1 Introduction

In recent years, the wide application of optical solitons in optical communication has attracted wide attention [1,2,3]. Spatial optical solitons [4, 5] is a state of optical transmission that occurs when the diffraction and self-focusing effect reach a precise balance. In the early research, the focus was mainly on the transmission of spatial optical solitons in local nonlinear media. Until the Snyder-Mitchell model is proposed to deal with the propagation of beams in nonlocal nonlinear media, “accessible solitons” are obtained [6]. Initially, due to limitations in experimental conditions, research on optical solitons in strongly nonlocal nonlinear media only stayed in the theoretical stage. Later, Conti et al. discovered the strong nonlocality in nematic liquid crystals and established a model [7]. Rotschild et al. discovered stable solitons in lead glass [8]. which sparked widespread attention to spatial optical solitons. The evolution of light beams in nonlocal nonlinear media is governed by the nonlocal nonlinear Schrödinger equation (NNLSE). Since Guo et al. improved the Snyder-Mitchell model by dealing with nonlinear terms in NNLSE [9], the propagation characteristics of the beam in nonlocal nonlinear media have been extensively studied both theoretically and experimentally [10,11,12,13,14,15].

In fact, it is not easy to solve NNLSE accurately. At present, researchers have investigated various mathematical methods, such as the tanh-function method [16], sine-cosine method [17], perturbation method [18] and variational method [19], etc. The study of the variational method can be traced back to 1983, when Anderson used the variational method to obtain approximate expressions for the parameters describing optical pulses in nonlinear fibers and proved that the variational results are in good agreement with numerical simulations [20]. Subsequently, the variational method was applied to solve NNLSE by Guo et al. [21]. To date, the variational method has been applied to solve many types of solitons and breathers [22, 23]. It can be demonstrated that the variational method can be used as a feasible approximation to study the propagation properties of beams in strongly nonlocal nonlinear media.

Since the dipole solitons were predicted theoretically and observed experimentally in 2000 [24], the research on multipole spatial optical solitons and their interactions has continued, and significant progress has been made [25,26,27]. However, multimode solitons in nonlocal nonlinear systems often exhibit instability and collapse [28,29,30]. Therefore, realizing stable multimode solitons in nonlocal media remains a challenging issue, and more and more researchers focus on the formation and stable transmission of multimode solitons in different media [31,32,33,34]. When nonlinear media respond to nonlocal conditions, soliton instability is improved. The nonlocality of nonlinear media can inhibit the instability of soliton transmission. Zeng et al. used numerical methods to analyze how quintic nonlinearity affects the formation of clusters of different surface gap solitons, including fundamental solitons, dipole solitons, tripole solitons, and quadrupole solitons. The stability of these soliton clusters is also studied by the linear stability analysis method and verified by numerical simulation [35]. Zhou et al. [36] obtained stable transmission of high-dimensional dipole and quadrupole solitons in competitive nonlocal triple-quintic nonlinear media by controlling cubic and quintic nonlinear coefficients and propagation constants. Dai et al. [22] obtained approximate analytical expressions for tripole and quadrupole solitons in (1+1)-dimensional nematic liquid crystals using the variational method. It is found that the power of four-pole solitons is greater than that of three-pole solitons under the same parameters. Saha et al. present analytical periodic (elliptic) solitary waves, which exhibit the dipole and quadrupole structure within a period. The stability and robustness of the solitary waves are discussed numerically [37].

Studying the propagation characteristics of spatial optical solitons in nonlocal media is helpful to realize all-optical devices. Currently, most studies on beam transmission in media are under ideal conditions without considering the losses [38,39,40]. However, losses are not negligible in many nonlocal materials [41, 43]. For example, in the strongly nonlocal nonlinear media nematic liquid crystals experiment, the measured loss value is about 0.47–5 cm\(^{-1}\) [42]. In recent years, the interaction between spatial optical solitons can be widely used in optical switching technology; in the design process of all-optical devices, the loss of the media must be considered [43, 44]. Therefore, in the study of the transmission and interaction characteristics of various spatial optical solitons, the loss of the media has attracted extensive attention from researchers recently. Huang et al. studied beam propagation in (1+1)-dimensional lossless nonlocal Kerr media by using variational method [45]. Liang et al. studied the propagation dynamics of helical elliptical beams in lossy nonlocal nonlinear media based on the variational method [44]. Guo et al. studied the evolution of dipole respiratory waves in lossy nonlocal nonlinear media [46]. Liu et al. investigated multi-stable multipole solitons’ existence, stability, and propagation dynamics in cubic-quintic competing media modulated by a harmonic-Gaussian double potential [47]. At the same time, the transmission characteristics of light beams with gain greater than loss in media have also been reported [48]. Wang et al. investigated the structure and stability of multipole and vortex solitons in the nonlocal nonlinear fractional Schrödinger equation with a gradually decreasing Lvy index are numerically studied. It is found that the solitons adiabatically compress with the decrease of the Lvy index, new species of stable ones are produced using this technique, and weak dissipation does not essentially affect the observed results [49].

Up to now, the transmission characteristics of quadrupole beams in strong nonlocal media with loss have not been reported in the literature. Based on the above research background, we will take the quadrupole beam in the nematic liquid crystal as the research object and use the variational method to study its transmission characteristics in lossy nonlocal nonlinear media.

