demonstrates that Q 0 > 1 4 , as remarked in Section 3, does imply that κ > 0. We also analyze the behavior of the solution pair (u, κ) for a fixed value of the energy flux Q 0 by varying the vortex winding number n (see Figure 2). Particularly, when α = 0.1 and R = 8, Theorem 2.1 states that the wave propagation constant κ must satisfy κ < 10 − (r 2 0 + n 2 )/128, which imposes an upper limit on the vortex winding number of |n| < 1280 − r 2 0 ≈ 35.6962 (i.e., |n| ≤ 35). However, for exponentially decaying solutions, i.e., κ > 0, owing to Theorem 3.2(ii), the vortex winding number is bounded above by Q 0 /π.
Consequently and for example, using Q 0 = 10π, we get |n| < 10 as a necessary condition for positive exponentially decaying solutions. We remark that our numerical approach is in contrast with that of Skryabin and Firth [36]. We compute the wave propagation constant for a prescribed energy flux (see Figure   2). On the other hand, Skryabin and Firth in [36], compute the soliton's amplitude for a prescribed propagation constant and then use (1.6) to determine its corresponding energy flux. Moreover, the vortex winding number must satisfy |n| < Q 0 /π (see Theorem 3.2(ii)) and the prescribed energy flux Q 0 > 1/4 (see Theorem 3.1(ii)). Further, no smallenergy-flux solutions exists for κ > 0 when Q 0 ≤ 1/4 (see Theorem 3.1(ii)).
2. The existence of a positive solution is guaranteed by Theorem 3.1(i), however, the propagation constant κ is undetermined. A lower bound for κ is provided by Theorem 3.2(i), and an upper bound by Theorem 2.1.
3. On a Nehari manifold, if the distance from the vortex core R is sufficiently large, then for any propagation constant satisfying (6.1), a positive exponentially decaying solution exists (see Theorem 4.7 and Theorem 3.2(iii)).
4. Using a finite element formalism, we compute the soliton's amplitude and wave propagation constant for a prescribed energy flux. The numerical analysis shows that the wave propagation constant increases as the energy flux increases and decreases as the vortex winding number increases. Moreover, for given parameter values α, n, and R, we are able to numerically obtain a necessary condition for the existence of positive exponentially decaying solutions in terms of a prescribed energy flux (see (5.7)).