Dynamics of nonlinear waves in two-dimensional cubic-quintic nonlinear Schr&#x00F6;dinger equation with spatially modulated nonlinearities and potentials

Si-Liu Xu; Jia-Xi Cheng; Milivoj R. Belić; Zheng-Long Hu; Yuan Zhao

doi:10.1364/OE.24.010066

1. Introduction

Solitons are prominent examples of nonlinear solitary waves. They emerge from the balance between nonlinearity and dispersion during propagation, and preserve their localized shapes and velocities in propagation and after collisions [1]. In Kerr media, the nonlinear Schrödinger equation (NLSE) is the most often used model to describe optical spatial solitons of nonspreading wave packets [1,2]. In (1 + 1) dimensions (1D) NLSE, solitons have been studied extensively and are understood quite well, both in theoretical and experimental aspects [3]. In contrast to 1D, relatively less research work on the higher-dimensional NLSE has been carried out. Perhaps a part of the reason is that all localized solutions are usually unstable in the 2D and 3D NLSEs with constant-coefficients, due to strong or weak collapse [4].

However, the stability of solitons in 2D and 3D can be improved using different methods, such as soliton management [5] or by using optical lattices in BECs [6] and periodic lattice potentials in optical media [7,8]. Up to now, various types of robust soliton clusters have been found in both 2D and 3D physical settings [9–12]. Many of these works have been utilizing the cubic nonlinear Schrödinger equation (CNLSE), which models a wealth of phenomena in nonlinear science and physics. On the other hand, when the intensity of an optical pulse exceeds a certain value, higher-order nonlinear effects such as the quintic nonlinearity should be taken into account [13,14]. This may arise due to an intrinsic nonlinear resonance in the material, which also occurs in the strong two-photon absorption [15]. Compared with the CNLSE, the cubic-quintic nonlinear Schrödinger (CQNLS) equation has received less attention, although there is a considerable relevance of the equation, both from the mathematical and physical points of view.

During the past several years, many interesting works have been reported constructing exact analytical solutions to the CQNLSE. For example, the works of Serkin et al. [16] and Hao et al. [17], which reported analytical solitary wave solutions by supposing an ansatz solution. J.D. He et al. [18] found the explicit spatial self-similar bright and dark soliton solutions of the cubic-quintic nonlinear Schrdinger equation with distributed coefficients and an external potential, and R.M. Caplan et al. discussed the issues of existence, interactions, and stability of solitary vortices in the two-dimensional CQNLS equation, using both analytical and numerical methods [19].

Here, we extend the previous work on the CNLS equation and study analytical solutions of the 2D CQNLS equation with space-dependent diffraction and nonlinearity coefficients. We employ the self-similarity transformation to transform the space-dependent model into the standard CQNLS equation with constant coefficients and find analytic solutions.

The paper is structured as follows. In Sec. II, the similarity method given in Refs [20,21]. is extended to treat Eq. (1). The generalized CQNLS equation with variable coefficients is reduced to the standard CQNLS equation and analytical resonance solitary wave solutions are found. In Sec. III, these solitary waves are discussed in some detail, for different choices of the solution parameters. Numerical simulation and comparison with the analytical results is also carried out. In Sec. IV, the conclusion of the paper is presented briefly.

2. The model and exact soliton solutions

The propagation of an optical electromagnetic field in a bulk nonlinear medium in the presence of a space-modulated photonic lattice is described by the following CQNLS equation in appropriately chosen dimensionless coordinates [22]:

i \frac{\partial ψ}{\partial z} + \frac{1}{2} \nabla ψ + χ_{1} (z, r) | ψ |^{2} ψ + χ_{2} (z, r) | ψ |^{4} ψ + V (z, r) ψ = 0,

where z is the propagation (or the evolution) coordinate,

\nabla = \frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial φ^{2}}

is the transverse 2D Laplacian with the transverse radial coordinate

r = \sqrt{x^{2} + y^{2}}

and the azimuthal angle

Φ (φ) = \cos (s φ) + i μ \sin (s φ)

, and

V (z, r)

describes a real external potential, which is to be specified later. The real functions

χ_{1} (z, r)

and

χ_{2} (z, r)

stand for the cubic and quintic variable nonlinearity coefficients, which will also be specified later. The diffraction coefficient in the second term in Eq. (1) has been normalized.