2 Theoretical model and derivation of variational method

As we all know, the beam propagation in nonlinear media with losses can be governed by NNLSE [9, 19, 22, 23]

$$\begin{aligned} & i\frac{\partial \varPsi }{\partial \tau }+\frac{1}{2}\frac{\partial ^2\varPsi }{\partial \xi ^2}+\varPsi \int ^{+\infty }_{-\infty } R(\xi -\xi ')|\varPsi (\xi ',\tau )|^2d\xi ' \nonumber \\ & \quad +i\frac{\delta }{2}\varPsi =0, \end{aligned}$$
(1)

where \({\varPsi (\xi ,\tau )}\) denotes the beam field distribution of paraxial slow-varying light envelope; \(\xi \) and \(\tau \) denote the normalized transversal and longitudinal coordinates of the quadrupole beams, respectively; \(R(\xi )\) represents the nonlocal response function, whose width determines the degree of nonlocality and satisfies the normalization condition \(\int ^{+\infty }_{-\infty }{R(\xi )}d\xi =1\). \(\delta \) is the loss coefficient of the transmission media. Although the exact solution of the Eq. (1) is difficult to calculate, the approximate solution can be obtained using the variational method, which provides us with a basic equation form that qualitatively and quantitatively describes the dynamic characteristics of a specific system. Following, we will use the variational method to investigate the evolution of quadrupole beam in Eq. (1). Firstly, perform the following substitution

$$\begin{aligned} {\varPsi } (\xi ,\tau )=A(\xi ,\tau )\exp \left( -\frac{\delta \tau }{2}\right) . \end{aligned}$$
(2)

Substituting Eq. (2) into Eq. (1), the NNLSE with an exponential nonlocal response can be restated as follows,

$$\begin{aligned} & i\frac{\partial A}{\partial \tau }+\frac{1}{2}\frac{\partial ^2\partial A}{\partial \xi ^2}\nonumber \\ & \quad + \exp (-\delta \tau ) A\int ^{+\infty }_{-\infty } R(\xi -\xi ')| A(\xi ',\tau )|^2d\xi '=0.\nonumber \\ \end{aligned}$$
(3)

Equation (3) is the NNLSE which describes the propagation of quadrupole beams in lossy media, and it can be regarded as an Euler-Lagrange equation, which corresponds to the variational problem

$$\begin{aligned} J= & \int _{0}^{+\infty }\int _{-\infty }^{+\infty }{\mathcal {L}}\nonumber \\ & \times \left( A,A^*,\frac{\partial {A}}{\partial {\xi }},\frac{\partial {A^*}}{\partial {\xi }},\frac{\partial {A}}{\partial {\tau }},\frac{\partial {A^*}}{\partial {\tau }}\right) d{\xi }d{\tau }, \end{aligned}$$
(4)

where \(*\) represents compound conjugation. Its variational equation is expressed as

$$\begin{aligned} & \delta \int _{0}^{+\infty }\int _{-\infty }^{+\infty }{\mathcal {L}}\left( A,A^*,\frac{\partial A}{\partial \xi },\frac{\partial A^*}{\partial \xi },\frac{\partial A}{\partial \tau },\frac{\partial A^*}{\partial \tau }\right) \nonumber \\ & \quad d{\xi }d{\tau }=0, \end{aligned}$$
(5)

and the Lagrange function density \({\mathcal {L}}\) is

$$\begin{aligned} {\mathcal {L}}= & \frac{i}{2}\left( A^*\frac{\partial A}{\partial \tau }-A\frac{\partial A^*}{\partial \tau }\right) -\frac{1}{2}\left| \frac{\partial A}{\partial \xi }\right| ^2+\frac{1}{2}\exp (-\delta \tau )\nonumber \\ & \times \left| A\right| ^{2}\int ^{+\infty }_{-\infty }R(\xi -\xi ')\left| A(\xi ',\tau )\right| ^2d\xi '. \end{aligned}$$
(6)

The trial solution of quadrupole beams takes the following form,

$$\begin{aligned} A(\xi ,\tau )= & a(\tau )\bigg [\frac{2\xi ^3}{w^2(\tau )}-3\xi \bigg ]\nonumber \\ & \exp \left[ -\frac{\xi ^2}{2w^2(\tau )}+ic(\tau )\xi ^2+i\theta (\tau ) \right] . \end{aligned}$$
(7)

The \(\omega (\tau )\), \(a(\tau )\), \(\theta (\tau )\), and \(c(\tau )\) are the beam width, real amplitude, phase, and equiphase wavefront curvature of the quadrupole beams, respectively, and these four parameters describing the beam transmission characteristics are functions of the longitudinal transmission distance \(\tau \).

By substituting the tentative solution Eq. (7) into the variational Eq. (5), we can get the mean variational process

$$\begin{aligned} \delta \int _{0}^{+\infty }\langle {\mathcal {L}}\rangle {d{\tau }}=0, \end{aligned}$$
(8)

here \(\langle {\mathcal {L}}\rangle \) is the average Lagrangian density, which can be written as

$$\begin{aligned} \langle {\mathcal {L}}\rangle&=\int _{-\infty }^{+\infty }{\mathcal {L}}d{\tau }. \end{aligned}$$
(9)

The normalized nonlocal response function \(R(\xi )\) takes an exponential-decay function as follows [22, 28, 49]

$$\begin{aligned} R(\xi )=\frac{1}{2\omega _m}\exp \left( -\frac{|\xi |}{\omega _m}\right) , \end{aligned}$$
(10)

where \(\omega _m\) denotes the characteristic width of the nonlocal material response function, which represents the nonlocal degree of the optical transmission media. A higher value of \(\omega _m\) indicates a stronger nonlocal property of the media. The nonlocal nonlinear response function of 1+1 dimensional nematic liquid crystals is exponential, while the response function of 1+2 dimensional nematic liquid crystals is a zero-order modified Bessel function. In practice, calculating the integrals in the averaged Lagrangian based on the trial function explicitly is too complicated. However, for the strongly nonlocal nonlinear case, we can calculate it by expanding Eq. (10) to the second order, Therefore, Eq. (10) can be written as

$$\begin{aligned} R(\xi )\simeq \frac{1}{2\omega _m}\left( 1-\frac{|\xi |}{\omega _m}+\frac{\xi ^2}{2\omega ^2_m}\right) . \end{aligned}$$
(11)

