1. Introduction

Nonlocal nonlinearity has attracted considerable interest in recent years in the fields of optical spatial shock waves [1] and optical spatial solitons [2], which is featured by nematic liquid crystal [3], lead glass [4], thermal nonlinear liquid [5], and nonlinear ion gas [6]. The evolution of optical beams in nonlocal nonlinear media is modeled by the nonlocal nonlinear Schrödinger equation (NNLSE), in which the nonlinear term is expressed by a convolution between optical intensity and the response function. The ratio of the characteristic length occupied by response function to beam width is termed as the degree of nonlocality. Particularly, when the degree of nonlocality approaches to zero, the NNLSE is reduced to the well known (local) nonlinear Schrödinger equation. The nonlocality plays a crucial role in both solitons’ structure and their interactions [7]. Thanks to the nonlocality, a variety of solitons, including vortex solitons [8], multipole solitons [9], spiralling elliptic solitons [10], chaoticons [11] and so on, can stably exist. Furthermore, the nonlocality can make it possible to overcome repulsion between out-of-phase bright [12,13] or in-phase dark solitons [14] that can form bound complex states.

Generally speaking, for a given nonlinear media the characteristic length of the response function is fixed, which is a constant independent of the propagation distance. However, in actual nonlinear system the characteristic length of response is in fact quite handy to be modulated. For example, in the two representative nonlocal nonlinear media, the nematic liquid crystals and the lead glass, the characteristic length of the response function is found to depend closely on the bias voltage [15] and the sample sizes [4], respectively. Therefore, it will supply a chance to design a $z$-dependent response in experiments along the propagation directions of beams. Evolutions of optical beams in the nonlocal nonlinear media with the $z$-varied response is seldom explored. We find that in the nonlocal nonlinear media of gradual nonlocality optical beams can exhibit a kind of adiabatic evolution, which may have potential use in the light controlling and the beam engineering.

2. Theoretical model

Evolutions of optical beams propagating in nonlocal nonlinear media can be modeled by the following NNLSE [16,17]

Equation (1) can be exactly solved only under two limit cases: the local limit ($w_m\rightarrow 0$) and strongly nonlocal limit ($w_m\rightarrow \infty$). In the former case, the NNLSE will be reduced into the well-known nonlinear Schrödinger equation and has the stable sech-form bright soliton [21]. In the latter case, the NNLSE becomes the Snyder-Mitchell Model and owns the Accessible Solitons [16]. Although the exact solutions of Eq. (1) are difficult to find, the approximate solutions can be obtained by a few techniques, among which the variational approach is the most frequently used method [22].

3. Results and discussions

For the variational approach [22], the choice of the trial function is crucial, which should be close to the exact solution of the partial differential equation. In the case of the Gaussian response, we have demonstrated that the soliton solutions of the NNLSE exhibit the profiles similar to Gaussian ones when the degree of nonlocality goes from the strongly nonlocal limit down to the local limit [2]. Therefore, the ideal trial function is the Gaussian one in the form of

The Lagrangian density of the system described by the NNLSE (1) is [2,23] $\mathcal {L}=\frac {i}{2}(\varphi ^{\ast }\frac {\partial \varphi }{\partial z}-\varphi \frac {\partial \varphi ^{\ast }}{\partial z})-\frac {1}{2}|\frac {\partial \varphi }{\partial x}|^{2}+\frac {1}{2}|\varphi |^{2}\Delta n.$ Substitution of the trial solution (3) into the Lagrangian density yields the $L=\int _{-\infty }^{\infty }\mathcal {L}dx=\sqrt {\pi } A^2 \left (-1-4 c^2 w^4+\frac {\sqrt {2} A^2 w^3}{\sqrt {w^2+w_m^2}}-2 w^4 c'-4 w^2 \psi '\right )/2 w,$ where the primes indicate derivatives with respect to the variable $z$. Detailed applications of the variational approach into the optical beams propagations in nonlocal nonlinear media can be found in Refs. [2,23,24]. The variation of $L$ with respect to $\psi$ yields the $dP/dz=0$, indicating that the optical power $P=\int |\varphi |^2dx=\sqrt {\pi }A^2w$ is conservative. The variation to $c$ results in

While, from variations to $A$ and $w$ we can obtain

Substitution of Eq. (5) into the first derivative of Eq. (4) gives the variational results about the evolutions of beam width

It is a little difficult to find the analytical expression of the solution to the differential Eq. (6) due to the dependence of $w_m$ on $z$. Even so, Eq. (6) can be easily solved numerically, which also can reveal the evolution characteristics of optical beams in nonlocal nonlinear media with gradual nonlocality.

For a constant $w_m$, the solutions of the NNLSE (1) can be solitons when the optical power $P_0$ equals to the critical one $P_s=\sqrt {2\pi }\left (w_m^2+w^2\right )^{3/2}/w^4$ or breathers when $P_0\neq P_s$ [20]. In other words, a beam can stay in soliton or breather states all the time as long as $w_m$ does not change. However, when $w_m$ varies, the beam can not maintain its status any more. One interesting issue naturally arises: can a beam recover to its initial status if $w_m$ periodically changes to its starting values.