We search for the axisymmetric cylindrical-beam solutions of Eq. (1) in the form $ψ (z, r, φ) = u (z, r) Φ (φ)$ . The azimuthal part of the solution is of the form $Φ (φ) = \cos (m φ) + i q \sin (m φ)$ [23], where $m$ is a non-negative integer (topological charge, TC), which can be considered as an azimuthal mode number, and $q \in [0, 1]$ is the modulation depth parameter of the beam intensity. It should be noted that the azimuthal part is only an approximate solution of Eq. (1), valid for weak nonlinearities or for large values of $q$ (close to 1); it is exact only for q = 1, when Φ = exp(imφ). This is because the ${| u |}^{2}$ and ${| u |}^{4}$ terms in the nonlinearity retain the $φ$ -dependence and spoil the assumed separation of variables. We still employ this general form of the function Φ(φ) and derive an equation for u in which the inﬂuence of φ is averaged out, in the spirit of the mean-ﬁeld approximation. Inserting the ansatz for $ψ$ into Eq. (1), integrating over $Φ_{j} (φ) = A_{n} \cos (m φ) + B_{n} \sin (m φ)$ from $A_{n}$ to $B_{n}$ , one readily derives an averaged equation for u:

i \frac{\partial u}{\partial z} + \frac{1}{2} (\frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} - \frac{m^{2}}{r^{2}} u) + χ_{1}^{'} (z, r) | u |^{2} u + χ_{2}^{'} (z, r) | u |^{4} u + V (z, r) u = 0.

Here,

χ_{1}^{'} (z, r) = (1 + q^{2}) χ_{1} (z, r) / 2

and

χ_{2}^{'} (z, r) = (3 + 2 q^{2} + 3 q^{4}) χ_{2} (z, r) / 8

are the redefined nonlinearity coefficients. For q = 1, the redefined coefficients are the same as the original coefficients.

To transform Eq. (2) into the standard CQNLS equation [24] with constant coefficients, we use the following form of this equation:

U_{R R} = - E U + g_{0} | U |^{2} U + G_{0} | U |^{4} U,

where both the solution

U = U (R)

and the self-similar variable

R = R (z, r)

are real functions (to be specified), E corresponds to the eigenvalue of the nonlinear equation, and

g_{0}

and G₀ are constant. As it is well known, Eq. (3) possesses various exact solutions, formed from different combinations of the Jacobi elliptic functions [25].

Henceforth, we explore the ground energy case, $E = 0$ , for which one obtains [25]

U (R) = \frac{3 s n (μ R, m_{0})}{\sqrt{3 a_{0} [3 - (m_{0}^{2} + 1) s n^{2} (μ R, m_{0})]}} .

Here

g_{0} = - \frac{2}{3} a_{0} μ^{2} (m_{0}^{4} - m_{0}^{2} + 1)

,

G_{0} = - \frac{1}{9} a_{0}^{2} μ^{2} (m_{0}^{2} - 2) (2 m_{0}^{2} - 1) (m_{0}^{2} + 1)

. If we consider the boundary conditions

ψ (r \to 0) = ψ (r \to \infty) = 0

for the localized solution,

μ

should satisfy

μ R (\infty) = 2 n K (m_{0})

, where

K (m_{0}) = {\int_{0}^{π / 2} [1 - m_{0}^{2} \sin^{2} τ]}^{- 1 / 2} d τ

is the complete elliptic integral of the first kind. Of the other two parameters,

a_{0}

is a constant and m₀ is the modulus of the Jacobi elliptic function. From Eq. (4a), one can see that if

a_{0} > 0

,

g_{0} < 0

is given for any m₀. Further, it is found that

G_{0}

is negative for

0 < m_{0} < \sqrt{2} / 2

and positive for

\sqrt{2} / 2 < m_{0} < 1

.

On the other hand, if $E \neq 0$ , the following solution of Eq. (3) is constructed [25]:

U (R) = \frac{s n (μ R, m_{0})}{\sqrt{a_{0} + a_{1} d n^{2} (μ R, m_{0})}},

where

μ = \sqrt{- E (a_{0} + a_{1}) / 3 (2 m_{0}^{2} a_{1} - m_{0}^{2} a_{0} - a_{0} - a_{1})},

g_{0} = 2 E m_{0}^{2} (a_{0}^{2} - 2 a_{1} a_{0} m_{0}^{2} - a_{1}^{2} + a_{1}^{2} m_{0}^{2}) / (m_{0}^{2} a_{0} - 2 m_{0}^{2} a_{1} + a_{0} + a_{1}),