Based on the above derivation, combining Eqs. (6), (7), (9), and (11), the analytical expression of average Lagrange density \(\langle {\mathcal {L}}\rangle \) is obtained as follows

$$\begin{aligned} \langle {\mathcal {L}}\rangle= & \frac{9a^4\omega ^6}{1024\omega _m^3}\exp (-\tau \delta )\nonumber \\ & \left( 256\pi {\omega _m^2}-687\sqrt{2\pi }\omega _m\omega +896\pi {\omega ^2}\right) \nonumber \\ & -\frac{21\sqrt{\pi }}{4}a^2\omega \left( 4c^2\omega ^4+1\right) \nonumber \\ & -\frac{3\sqrt{\pi }}{2}{a^2}\omega ^3\left( 7\omega ^2c_\tau '+2\theta _\tau '\right) , \end{aligned}$$
(12)

where \(c_\tau '\) and \(\theta _\tau '\) represents the first derivative of wavefront curvature and phase, respectively.

Plugging Eq. (12) into the corresponding Euler equation

$$\begin{aligned} & \frac{\partial \langle {\mathcal {L}}\rangle }{\partial \theta }-\frac{\partial }{\partial \tau }\big (\frac{\partial \langle {\mathcal {L}}\rangle }{\partial \theta _\tau }\big )=0,\end{aligned}$$
(13)
$$\begin{aligned} & \frac{\partial \langle {\mathcal {L}}\rangle }{\partial c}-\frac{\partial }{\partial \tau }\big (\frac{\partial \langle {\mathcal {L}}\rangle }{\partial c_\tau }\big )=0,\end{aligned}$$
(14)
$$\begin{aligned} & \frac{\partial \langle {\mathcal {L}}\rangle }{\partial a}-\frac{\partial }{\partial \tau }\big (\frac{\partial \langle {\mathcal {L}}\rangle }{\partial a_\tau }\big )=0,\end{aligned}$$
(15)
$$\begin{aligned} & \frac{\partial \langle {\mathcal {L}}\rangle }{\partial \omega }-\frac{\partial }{\partial \tau }\big (\frac{\partial \langle {\mathcal {L}}\rangle }{\partial \omega _\tau }\big )=0. \end{aligned}$$
(16)

Following the general procedure of the variational method, one can obtain

$$\begin{aligned} & 3\sqrt{\pi }a(\tau )\omega ^3(\tau )\bigg [2a'(\tau )\omega (\tau )+3a(\tau )\omega '(\tau )\bigg ]=0,\nonumber \\ \end{aligned}$$
(17)
$$\begin{aligned} & \frac{21}{2}\sqrt{\pi }a(\tau )\omega ^4(\tau )\nonumber \\ & \quad \bigg [2a'(\tau )\omega (\tau ){+}a(\tau )\bigg ({-}4c(\tau )\omega (\tau ){+}5\omega '(\tau )\bigg )\bigg ]{=}0,\nonumber \\ \end{aligned}$$
(18)
$$\begin{aligned} & \frac{9a^3(\tau )\omega ^6(\tau )\exp (-\tau \delta )}{256\omega _m^3}\nonumber \\ & \quad \bigg (256\pi \omega _m^2-687\sqrt{2\pi }\omega _m\omega (\tau )+896\pi \omega ^2(\tau )\bigg )\nonumber \\ & -\frac{3}{2}\sqrt{\pi }a(\tau )\omega (\tau ) \nonumber \\ & \quad \bigg (7{+}28c^2(\tau )\omega ^4(\tau ){+}14\omega ^4(\tau )c'(\tau ){+}4\omega ^2(\tau )\theta '(\tau )\bigg )\nonumber \\ & \qquad =0,\end{aligned}$$
(19)
$$\begin{aligned} & -\frac{3\sqrt{\pi }a^2(\tau )\exp (-\tau \delta )}{1024\omega _m^3}\bigg [-4608\sqrt{\pi }a^2(\tau )\omega _m^2\omega ^5(\tau )\nonumber \\ & +14427\sqrt{2}a^2(\tau )\omega _m\omega ^6(\tau )-21504\sqrt{\pi }a^2(\tau )\omega ^7(\tau )\nonumber \\ & +256\exp (\tau \delta )\omega _m^3 \bigg (7+140c^2(\tau )\omega ^4(\tau )\nonumber \\ & +70\omega ^4(\tau )c'(\tau )+12\omega ^2(\tau )\theta '(\tau )\bigg )\bigg ]=0. \end{aligned}$$
(20)

Based on Eqs. (17)–(20), the following expression can be obtained:

$$\begin{aligned} & \frac{d}{d\tau }\left( 3\sqrt{\pi }a^2(\tau )\omega ^3(\tau )\right) =0,\end{aligned}$$
(21)
$$\begin{aligned} & \frac{d\omega (\tau )}{d\tau }=2c(\tau )\omega (\tau ),\end{aligned}$$
(22)
$$\begin{aligned} & \frac{d\theta (\tau )}{d\tau }=\frac{1}{\omega ^3(\tau )\left( 896\sqrt{2\pi }\omega (\tau )-687\omega _m\right) }\nonumber \\ & \times \left\{ {-}512\sqrt{2\pi }\omega _m^2{+}w(\tau )\left[ 512\sqrt{2\pi }\omega _m^2\omega ^2(\tau )\frac{d^2\omega (\tau )}{d\tau }\right. \right. \nonumber \\ & \left. \left. +896\sqrt{2\pi }\omega (\tau )\left( \omega ^3(\tau )\frac{d^2\omega (\tau )}{d\tau }-3\right) \right. \right. \nonumber \\ & \quad \left. \left. -687\omega _m\left( 3\omega ^3(\tau )\frac{d^2\omega (\tau )}{d\tau }-5\right) \right] \right\} ,\end{aligned}$$
(23)
$$\begin{aligned} & \frac{d^2\omega }{d\tau ^2}=\frac{\exp (-\tau \delta )}{3584\sqrt{\pi }\omega _m^3\omega ^3(\tau )}\nonumber \\ & \quad \bigg (3584\exp (\tau \varepsilon )\sqrt{\pi }\omega _m^3-687\sqrt{2}P_0\omega _m\omega ^3(\tau )\nonumber \\ & +1792\sqrt{\pi }P_0\omega ^4(\tau )\bigg ). \end{aligned}$$
(24)

