Wave-speed management of dipole, bright and W-shaped solitons in optical metamaterials

Soliton formation in nonlinear metamaterials is presently a very active area of research PLi ; Pendry ; Reed ; Zharov ; Parazzoli ; Shen . Such new materials are known as negative-index materials and more commonly referred to as left-handed metamaterials Kozyrev1 . To demonstrate solitons experimentally, left-handed nonlinear transmission lines, employed as nonlinear metamaterials, have been recently used to study the generation of envelope solitons Kozyrev1 ; Kozyrev2 ; Kozyrev3 ; Kozyrev4 . In addition, the stable generation of soliton pulses has been also demonstrated experimentally in an active nonlinear metamaterial formed by a left-handed transmission line inserted into a ring resonator Kozyrev3 . Moreover, dark envelope solitons in a practical left-handed nonlinear transmission line with series nonlinear capacitance are demonstrated by circuit analysis, which showed that the left-handed nonlinear transmission lines could support dark solitons by tailoring the circuit parameters Kozyrev4 . Furthermore, a left-handed nonlinear electrical lattice has been shown to support the formation of discrete envelope solitons of the bright and dark type English . These significant results indicate that the soliton propagation is one of the physically relevant phenomena associated with practical nonlinear metamaterials.

To describe the transmission of an ultrashort optical pulse through an homogeneous nonlinear metamaterial, a generalized nonlinear Schrödinger equation (NLSE) for a dispersive dielectric susceptibility and permeability has been introduced by Scalora et al. Scalora . One should note here that the utilization of the NLSE to describe the nonlinear wave dynamics is not only restricted to nonlinear metamaterials, but also to other important physical media like optical fibers Agra , Bose-Einstein condensates Beitia , plasma physics Dodd , biomolecular dynamics Davydov , etc. Regarding realistic optical waveguiding media, they are actually inhomogeneous because of the existence of some nonuniformities which arise due to various factors such as the variation of lattice parameters in the optical medium, the imperfection of manufacture and fluctuation of the system diameters Abdullaev ; Lei . With consideration of the inhomogeneities in an optical material, the theoretical description of the light pulse dynamics is mainly based on the generalized NLSE equation with varying group velocity dispersion, nonlinearity, and gain (absorption) coefficients Serkin ; Kruglov3 ; Kruglov4 . Especially in optics, the application of such generalized NLSE has stimulated further studies of the integrable inhomogeneous equations giving rise to the concepts of nonautonomous and self-similar solitons Serkin ; Kruglov3 ; Kruglov4 ; Chen ; Ponomarenko ; Serkin2 . For the soliton transmission in an optical medium within the femtosecond duration range, however, the higher-order effects influenced by the variations of material parameters should be also taken into account Abdullaev1 .

From a physical viewpoint, a soliton structure can be described by four parameters which are the frequency (or velocity), amplitude (or width), time position and phase Has5 . An appropriate modulation of these parameters allows efficient control of the soliton dynamics. Recently, the problem of optimal control the parameters of optical solitons which is also called the problem of soliton management becomes a subject of significant interest Belya ; R1 ; R2 . This because concepts of soliton dispersion management and soliton control in optical fiber systems constitute a physically relevant developments in the practical application of envelope solitons for optical transmissions Hasegawa1 ; Hasegawa2 ; Hasegawa3 ; Hasegawa4 . Interestingly, studies of dispersion management have demonstrated that several effects can be reduced by utilizing this technique such as the modulational instability Doran1 , Gordon-Haus effect resulting from the interaction with noise Doran2 , radiation due to lumped amplifiers compensating the fiber loss Doran3 , and time jitters caused by the collisions between signals Doran4 .