G_{0} = 3 E a_{1} a_{0} m_{0}^{4} (a_{0} + a_{1} - a_{1}^{2} m_{0}^{2}) / (m_{0}^{2} a_{0} - 2 m_{0}^{2} a_{1} + a_{0} + a_{1}),

with constant

a_{0}

and

a_{1}

. To concatenate the solution of Eq. (2) with the one of Eq. (3), we use the similarity transformation

u (z, r) = A (z, r) U [R (z, r)] e^{i Θ (z, r)}

, where the amplitude

A (z, r)

and the phase

Θ (z, r)

are real functions of z and r [26]. Note that

u (z, r)

is a solution of Eq. (2) and

U (R)

satisfies Eq. (3). Substituting u into Eq. (2), one obtains:

\frac{\partial A}{\partial z} + \frac{1}{2} (A \frac{\partial^{2} Θ}{\partial r^{2}} + 2 \frac{\partial A}{\partial r} \frac{\partial Θ}{\partial r} + \frac{A}{r} \frac{\partial Θ}{\partial r}) = 0, \frac{\partial R}{\partial z} + \frac{\partial R}{\partial r} \frac{\partial Θ}{\partial r} = 0

- A \frac{\partial Θ}{\partial z} + \frac{1}{2} [\frac{\partial^{2} A}{\partial r^{2}} - A {(\frac{\partial Θ}{\partial r})}^{2} + \frac{1}{r} \frac{\partial A}{\partial r} - A \frac{m^{2}}{r^{2}}] + A V - E A {(\frac{\partial R}{\partial r})}^{2} = 0,

2 \frac{\partial A}{\partial r} \frac{\partial R}{\partial r} + A \frac{\partial^{2} R}{\partial r^{2}} + \frac{A}{r} \frac{\partial R}{\partial r} = 0, χ_{1}^{'} A^{2} = g_{0} \frac{\partial^{2} R}{\partial r^{2}}, χ_{2}^{'} A^{4} = G_{0} \frac{\partial^{2} R}{\partial r^{2}}

To find exact solutions of Eqs. (6a)-(6c), we introduce another self-similar transformation [27,28]

A (z, r) = k_{1} F (θ) / w (z)

,

Θ (z, r) = a (z) r^{2} + b (z)

. Here,

k_{1}

is the normalization constant,

w (z)

is the beam width,

θ (z, r)

is the similarity variable to be determined,

a (z)

is the wave front curvature, and

b (z)

represents the phase offset. These variables vary with the propagation distance z. Substituting the presumed solutions for the amplitude and phase into Eqs. (6a)-(6e), one obtains

θ (z, r) = r^{2} / w^{2} (z)

,

a (z) = \frac{d w}{2 w d z}

,

R (z, r) = \int_{0}^{r^{2} / w^{2}} \frac{1}{τ F^{2} (τ)} d τ

,

χ_{1} (z, r) = - \frac{2 g_{0}}{(1 + q^{2}) A^{2}} {(\frac{\partial R}{\partial r})}^{2}

, and

χ_{2} (z, r) = - \frac{8 G_{0}}{(3 + 2 q^{2} + 3 q^{4}) A^{4}} {(\frac{\partial R}{\partial r})}^{2}

. Thus, the nonlinearity coefficients are specified in terms of the beam characteristics and modulation depth.

Further, from Eq. (6b), one can find the amplitude A(z,r), which is transformed into the following nonlinear differential equation for $F (θ)$

θ \frac{d^{2} F}{d θ^{2}} + \frac{d F}{d θ} - (θ \frac{w^{3}}{4} \frac{d^{2} w}{d z^{2}} + \frac{m^{2}}{4 θ}) F - \frac{w^{2} F}{2} \frac{d b}{d z} - \frac{w^{2}}{2} [V + E {(\frac{\partial R}{\partial r})}^{2}] F = 0.

At this point, we also specify the trapping potential as:

V = s r^{2} - E {(d R / d r)}^{2}

, where s is a positive constant. This leads to the simplification of Eq. (7) and allows its solution. After a variable transformation

F (θ) = θ^{\frac{m}{2}} e^{- \frac{θ}{2}} f (θ)

, Eq. (7) transforms into:

θ \frac{d^{2} f}{d θ^{2}} + (m + 1 - θ) \frac{d f}{d θ} - n f = 0,

with

- \frac{w^{2}}{2} \frac{d b}{d z} - \frac{m + 1}{2} = n

, and

- \frac{w}{2} \frac{d w^{2}}{d z^{2}} - s w^{2} + \frac{1}{2 w^{2}} = 0

. Here, n is assumed to be a non-negative integer, which can be considered as the radial mode number. Differential Eq. (8) is known as the confluent hypergeometric differential equation and its solutions are the Sonine functions [29], namely

f (θ) = S_{n}^{m} (θ)

with

S_{n}^{m} (θ) = \sum_{k = 0}^{n} {(- 1)}^{k} \frac{1}{k! (n - k)!} \frac{(m + n)!}{(m + k)!} θ^{k}

. We take

w (z) |_{z = 0} = w_{0}

and

\frac{d w (z)}{d z} |_{z = 0} = 0

, where the subscript '0' denotes the value of the corresponding quantity at

z = 0

. Hence, the exact soliton solution is obtained with

w = w_{0}

, thus the beam diffraction is exactly balanced by the nonlinearity. Other parameters in this case are:

s = \frac{1}{\sqrt{2} w_{0}^{2}}

,

a (z) = 0

, and

b (z) = b_{0} - (2 n + m + 1) \frac{z}{w_{0}^{2}}

.

Plugging these results into Eq. (1), one obtains the following analytical resonance soliton solution:

ψ (z, r, φ) = [\cos (m φ) + i q \sin (m φ)] \frac{k_{1}}{w_{0}} {(\frac{r}{w_{0}})}^{m} e^{- \frac{r^{2}}{2 w_{0}^{2}}} S_{n}^{m} (\frac{r^{2}}{w_{0}^{2}}) U [R (r)] e^{i [b_{0} - \frac{(2 n + m + 1)}{w_{0}^{2}} z]} .

where

k_{1} = \sqrt{\frac{n!}{Γ (n + m + 1)}}

. It is an approximate solution, as long as it concerns the value of parameter q, as explained above. We take b₀ = 0, without loss of generality.

A note of explanation is needed here. The nonlinearity coefficients and the external potential in Eq. (1) are given in terms of the radial solution of the equation. This requires a self-consistent solution procedure, in which the conditions put on the coefficients and the potential can be considered as integrability constraints. In other words, it is an inverted solution procedure, in which one first finds the solution of the equation and then defines the coefficients and the potential in the equation in terms of the solution itself. Thus, it is a convoluted procedure whose generality may be questioned, but which nonetheless offers the convenience of having localized solutions with desirable features of equations with coefficients and potentials that maybe are related to the solutions, but in a self-consistent procedure might lead to interesting models with realistic space-varying coefficients and potentials. While one may argue about the availability of such coefficients and potentials, we believe that there is enough freedom in the solution method, to provide interesting solutions to viable material models.

3. The stability of the analytical solution

Next, we study the linear stability of solutions of Eq. (1). The perturbations in the solutions are chosen in the form [30]:

ψ (x, y, z) = e^{i λ z} {\bar{ψ} (x, y) + ε [g (x, y) + h (x, y)] e^{i δ z}},

where

\bar{ψ} (x, y) = ψ (r, φ)

is the steady soliton solution of Eq. (1), λ is the propagation constant,

ε

is an infinitesimal amplitude,

g (x, y)

and

h (x, y)

are the real and imaginary parts of the perturbation, and

δ

denotes the perturbation growth rate. Substituting the perturbed

ψ (x, y, z)

into Eq. (1) and then linearizing it around the unperturbed solution (to the first-order in

ε

), one obtains the following eigenvalue equations:

(\begin{array}{l} L_{+} \\ 0 \end{array} \begin{array}{l} 0 \\ L_{-} \end{array}) (\begin{array}{l} g \\ h \end{array}) = δ (\begin{array}{l} g \\ h \end{array}),

where

L_{+} = - \frac{1}{2} (\partial_{x x} + \partial_{y y}) + 3 χ_{1} {\bar{ψ}}_{}^{2} + 5 χ_{2} {\bar{ψ}}_{}^{4} + V - λ, L_{-} = - \frac{1}{2} (\partial_{x x} + \partial_{y y}) + χ_{1} {\bar{ψ}}_{}^{2} + χ_{2} {\bar{ψ}}_{}^{4} + V - λ

This eigenvalue problem can be computed by the Fourier Collocation Method for discrete eigenvalues, hence the whole spectrum can be calculated at once [31]. From Eq. (9), one can find the propagation constant $λ = - (2 n + m + 1) / w_{0}^{2}$ . Without loss of generality, here we let $w_{0} = 1$ . Once parameters m, and n are chosen, from Eq. (9) one can get the imaginary and real parts of $δ$ . If imaginary parts of all eigenvalues $δ$ are equal to zero or are positive, the soliton solutions can be stable; otherwise, the perturbed solution would grow exponentially with z, and thus, the corresponding solitons would become linearly unstable.