Equations (21)–(24) is the parametric coupling equation of the quadrupole beams trial solution, \(\omega (\tau )\), \(a(\tau )\), \(\theta (\tau )\) and \(c(\tau )\) can be theoretically obtained from the above equation, respectively.

According to Eq. (21), it is evident that \(3\sqrt{\pi }a^2\)\((\tau )\omega ^3(\tau )\) is a constant; in addition, due to the initial incident power of the quadrupole beam is defined by

$$\begin{aligned} P_0= & \int _{-\infty }^{+\infty }\mathcal |A(\xi ,\tau )|^2d{\xi }\nonumber \\= & 3\sqrt{\pi }a^2(\tau )\omega ^3(\tau ). \end{aligned}$$
(25)

Therefore, Eq. (21) expresses the fact that the beam’s energy does not change.

3 Solution of parametric coupling equations and comparison with numerical simulation results

First, consider that quadrupole beams can be stably transmitted in lossless media, forming spatial optical solitons with constant light intensity and beam width. And the beam width \(\omega (\tau )\) is the initial incident width \(\omega _0\). Make use of

$$\begin{aligned} \frac{d\omega (\tau )}{d\tau } =\frac{d^2\omega (\tau )}{d\tau ^2}=0, \end{aligned}$$
(26)

the corresponding incident power is obtained as [40]

$$\begin{aligned} P_0=\frac{3584\sqrt{\pi }\omega _m^{3}}{\omega _0^3\left( -678\sqrt{2}\omega _m+1792\sqrt{\pi }\omega _0\right) }. \end{aligned}$$
(27)

In this paper, Eq. (27) is defined as the soliton power \(P_c\).

If the media loss is considered, Eqs. (22) and (24) are equal to zero, and the beam power of the quadrupole beams transmitted in lossy media is obtained as

$$\begin{aligned} P_0=\frac{3584\exp (-\tau \delta )\sqrt{\pi }\omega _m^{3}}{\omega _0^3\left( -678\sqrt{2}\omega _m+1792\sqrt{\pi }\omega _0\right) }. \end{aligned}$$
(28)

From Eq. (28), it can be found that the beam power of the quadrupole beam decays with the longitudinal transmission distance. The reason is that when the beam is transmitted in nonlocal nonlinear media with loss, the light intensity attenuates due to the absorption effect of the media on the light. The diffraction effect and self-focusing effect at this time cannot reach an accurate equilibrium state, so standard solitons cannot be formed, which can be defined as lossy spatial optical solitons.

Next, the case of \(d^2\omega (\tau )/d\tau ^2\ne 0\) is discussed, that is, the beam width \(\omega (\tau )\) changes with the longitudinal transmission distance, forming a lossy quadriauroral breather. Equation (24) is similar to Newton’s second law in classical mechanics, so it can be regarded as the motion of a particle with a mass of 1 under the action of equivalent force \(\textrm{F}(\omega )\). The expression of equivalent potential \(\textrm{F}(\omega )\) can be written as

$$\begin{aligned} \textrm{F}(\omega )= & \frac{1}{\omega ^3(\tau )}\nonumber \\ & {+}\frac{\exp ({-}\tau \delta )P_0\left[ {-}687\sqrt{2}\omega _m{+}1792\sqrt{\pi }\omega (\tau )\right] }{3584\sqrt{\pi }w_m^3}.\nonumber \\ \end{aligned}$$
(29)

According to Newtonian mechanics, the relationship between equivalent force \(\textrm{F}(\omega )\) and equivalent potential energy \(\textrm{V}_p(\omega )\) is as follows

$$\begin{aligned} \textrm{V}_p(\omega )=-\int {\textrm{F}(\omega )}d\omega . \end{aligned}$$
(30)

After further calculation and simplification, we obtain

$$\begin{aligned} \textrm{V}_p(\omega )= & \frac{1}{2\omega ^2(\tau )}+\frac{687\exp (-\tau \delta )P_0\omega (\tau )}{1792\sqrt{2\pi }\omega _m^2}\nonumber \\ & -\frac{\exp (-\tau \delta )P_0\omega ^2(\tau )}{4\omega _m^3}+{\mathcal {A}}_0. \end{aligned}$$
(31)

Suppose the equilibrium point of the equivalent potential energy is \(\textrm{V}_p(\omega _0)=0\), the expression for the integral constant \({\mathcal {A}}_0\) can be obtained

$$\begin{aligned} {\mathcal {A}}_0= & -\frac{\exp (-\tau \delta )}{3584\sqrt{\pi }\omega _0^2\omega ^3_m}\left[ -896\sqrt{\pi }P_0\omega ^4_0\right. \nonumber \\ & \left. +687\sqrt{2}P_0\omega _0^3\omega _m+1792\exp (\tau \delta )\sqrt{\pi }\omega ^3_m\right] ,\nonumber \\ \end{aligned}$$
(32)