While the control of solitons shape or amplitude under dispersion and nonlinearity managements have been demonstrated in physical systems within the framework of both cubic and higher-order NLSE models, the control of soliton wave speed in nonlinear media has been studied thus far only for the cubic NLSE case in Luke . With use of this envelope equation which includes only two physical effects, it is shown that one can control the wave speed of bright and dark NLS solitons by appropriate modification of the dispersion and nonlinearity coefficients Luke . A challenging problem is the study of wave-speed management of solitons in the presence of higher-order effects, which come into play as pulse durations get shorter and peak powers increase. Also, a more significant issue is to examine the wave-speed management of various types of solitons under the contributions of these higher-order processes. As is known, in addition to the bright and dark soliton types, solitons could also display other complex shapes such as dipole and W-shaped structures. It is worth noting that dipole-mode solitons which are consisting of two peaks Susanto , have been recently observed in a three-level cascade atomic system where it has been experimentally demonstrated that the key to observe them is to create via Kerr nonlinearity, a high enough index contrast in the atomic medium by laser-induced index gratings Yanpeng . As concerns solitons that take the shape of W, these have been firstly presented in optical fibers described by the higher-order NLSE with third-order dispersion, self-steepening, and self-frequency shift effects in Zhoou . Up to the present time, the control of soliton wave speed in optical metamaterials supporting higher-order effects has not been reported to our knowledge. In this paper, we present the analysis of wave-speed management of dipole, bright and W-shaped solitons in an inhomogeneous optical metamaterial exhibiting not only the group velocity dispersion and self-phase-modulation, but also a rich variety higher-order effects. One should note that studying on the control of soliton wave speed is not only of scientific relevance but also of practical significance.

This paper is organized as follows. In Sec. II, we present the generalized higher-order NLSE that governs the few-cycle pulse propagation through a nonlinear metamaterial with higher-order nonlinear dispersion effects and derive its general traveling wave solution. New types of exact analytical periodic wave solutions that are composed of the product of pairs of Jacobi elliptic functions are derived in Sec III. Results for dipole soliton solutions is also presented here in the long-wave limit of periodic wave solutions. We also present the exact bright and W-shaped soliton solutions of the model equation. In Sec IV, we introduce the similarity transformation method for solving the generalized NLSE with varying coefficients. The application of developed method for the wave-speed management of dipole solitons is presented in Sec V. We also apply the developed technique to the management of bright and W-shaped solitons in Sec VI. Finally, in Sec. VII, we give some concluding remarks.

The generalized NLSE describing the propagation of a few-cycle pulse in nonlinear metamaterials has the following form for the optical pulse envelope $U(z,t)$ M1 ; M2 ; M3 ; M4 ; M5 ; M6 ; M7 ; M8 ; M9 ,

$$iU_{z}+i\alpha_{1}U_{t}-\alpha_{2}U_{tt}+\gamma\left|U\right|^{2}U=i\lambda(\left|U\right|^{2}U)_{t}+i\epsilon(\left|U\right|^{2})_{t}U+\sigma_{1}(\left|U\right|^{2}U)_{tt}+\sigma_{2}\left|U\right|^{2}U_{tt}+\sigma_{3}U^{2}U_{tt}^{\ast},$$

(1)

where $U(z,t)$ represents the complex envelope of the electrical field, $z$ and $t$ are the propagation distance and time, respectively, while the parameters $\alpha_{2},$ $\gamma,$ $\alpha_{1},$ $\lambda,$ and $\epsilon$ represent the group velocity dispersion, cubic nonlinearity, intermodal dispersion, self-steepening, and nonlinear dispersion coefficients, respectively. Also, $\sigma_{i}$ (with $i=1,2,3$) are higher-order terms that appear in the context of metamaterials.

This equation has gathered significant attention recently for its importance from various physical view points M3 ; M4 ; M5 ; M6 ; M7 ; M8 . In particular, the existence of bright and dark soliton solutions of Eq. (1) have been recently investigated using different methods M1 ; M9 . Super-Gaussian envelope solitons M3 as well as singular solitons M4 have been also presented for this model. Here, we present the important results showing the wave-speed management of different types of solitons obtained within the generalized NLSE (1) framework. It will be demonstrated that these soliton modes can be deceleraed and accelerated through appropriate modification of the distributed parameters.

To start with, we consider the solution of generalized NLSE (1) in the form Kruglov1 ; Kruglov2 ,

$$U(z,t)=u(x)\exp[i(\kappa z-\delta t+\theta)],$$

(2)

where $u(x)$ is a real amplitude function depending on the traveling coordinate $x=t-qz$, and $q=v^{-1}$ is the inverse velocity. Also, $\kappa$ and $\delta$ are the respective real parameters describing the wave number and frequency shift, while $\theta$ represents the phase of pulse at $z=0$.