4. Characteristic distributions of resonance solitons

From Eq. (9), one can see that the novel solitons are characterized by three parameters: the mode numbers n and m, and the modulation depth q. When $w = w_{0}$ , the auxiliary function R, the cubic-quintic nonlinearity coefficients $χ_{1}$ , $χ_{2}$ , and the trapping potential V are the functions of the radial variable r only. Typical distributions of the nonlinearity coefficients $χ_{1} (r)$ , $χ_{2} (r)$ and the external potential V (r) with respect to the radial coordinate r are shown in Fig. 1. Without much loss of generality, in this paper we will study the cases when n and m are different non-negative integers. On the account of values of these parameters, the characteristics of the new 2D resonance solitons are analyzed. Furthermore, from Eq. (9) it is evident that $\lim_{| r | \to \infty} ψ (z, r, φ) = 0$ , thus, the solutions are localized. In this paper we will consider the cases $E = 0$ , and $E \neq 0$ , with the parameters n, m and $q$ being varied. Note that the resonance solitons presented by Eq. (9) have intensity distributions that do not depend on z, and the phase that changes linearly with z. These solutions might be stable or unstable, depending on the spectrum of the eigenvalue $δ$ . Thus, in all the figures we will consistently present the intensity and phase distributions of resonance solitons, usually at z = 0, and the spectra of $δ$ .

Fig. 1 Distributions of the nonlinearity coefficients $χ_{1} (r)$ , $χ_{2} (r)$ and the external potential $V (r)$ for the resonance soliton from Eq. (9). Parameters: $w_{0} = a_{0} = a_{1} = 1$ , $q = 0.2$ , $m_{0} = 0.1$ , $m = 0$ , $n = 1$ and $s = 0.1$ .

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First, we begin by analyzing the 2D resonance soliton with different m, when $n = 2$ , $n_{0} = m_{0} = 1$ , and $0 \geq q \geq 1$ . Figures 2 and 3 display the intensity profiles of vortex resonance solitons for $E = 0$ (Fig. 2) and $E \neq 0$ (Fig. 3). The profiles of vortex solitons possess several amplitude peaks covering a ring-like substrate. The number of amplitude peaks is determined by the azimuthal index m. Such a localized solution displays a necklace-type self-trapped structure; the number of “petals” is 2m in each necklace ring and the total intensity distribution exhibits similar vortex profile. Further, the intensity of rings surrounding the center decreases with the increasing radial distance. A noticeable difference between Figs. 2 and 3 is that the intensity of $E \neq 0$ is reduced faster and is more strongly localized in the transverse plane, and the thickness of “petals” in each necklace ring is smaller than that of $E = 0$ . In the middle row of panels in Figs. 2 and 3 we depict the phase distributions of the soliton solution. The phase pattern changes in the course of evolution similar to the vortex soliton, developing a gradient phase change in a spiral form. It is clear that the vortex solitons exhibit spatially modulated patterns; here the amplitude and phase distributions of multipeaked vortex solitons are similar to those of azimuthons [32]. The linear-stability spectra are displayed in the bottom row of Figs. 2 and 3. It is seen that when $m = 1$ , eigenvalue $δ$ develops real values, hence these types of solitons might be stable. On the other hand, if $m = 2, 3$ , we note a pair of or a quadruple of complex eigenvalues $δ$ visible in Figs. 2 (h)-(i) and Figs. 3 (h)-(i), thus these solitons might be linearly unstable.

Fig. 2 Intensity profiles (top row) of resonance solitons for the cubic-quintic nonlinearities and potentials at the propagation distance $z = 50$ for E=0 and different m; Phase distributions (middle row); Linear-stability spectra (bottom row). The parameters are: $n = 2$ , $q = 0.5$ , $w_{0} = 1$ and $m = 1, 2, 3$ from left to right.

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Fig. 3 Same as Fig. 2, but for $E = - 1$ , and $a_{1} = 1$ .