Substituting Eq. (32) into Eq. (31), the expression of the equivalent potential energy \(\textrm{V}_p(\omega )\) can be obtained as

$$\begin{aligned} \textrm{V}_p(\omega )= & \frac{1}{2\omega ^2(\tau )}+\frac{687\exp (-\tau \delta )P_0\omega (\tau )}{1792\sqrt{2\pi }\omega _m^2}\nonumber \\ & -\frac{\exp (-\tau \delta )P_0\omega ^2(\tau )}{4\omega _m^3}-\frac{\exp (-\tau \delta )}{3584\sqrt{\pi }\omega _0^2\omega ^3_m}\nonumber \\ & \times \left[ -896\sqrt{\pi }P_0\omega ^4_0+687\sqrt{2}P_0\omega _0^3\omega _m \right. \nonumber \\ & \left. +1792\exp (\tau \delta )\sqrt{\pi }\omega ^3_m\right] . \end{aligned}$$
(33)

Based on the above calculation, as shown in Fig. 1, the evolution law of equivalent force and equivalent potential energy of the quadrupole beam with beam width is given when the incident power is less than, equal to and greater than the soliton power \(P_c\), respectively.

Fig. 1
figure 1

Evolution of equivalent force (first row) and equivalent potential energy (second row) with beam width at different incident power. The incident power: \(P_0=0.6P_c\) for (a1) and (b1), \(P_0= 1.0P_c\) for (a2) and (b2), \(P_0=1.6P_c\) for (a3) and (b3). Other parameters: \(\omega _m=100, \omega _0=1, \delta =0.01\)

Full size image

When \(P_0<P_c\) (see Fig. 1a1, b1), the diffraction effect is greater than the self-focusing effect at the incident position, the equivalent force is bigger than zero, and the beam begins to broaden until the equilibrium point of equivalent potential energy is reached. Afterward, the equivalent force is less than zero, and the beam begins to compress. Unlike the case of ignoring media losses, the evolution pattern of the equivalent potential energy is no longer symmetric about the equilibrium point of the equivalent potential energy. The expansion rate of the beam width is faster than the compression rate, resulting in the overall slow broadening trend of the beam width. When \(P_0=P_c\) (see Fig. 1a2 and b2), the beam diffraction effect is balanced with the self-focusing effect, and the equivalent force is zero at the equilibrium point of the equivalent potential energy, the beam width is approximately unchanged, the beam is transmitted in a soliton-like state. When \(P_0>P_c\) (see Fig. 1a3 and b3), although the periodic variation pattern of the beam is opposite to the case of \(P_0<P_c\): the beam is first compressed and then broadened, but the broadening rate is still greater than the compression rate. Namely, The width of the quadrupole breather is still broadened as a whole. In this paper, we only give the relation of equivalent force and equivalent potential energy with beam width under certain small losses. However, it can be proved that the change rule is basically unchanged under other loss conditions.

Next, we will calculate the variational solution of the beam width. When the quadrupole breather is transmitted in nonlocal nonlinear media with losses. The effect of losses on beam width is regarded as a perturbation term, and the analytical solution of beam width can be defined as

$$\begin{aligned} \omega (\tau )=\omega _1(\tau )+\omega _2(\tau ), \end{aligned}$$
(34)

where \(\omega _1\) is the analytical solution of the beam width when the quadrupole breather propagates in a medium without loss, and its expression is [40]

$$\begin{aligned} \omega _1(\tau )=\omega _d+(\omega _0-\omega _d)\cos \left( \beta \tau \right) , \end{aligned}$$
(35)

where

$$\begin{aligned} \omega _d= & \frac{1}{2}T-\frac{1}{2}\sqrt{\varLambda },\\ \varLambda= & T^2-4\Bigg (\frac{t}{2}+\sqrt{\frac{t^2}{4}-\frac{2\omega _m^3}{P_0}}\Bigg ),\\ T= & \frac{687\omega _m}{1792\sqrt{2\pi }}+\sqrt{t+\frac{471969\omega _m^2}{6422528\pi }},\\ t= & \left( \frac{b}{2}+\sqrt{\frac{b^2}{4}+\frac{a^3}{27}}\right) ^{\frac{1}{3}} +\left( \frac{b}{2}-\sqrt{\frac{b^2}{4}+\frac{a^3}{27}}\right) ^{\frac{1}{3}},\\ a= & -\frac{8\omega _m^3}{P_0},\\ b= & \frac{471969\omega _m^5}{802816P_0\pi },\\ \beta= & \sqrt{\frac{3}{\omega _d^4}-\frac{P_0}{2\omega _m^3}}. \end{aligned}$$

The analytical expression of the beam width perturbation term \(\omega _2\) caused by media loss is derived below. Bring Eq. (34) into the Eq. (24), considering the loss coefficient \(\delta \ll 1\) of strongly nonlocal nonlinear media, and some approximate processing can be done in the calculation process: \(\exp (-\delta \tau )\approx 1-\delta \tau \),\(1/(\omega _1+\omega _2)^3\approx 1/\omega ^3_1-3\omega _2/\omega ^4_1\), ignoring the small amount \(\delta \omega _2\), the final calculation is

$$\begin{aligned} \frac{d^2\omega }{d\tau ^2}= & \frac{1}{\omega _1^3}-\frac{3\omega _2}{\omega _1^4}+\frac{\omega _1P_0}{2\omega _m^3}+\frac{\omega _2P_0}{2\omega _m^3}-\frac{\omega _1P_0\tau \delta }{2\omega _m^3}\nonumber \\ & -\frac{687P_0}{1792\sqrt{2\pi }\omega _m^2}-\frac{687P_0\tau \delta }{1792\sqrt{2\pi }\omega _m^2}. \end{aligned}$$
(36)
Fig. 2
figure 2

Influence of nonlocal degree on beam width evolution at different incident power. Solid line: analytic result; Dotted line: numerical integration. a \(\omega _m=70\), b \(\omega _m=100\), c \(\omega _m=140\). The first column: \(P_0=0.8P_c\), the second column: \(P_0=1.0P_c\), and third column: \(P_0=1.1P_c\). Other parameters are the same as those in Fig. 1