On substitution of the waveform solution (2) into the model (1), one obtains the system of ordinary differential equations:

$$(q-2\alpha_{2}\delta-\alpha_{1})\frac{du}{dx}+\left[3\lambda+2\epsilon-2\delta\left(3\sigma_{1}+\sigma_{2}-\sigma_{3}\right)\right]u^{2}\frac{du}{dx}=0,$$

(3)

$$\alpha_{2}\frac{d^{2}u}{dx^{2}}+(3\sigma_{1}+\sigma_{2}+\sigma_{3})u^{2}\frac{d^{2}u}{dx^{2}}+6\sigma_{1}u\left(\frac{du}{dx}\right)^{2}-\left[\gamma-\lambda\delta+\delta^{2}\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)\right]u^{3}+(\kappa-\alpha_{1}\delta-\alpha_{2}\delta^{2})u=0.$$

(4)

In the general case the system of Eqs. (3) and (4) is overdetermined because we have two differential equations for the function $U(z,t)$. However, if some constraints for the parameters in Eq. (3) are fulfilled the system of Eqs. (3) and (4) has non-trivial solutions. From Eq. (3), one finds the relations:

$$q=\alpha_{1}+2\alpha_{2}\delta,~{}~{}~{}~{}\delta=\frac{3\lambda+2\epsilon}{2\left(3\sigma_{1}+\sigma_{2}-\sigma_{3}\right)}.$$

(5)

The latter relations lead to the following expression for the wave velocity $v=1/q$,

$$v=\frac{3\sigma_{1}+\sigma_{2}-\sigma_{3}}{\alpha_{1}(3\sigma_{1}+\sigma_{2}-\sigma_{3})+\alpha_{2}(3\lambda+2\epsilon)}.$$

(6)

Moreover, the relations in Eq. (5) reduce the system of Eqs. (3) and (4) to the ordinary nonlinear differential equation,

$$(\alpha_{2}+bu^{2})\frac{d^{2}u}{dx^{2}}+cu\left(\frac{du}{dx}\right)^{2}-du^{3}+\Gamma u=0,~{}~{}~{}~{}~{}~{}~{}$$

(7)

where the parameters $a,$ $b,$ $c$ and $d$ are given by

	$\displaystyle b$	$\displaystyle=$	$\displaystyle 3\sigma_{1}+\sigma_{2}+\sigma_{3},~{}~{}~{}~{}~{}~{}~{}~{}c=6\sigma_{1},$		(8a)
	$\displaystyle d$	$\displaystyle=$	$\displaystyle\gamma+\frac{(3\lambda+2\epsilon)\left[2\epsilon\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)+\lambda(5\sigma_{3}+\sigma_{2}-3\sigma_{1})\right]}{4(3\sigma_{1}+\sigma_{2}-\sigma_{3})^{2}},$		(8b)
	$\displaystyle\Gamma$	$\displaystyle=$	$\displaystyle\kappa-\frac{(3\lambda+2\epsilon)\left[2\alpha_{1}\left(3\sigma_{1}+\sigma_{2}-\sigma_{3}\right)+\alpha_{2}(3\lambda+2\epsilon)\right]}{4(3\sigma_{1}+\sigma_{2}-\sigma_{3})^{2}}.$		(8c)
By analytically solving the amplitude equation (7), we can identify the various nonlinear waves that can propagate in the optical metamaterial described by the generalized NLSE (1). Noting that Eq. (7) can be written as

$$\frac{d^{2}u}{dx^{2}}+f(u)\left(\frac{du}{dx}\right)^{2}+g(u)=0,$$

(9)

where the functions $f(u)$ and $g(u)$ are given by

$$f(u)=\frac{cu}{bu^{2}+\alpha_{2}},~{}~{}~{}~{}g(u)=\frac{u(\Gamma-du^{2})}{bu^{2}+\alpha_{2}}.$$

(10)

We introduce new function $Y(u)$ by equation,

$$\frac{du}{dx}=Y(u).$$

(11)

Thus, Eq. (9) for new function $Y(u)$ has the following form,

$$\frac{dY}{du}+f(u)Y+g(u)Y^{-1}=0.$$

(12)