Let’s start with the simplest case, that is, $w_m=w_{m0}$ when $z\in [0,z_0]$; $w_{m0}-p(z-z_0)$ when $z\in [z_0,2z_0]$; $w_{m0}+p(z-3z_0)$ when $z\in [2z_0,3z_0]$; and $w_{m0}$ when $z\in [3z_0,4z_0]$. The variation of $w_m$ is linear with respect to $z$ as shown in Fig. 1(a). The beam is set to the soliton state by $P_0=P_s$, which evolves invariably until $w_m$ changes at $z=z_0$.After that point, $w_m$ is gradually increased, the balance between diffraction and focusing effects breaks down, and the beam can not sustain its soliton state. The diffraction effect dominates, then the beam intensity decreases, which is shown in Fig. 1(b), which can be briefly explained in the following. It is known that the nonlocality tends to average the nonlinear refraction [7]. Therefore, when $w_m$ increases, the nonlocality will become stronger, and the nonlinearity experienced by the beam will be averaged and become weaker, then the diffraction effect dominates. As $w_m$ decreases from its maximum, the inverse evolution appears. Interestingly, when $w_m$ recovers to its initial value, the soliton state comes back. It should be noted that Eq. (1) is a nonlinear differential equation rather than a linear one. Generally speaking, when the parameters of a nonlinear differential equation change, the solutions should change accordingly. For comparison, we investigate the other case shown in Fig. 1(c), in which $w_m$ is modulated abruptly. In this case, the beam can not evolve into its original state when $w_m$ recovers to its initial value, which is shown in Fig. 1(d).

 

Fig. 1. Gradual [(a)] and abrupt [(c)] variation of $w_m$, and the induced evolutions of initial solitons [(b) and (d)] when $P_0=P_s$.

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In the case of $P_0\neq P_s$, the solutions of the NNLSE (1) are breathers [16], the propagation states an optical beam exhibits periodically between spreading and contraction. Like the case of solitons discussed above, when the characteristic length $w_m$ varies abruptly, the initial breathers can not maintain either as shown in Fig. 2. Particularly, spiky structures appear during propagations after $w_m$ goes back to its initial value as shown in Fig. 2(b). While, if the changes of $w_m$ are slowing down, the breathers can also revert back to their original states as shown in Fig. 3. The amplitudes of $w$ before and after the changes of $w_m$ are a little different, which is mainly because the variation of $w_m$ is not slow enough. The oscillation frequency of breathers can be roughly obtained from Eq. (6)

Equation (7) reveals that the oscillation frequency $\Omega$ increases as the diminution of $w_m$, which can be confirmed in both Fig. 2(c) and Fig. 3(a).

 

Fig. 2. Gradual [blue dotted curve in (a)] and abrupt [red dashed curve in (a)] variation of $w_m$, and the induced evolutions of beam intensity [(b)] and beam width [(c)] for initial breathers when $P_0=1.5P_s$ in the case of abruptly varying $w_m$. The evolution shown in (b) and (c) correspond to the case marked by the blue dotted curve in (a).

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Fig. 3. Evolution of beam width [(a)] and intensity [(b)] at gradually varied $w_m$ given by Fig. 2(a) [blue dotted curve] for initial breathers when $P_0=1.5P_s$. The insets in (a) shows the comparison between numerical simulations [red solid curves] and variational results [blue dashed curves]. Inset in (b) indicates the periodically oscillating characteristic of breather intensity.

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From the discussions above, it can be found that both solitons and breathers can restore their initial states if the characteristic length $w_m$ gradually changes. We can term this phenomenon the "adiabatic evolution" by borrowing the concept related to the adiabatic theorem of Hamiltonian in quantum systems [25]. The adiabatic evolution essentially originates from the transform invariance of the nonlocal nonlinear system [26]. When $w_m$ gradually changes, the variation of $w_m$ can be considered as a perturbation to the system, then one solution of the NNLSE can gradually transit to the other one. If $w_m$ gradually recovers to its initial value, the solution will also restore again. This kind of adiabatic evolution of optical beams in nonlocal nonlinear media is not determined by the specific functional form of $w_m$ with respect to $z$, while the only requirement is that the variation of $w_m$ must be slow enough.For example, we can set $w_m$ to

 

Fig. 4. "sin-" [(a)] and "cos-" [(d)] modulated $w_m$, and the induced evolutions of beam intensity [(b) and (e)] and beam width [(c) and (f)] for initial breathers when $P_0=2P_s$.

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In order to further confirm the requirement by the adiabatic evolutions, we investigate the effect of the gradient of the characteristic length on the recovery of the soliton state, which is shown in Fig. 5. The larger is the gradient $dw_m/dz$, the quicker $w_m$ varies. It is found that a more rapidly changing $w_m$ results in a stronger fluctuation of beam width. Especially, when $w_m$ changes rapidly enough the beams can not return to their initial soliton states, but revolve into breathers [see the green curves in Fig. 5(b)]. We make a sequence of numerical simulations by using different changing rates of $w_m$, and find that beam width fluctuation $\sigma$ increases with the gradient $dw_m/dz$ shown in Fig. 5(c), which agrees well with the results in Fig. 5(b) in a more general sense.

 

Fig. 5. $w_m$ of three different changing speeds (a), the corresponding evolutions of optical solitons when $P_0=P_s$ (b), and the dependence of fluctuation $\sigma$ of beam width on the gradient $dw_m/dz$ (c). The fluctuation is defined by $\sigma =\left (w_{max}-\overline {w}\right )/\overline {w}$, where $w_{max}$ and $\overline {w}$ are the maximum and mean values of the beam width in the ending propagation zone after $w_m$ recovers to its initial valve.

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4. Conclusion

We have revealed a kind of adiabatic evolutions for the solitons and breathers in nonlocal nonlinear media of gradual nonlocality, which is modeled by the nonlocal nonlinear Schrödinger equation with variable coefficient. Both solitons and breathers can restore their initial states if the characteristic length of the response function gradually changes. Furthermore, this kind of adiabatic evolutions is not determined by the specific functional forms of the response. The results supply a tool to engineer the beam propagation states in actual nonlocal nonlinear media such as the lead glass and the nematic liquid crystal.