Full size image

The analytical expression of the beam width of a quadrupole breather in media without loss is

$$\begin{aligned} \frac{d^2\omega _1}{d\tau ^2}=\frac{687P_0\tau \delta }{1792\sqrt{2\pi }\omega _m^2}+\frac{1}{\omega (\tau )^3}+\frac{P_0\omega (\tau )}{2\omega _m^3}, \end{aligned}$$
(37)

from the relation \(\frac{d^2\omega }{d\tau ^2}=\frac{d^2\omega _1}{d\tau ^2}+\frac{d^2\omega _2}{d\tau ^2}\), one can get

$$\begin{aligned} \frac{d^2\omega _2}{d\tau ^2}= & \Bigg (-\frac{3}{\omega _1^4}+\frac{P_0}{2\omega _m^3}\Bigg )\omega _2\nonumber \\ & -\Bigg (\frac{\omega _1P_0}{2\omega _m^3}-\frac{687P_0}{1792\sqrt{2\pi }\omega _m^2}\Bigg )\tau \delta . \end{aligned}$$
(38)

When \(P_0=P_c\), by \(\omega _1=1\), \(\omega _2 \) is calculated through direct integral

$$\begin{aligned} \omega _2(\tau )=\frac{\Big [\sqrt{\chi }\tau -\sin \Big (\sqrt{\chi }\tau \Big )\Big ]\delta }{\chi ^{3/2}}, \end{aligned}$$
(39)

where the parameter is

$$\begin{aligned} \chi= & -3-\frac{1792\sqrt{\pi }}{\omega _0^3\Big (1792\sqrt{\pi }\omega _0-687\sqrt{2}\omega _m\Big )}. \end{aligned}$$

When \(P_0\ne P_c\), because of the loss is very small, \(\omega _2\) actually changes slowly, thus make \(\frac{d^2\omega _2}{d\tau ^2}=0\), the expression for \(\omega _2\) is

$$\begin{aligned} \omega _2(\tau )=\frac{\omega _1^4P_0\tau \delta \Big (1792\sqrt{\pi }\omega _1- 687\sqrt{2}\omega _m\Big )}{1792\sqrt{\pi }\Big (\omega _1^4P_0-6\omega _m^3\Big )}. \end{aligned}$$
(40)

So far, we have obtained the analytical solution of the beam width under different incident power. Because some approximate calculation is used in the whole calculation process, we need to compare the analytical solution with the numerical solution to verify the correctness and feasibility of the solution of the variational method. The evolution process of beam width with longitudinal transmission distance was obtained by numerical integration using Eq. (24), and compared with the results obtained by analytical solution, as shown in Fig. 2.

In Fig. 2, the influence of nonlocal degree on beam width evolution is obtained by changing the incident power with a fixed loss coefficient. It can be seen from the figure that at a certain longitudinal transmission distance, no matter whether the incident power is \(P_0=P_c\) or \(P_0\ne P_c\), the results of the analytical and numerical solutions show an approximate consistency. Due to the loss of the media, the optical power decreases gradually, and the beam width expands slowly with the longitudinal transmission distance. When \(P_0=P_c\) (see the second column of Fig. 2), due to the continuously weakening nonlinearity that cannot balance out the diffraction effect, the beam width widens slowly with longitudinal transmission distance, that is, non-standard solitons are formed. We call them lossy optical solitons. In addition, the fitting degree of the analytical and numerical solutions is less affected by the longitudinal transmission distance. When \(P_0>P_c\) (see the third column of Fig. 2), the beam width evolution period is compared with \(P_0<P_c\) (see the first column of Fig. 2) will shorten, and the degree of beam width change is relatively small. Whether the beam is transmitted in the soliton-like or the breather, the fitting degree of the analytical solution and the numerical solution will gradually weaken with the beam longitudinal transmission distance. However, the evolution law of the analytical and numerical solutions is consistent. Compared with the propagation of low-power incident beams, the analytical and numerical solutions of the high-power incident beam agree better. Except that the initial incident light power and beam propagation distance affect the fit degree of numerical and analytical solutions to some extent. It can also be seen from the figure that the degree of non-locality is also a very important influencing factor. When other parameters are the same, and the degree of non-locality is increased, the two results fit more and more perfectly. Moreover, the greater the degree of nonlocality, the smaller the broadening of the beam width of quadrupole lossy solitons and breathers with propagation distance. Therefore, whether in the experimental observation or practical application of quadrupole solitons, to avoid errors caused by low power and long longitudinal distances and ensure effective signal transmission without distortion, we can select critical power to transmit information and energy in stronger nonlocal media.

Figure 3 shows the media loss coefficient’s effect on the beam width variation. As can be seen from the figure, when \(P_0<P_c\) (see Fig. 3a), at the initial incident position of the light, the diffraction effect is larger than the self-focusing effect, and the width of the quadrupole breathers show an oscillatory broadening evolution law of first broadening, then compression, and then broadening. Under the same longitudinal transmission distance, with the increase of dielectric loss coefficient, the beam broadening degree also increases. In addition, the evolution period of the beam width also increases with the increase of the loss. When \(P_0=P_c\) (see Fig. 3b), the diffraction effect is equal to the self-focusing effect at the initial incident position of the light. However, due to the dielectric loss, the light intensity is weakened, resulting in the self-focusing effect less than the diffraction effect, and the quadrupole beam slowly widens with the transmission distance. With the increase of media loss, the broadening degree of beam width increases obviously when the same transmission distance is reached. When \(P_0>P_c\) (see Fig. 3c), it can be found that the evolution law of the beam width is opposite to the case of \(P_0<P_c\), the beam width of the quadrupole breathers presents an oscillatory broadening law of first compression and then broadening. The broadening degree of beam width is significantly less than that of \(P_0<P_c\). In addition, as shown in Fig. 3, with the increase of media loss, the broadening rate of beam width with the propagation distance increases, and the evolution period of beam width becomes longer. When the same longitudinal propagation distance is reached, the larger the loss coefficient, the wider the beam width.