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Figures 4 and 5 display the lower order vortex-shaped distributions of the beams for $m = 1$ with $n = 0, 1, 2$ , and $q = 0.5$ . It is seen that the soliton profiles are self-similar and that the solitons are composed of two symmetrical half-moon petals in the top row in Fig. 4 ( $E = 0$ ) and Fig. 5 ( $E \neq 0$ ). Similar to Figs. 2 and 3, the number of outer bright rings is $n + 1$ , the optical intensity in the center is zero, and the intensity of rings surrounding the center decreases with the increasing radial distance. Further, one can see that the intensity of the soliton with $E = 0$ is smaller than that of $E \neq 0$ . The middle rows in Figs. 4 and 5 display the phase distributions of solitons. One can note that in the radial direction the phases are made of (n + 1) layers, the gradient change in phases can be seen along angular directions. The bottom row in Figs. 4 and 5 display the stability analysis of the solitons. It is visible that eigenvalue $δ$ has a real value, hence these types of solitons might be linearly stable.

Fig. 4 Same as Fig. 2, but for $m = 1$ , $n = 0, 1, 2$ from left to right.

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Fig. 5 Same as Fig. 4, but for $E = - 1$ , and $a_{1} = 1$ .

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Increasing the modulation depth q from 0 to 1, with fixed n and m, angularly modulated vortex rings are obtained (see the top rows of Figs. 6 and 7). One can observe that the intensity and phase of the beam is modulated by the modulation depth q. Increasing q, the distance between the petals decreases, and the multi-TC vortices change into vortex rings. Further, with increasing q, the changes, an angular gradient changing phase pattern is formed (see the middle of Figs. 6 and 7). The linear-stability spectra are displayed in the bottom rows of Figs. 6 and 7. In Fig. 6 with $E = 0$ , it is shown that the solitons are stable for any values of q, however, for $E = - 1$ , we note that a pair of real eigenvalues bifurcates from zero eigenvalue, thus the soliton can display instability for $q = 1$ .

Fig. 6 Same as Fig. 2, but for $n = 1$ , $m = 2$ , and $q = 0.1, 0.5, 1$ from left to right.

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Fig. 7 Same as Fig. 5, but for $E = - 1$ .

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A very important aspect of the present issue is the stability of the obtained exact solutions. This aspect of the problem can also be addressed numerically. In order to check the stability of the soliton, we perform direct numerical simulations using the split-step Fourier method [33] and solve Eq. (1) by taking the analytical solution (9) at $z = 0$ as an initial condition. In Fig. 8, we present the comparison of analytical (the first column) and numerical (the second and the third columns) intensity distribution contour plots in the x-y plane, with $E = 0$ . We find that only when the topological charge is $m = 0, 1$ and $n \leq 2$ , the numerical solution is stable against perturbation with an initial Gaussian noise level of 6%. But when the topological charge is $m \geq 2$ , the soliton solution (9) is unstable in propagation and splits into necklace ring-shaped structures.

Fig. 8 Comparison of analytical (the first column) and numerical (the second and the third columns) intensity distribution contour plots at different distances $z = 10, 100, 160$ in the x-y plane and with a white noise of variance $σ = 0.06$ added to the numerical input. The parameters are $E = 0$ , $q = 0$ , $n = m = 0$ (the first row), $n = m = 1$ (the second row), $n = m = 2$ (the third row).

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4. Conclusion

In summary, we have investigated the propagation behavior of space-modulated resonance solitons in CQNLS equation. Using the three parameters, the intensity and the phases of resonance solitons with $E = 0$ , and $E \neq 0$ are displayed. The stability of the solitons is examined by the linear stability analysis and by a direct numerical simulation. Our results show that the resonance solitons with $m \leq 1$ are stable, and for higher topological charges ( $m \geq 2$ ), they are unstable. These solitary waves propagate self-similarly. Besides nonlinear optics, our approach can be applied to other fields governed by similar CQNLSEs that may display similar localized solutions, e.g. Bose-Einstein condensation and light propagation in plasmas.

Acknowledgments

This work is supported in China by the Natural Science Foundation of Hubei Province in China (Grant no. 2013CFB38), and the Project with Hubei Province Department of Education, under Grant Q20142805. In addition, work in Qatar is supported by the NPRP 6-021-1-005 project with the Qatar National Research Fund (a member of the Qatar Foundation) and in China by the Natural Science Foundation of Guangdong Province, under Grant No. 1015283001000000. MRB also acknowledges support by the Al Sraiya Holding Group.

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Dynamics of nonlinear waves in two-dimensional cubic-quintic nonlinear Schrödinger equation with spatially modulated nonlinearities and potentials

Abstract

1. Introduction

2. The model and exact soliton solutions

3. The stability of the analytical solution

4. Characteristic distributions of resonance solitons

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (17)

Optics Express