Fig. 3
figure 3

Influence of loss coefficient of transmission media on beam width evolution. \(P_0=0.8P_c\) for (a), \(P_0=1.0P_c\) for (b), \(P_0=1.1P_c\) for (c). The solid line represents \(\delta =0.01\), the dashed lines represent \(\delta =0.02\), and the dotted lines represent \(\delta =0.03\). Other parameters are the same as those in Fig. 1

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Fig. 4
figure 4

Wavefront curvature evolution diagram of a lossy quadrupole respirator. \(P_0=0.8P_c\) for the first column, \(P_0=1.0P_c\) for the second column, and \(P_0=1.1P_c\) for the third column. \(\delta =0.01\) for (a), \(\delta =0.03\) for (b). Other parameters are the same as those in Fig. 1

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Next, we calculate and analyze the evolution characteristics of the wavefront curvature of the quadrupole breathers by analogy with the evolution of the beam width. Combining Eqs. (13) and (14), one can get

$$\begin{aligned} c(\tau )=\frac{\omega (\tau )'}{2\omega (\tau )} \end{aligned}$$
(41)

Combining the beam width analytical solution with Eq. (41), when \(P_0=P_c\), there is

$$\begin{aligned} c(\tau )= & \frac{\omega _d-1}{2\varGamma }\sin (\tau \beta )\sqrt{\frac{12}{\omega _d^4}-\frac{2P_0}{\omega _m^3}}\nonumber \\ & {+s}\frac{\delta \Big [{-}1{+}\cos (\tau \chi )\Big ]\omega _0^3\big (1792\sqrt{\pi }\omega _0{-}687\sqrt{2}\omega _m\big )}{\varGamma \Big [1792\sqrt{\pi }{+}5367\sqrt{\pi }\omega _0^4{-}2061\sqrt{2}\omega _0^3\omega _m\Big ]},\nonumber \\ \end{aligned}$$
(42)

where

$$\begin{aligned} \varGamma= & 2\Big [\omega _d-\big (\omega _d-1\big )\cos (\tau \beta )-\frac{\delta \sin (\tau \chi )}{\chi ^{3/2}}+\frac{\delta \tau }{\chi }\Big ]\nonumber \\ \chi= & -3-\frac{1792\sqrt{\pi }}{\omega _0^3\Big (1792\sqrt{\pi }\omega _0-687\sqrt{2}\omega _m\Big )}. \end{aligned}$$
(43)

When \(P_0\ne P_c\), there is

$$\begin{aligned} c(\tau )= & \frac{\big (1-\omega _d\big )\beta \sin (\tau \beta )}{K\varUpsilon }\nonumber \\ & \Bigg (-K-\tau \delta \zeta ^4P_0+\frac{\tau \delta \zeta ^7P_0^2\varLambda }{448\sqrt{\pi }K}-\frac{\tau \delta \zeta ^3P_0\varLambda }{448\sqrt{\pi }}\Bigg ) \nonumber \\ & +\frac{\delta \zeta ^4P_0\varLambda }{1792\sqrt{\pi }K\varUpsilon }, \end{aligned}$$
(44)

where the parameters are

$$\begin{aligned} \zeta= & \omega _d-\big (-1+\omega _d\big )\cos (\tau \beta ),\nonumber \\ \varLambda= & 1792\sqrt{\pi }\zeta -687\sqrt{2}\omega _m,\nonumber \\ K= & \zeta ^4P_0-6\omega _m^3,\nonumber \\ \varUpsilon= & 2\big (\zeta +\frac{\tau \delta \zeta ^4P_0\varLambda }{1792\sqrt{\pi }K}\big ). \end{aligned}$$
(45)
Fig. 5
figure 5

Influence of loss coefficient of transmission medium on amplitude evolution. a \(P_0=0.8P_c\), b \(P_0=1.0P_c\), c \(P_0=1.1P_c\). Other parameters are the same as those in Fig. 1

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Figure 4 compares the analytical and numerical solutions of the wavefront curvature of the quadrupole breathers under different power and dielectric losses. It can be seen that under strong nonlocal conditions, regardless of whether the incident light power is equal to the critical power, the analytical and numerical solutions are nearly identical within a certain longitudinal propagation distance. When \(P_0\ne P_c\) (see the first and third columns of Fig. 4), the curvature of the wavefront evolves according to the motion law of the spring-like harmonic oscillator. The evolution period is affected by the incident light energy; the stronger the incident light is, the shorter the evolution period. When \(P_0<P_c\) (\(P_0>P_c\)), the curvature radius at the initial incident position is positive (negative) so that the quadrupole breathers diverge (compresses) at first. This also explains the evolution law of beam width presented in Fig. 2 theoretically. When the incident power is equal to the critical power, the radius of curvature is equal to zero, and the lossy solitons appears when the loss coefficient is very small. With the propagation of the beam, the difference between the two results will gradually increase. In addition, the increase in the loss caused by the transmission media will also cause a greater difference between the analytical and numerical solutions. Based on the above analysis of the evolution of wavefront curvature, it can still be concluded that the solution of the trial equation obtained by the variational method can be approximately consistent with the numerical solution in a certain longitudinal transmission distance when high power incidence is applied in the strongly nonlocal media with low losses.

Next, the amplitude of the quadrupole breathers is calculated, and the law of its variation is analyzed. According to the Eq. (25), the amplitude can be expressed as

$$\begin{aligned} a(\tau )=\frac{P_0^{\frac{1}{2}}}{\sqrt{3}\pi ^{\frac{1}{4}}\omega ^{\frac{3}{2}}(\tau )}, \end{aligned}$$
(46)

where when \(P_0=P_c\), combine Eqs. (34), (39) and (46), one can get

$$\begin{aligned} a(\tau )=\frac{1}{{\sqrt{3}\pi ^{1/4}}}\frac{P_0^{1/2}}{\Bigg \{\omega _1(\tau )+\frac{\delta \big [\sqrt{\chi }\tau -\sin \big (\sqrt{\chi }\tau \big )\big ]}{\chi ^{3/2}}\Bigg \}^{3/2}}.\nonumber \\ \end{aligned}$$
(47)

When \(P_0\ne P_c\), combine Eqs. (34), (40) and (46), one can get

$$\begin{aligned}&a(\tau )=\frac{1}{\sqrt{3}\pi ^{1/4}}\nonumber \\&\quad \times \frac{P_0^{1/2}}{\Bigg \{\omega _1(\tau ){+}\frac{P_0\tau \delta \omega _1^4(\tau )\big [-687\sqrt{2}\omega _m+1792\sqrt{\pi }\omega _1(\tau )\big ]}{1792\sqrt{\pi }\big [-6\omega _m^3+P_0\omega _1^4(\tau )\big ]}\Bigg \}^{3/2}}. \end{aligned}$$
(48)
Fig. 6
figure 6

The light intensity evolution of quadrupole breathers corresponding to the parameters in Fig. 5 obtained by numerical integration of the original equation

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According to the above calculation results, the evolution law of the amplitude of the quadrupole breathers in the evolution process can be obtained. Figure 5 shows the evolution process of the amplitude of the quadrupole beams with longitudinal transmission distance under different losses.

As can be seen from Fig. 5, no matter whether the incident power is equal to the soliton power, due to the loss of the media, the light intensity is absorbed so that the amplitude of the quadrupole breathers appears oscillatory attenuation with the longitudinal propagation distance. Even when the incident power equals the soliton power (see Fig. 5b), due to the weakening of light intensity, the balance between the diffraction effect and self-focusing effect is broken, and the beam is transmitted in a soliton-like state with decreasing amplitude. It is also found that the amplitude attenuation rate of the quadrupole breathers increases with the increase of the loss coefficient. Figure 6 further confirms that the losses of optical transmission media has a significant impact on beam propagation. A large number of numerical simulations show that, as long as the strong nonlocal condition is met, the quadrupole beam can be transmitted stably over a long distance in its varying form regardless of whether it is transmitted as a lossy optical soliton or a lossy breather. The small loss of media affects the beam width extension degree, and the numerical results show the stable transmission distance of quadrupole beams can be observed in experiments.

Figure 6 shows the influence of the loss coefficient on the evolution of light intensity under different power. From the three-dimensional evolution diagram of light intensity, it is not difficult to find that the light intensity of the quadrupole breathers decays periodically with the propagation distance, further confirming the conclusion obtained in Fig. 5. Due to the loss of the media, the intensity of the quadrupole beam shows attenuation characteristics at different incident powers, but the attenuation is relatively slow when the transmission is at the critical power, so it is further confirmed that the transmission accuracy of the quadrupole beam at the soliton power is higher.

4 Conclusion and discussion

In conclusion, we investigate the evolution of quadrup-ole beam in the strong nonlocal nonlinear media with loss using the variational approach. The expressions of several important parameters are calculated analytically. The numerical simulation is carried out to illustrate the accuracy of the analytical solution. It is found that for the strongly nonlocal case, by expanding the response function to the second order, the approximate soliton solutions are in good agreement with the numerical results. The analytical expressions of beam width, wavefront curvature, and amplitude of quadrupole beam in lossy media are calculated, and the influence of the loss coefficient on its periodic evolution was analyzed. Due to the absorption effect of transmission media on the light intensity, the amplitude of the quadrupole beam decreases gradually, and the beam width expands oscillatively. The effect of the loss factor of the media on the degree of broadening of the quadrupole beam is given. When the loss is small, and the critical power is used, the beam width expands very slowly, which can be called a lossy quadrupole soliton. Although the beam width is widened due to the loss of the media, but the quadrupole soliton or the breather can propagation stably over a long distance under strong nonlocal conditions. This theoretical work found that under strong nonlocal conditions, the beam can be shaped by controlling the transmission distance, the loss coefficient, and the input power of the soliton. Our theoretical analysis further improves and enriches the soliton theory, and will pave the way for the experimental observation of quadrupole solitons in nonlocal, nonlinear media and provide a theoretical basis for practical optical system design, beam transmission transformation, and beam transmission trajectory control.

This article studies the transmission of quadrupole solitons in nonlocal nonlinear media with loss coefficients, with a focus on analyzing the influence of loss coefficients on solitons. In the absence of losses, quadrupole solitons can exist stably in nonlocal nonlinear media with exponential decay. This has been confirmed, including linear stability analysis [29]. Obviously, in the case of a loss coefficient, there will be energy loss during soliton transmission, so the width of the soliton will continue to increase, the light intensity will continue to decay, and theoretically, they will eventually disappear. In practical experiments, losses are inevitable, so this article focuses more on the evolution of various dynamic characteristics of quadrupole soliton transmission with small losses. In the case of small losses, through our numerical simulation verification, solitons can be transmitted over long distances, at least as observable in experiments. In fact, similar solitons in this article have also been obtained in experiments. The purpose of this article is to provide analytical solutions for solitons here, which can better analyze such solitons in theory. So what we emphasize in the article is long-distance transmission and observability, without focusing on its theoretical linear stability. This is due to the existence of losses, and theoretically, the solitons in the article can no longer maintain soliton transmission. Instead, their transverse width always has a breathing effect. Therefore, in this article, we refer to the beam as a breather with losses.