Now let us define the function $F(u)$ by relation,

$$F(u)=Y^{2}(u),$$

(13)

then the equation for new function $F$ is

$$\frac{dF}{du}+2f(u)F+2g(u)=0.$$

(14)

The general solution of this equation has the form,

$$F(u)=F_{0}e^{-G(u)}-2e^{-G(u)}\int_{u_{0}}^{u}g(u^{\prime})e^{G(u^{\prime})}du^{\prime},$$

(15)

where the function $G(u)$ is defined as

$$G(u)=2\int_{u_{0}}^{u}f(u^{\prime})du^{\prime}.$$

(16)

We note the solution in Eq. (15) satisfies to the boundary condition as $F(u_{0})=F_{0}$. In addition, the function $G(u)$ has the following explicit form,

$$G(u)=2c\int_{u_{0}}^{u}\frac{u^{\prime}du^{\prime}}{bu^{\prime 2}+\alpha_{2}}=\frac{c}{b}\ln\frac{|bu^{2}+\alpha_{2}|}{|bu_{0}^{2}+\alpha_{2}|}.$$

(17)

Also, Eqs. (11) and (13) yield the differential equation: $dx=\pm du/\sqrt{F(u)}$. Thus, we find that the general solution of Eq. (7) takes the form,

$$x-x_{0}=\pm\int_{u_{0}}^{u}\frac{du^{\prime}}{\sqrt{F(u^{\prime})}},$$

(18)

where the function $F(u)$ is defined by Eq. (15).

The general solution of Eq. (7) can also be written in the form: $x=x_{0}\pm[R(u)-R(u_{0})]$ with $R(u)=\int F^{-1/2}(u)du$. This solution defines the variable $x$ as a function of the amplitude $u$. Hence, we have the boundary condition: $x=x_{0}$ at $u=u_{0}$ or $u(x_{0})=u_{0}$. This general solution leads to many different particular solutions when we define some constrains to the coefficients of NLSE given in Eq. (1). In what follows, the different classes of periodic (elliptic) waves are shown to exist in the presence of all physical processes. Such kind of waves serves as a model of pulse train propagation in optical systems CDai . Moreover, results for dipole, bright and W-shaped soliton solutions are also obtained for the governing envelope equation.

A natural situation would be to consider the inhomogeneities of media in describing more realistic phenomena. The presence of such inhomogeneities modifies the propagation dynamics of nonlinear waves since in this case, the system parameters become functions of the propagation distance. Thus, the ultrashort light pulse transmission in a realistic nonlinear metamaterial can be described by the generalized NLSE with varied coefficients:

$$i\psi_{z}+iD_{1}(z)\psi_{t}-D_{2}(z)\psi_{tt}+R(z)\left|\psi\right|^{2}\psi=i\rho(z)(\left|\psi\right|^{2}\psi)_{t}+if(z)(\left|\psi\right|^{2})_{t}\psi+\chi_{1}(z)(\left|\psi\right|^{2}\psi)_{tt}+\chi_{2}(z)\left|\psi\right|^{2}\psi_{tt}+\chi_{3}(z)\psi^{2}\psi_{tt}^{\ast}.$$

(40)

Equation (1) is an important generalization of the model (1) which may be useful for further understanding of the general behavior of the realistic system. In what follows, we introduce a powerful similarity transformation method which enables us to study the soliton dynamics in a real optical metamaterial described by Eq. (40). One notes that studying the control and manipulation of the soliton pulses within the framework of variable-coefficient NLSE models may help to manage them experimentally not only in inhomogeneous nonlinear metamaterials but also in optical fibers and Bose-Einstein condensates where the NLS family of equations can be also applied to describe the field dynamics.

To obtain the exact analytical solutions for Eq. (40), we developed the method based on the following form of the wave function $\psi(z,t)$:

$$\psi(z,t)=u(y)\exp[i(\kappa Z(z)-\delta T(z,t)+\theta)],$$

(41)

$$y=T(z,t)-qZ(z),~{}~{}~{}~{}q=\alpha_{1}+2\alpha_{2}\delta,$$

(42)

where $T\left(z,t\right)$ and $Z\left(z\right)$ are two functions to be determined and the frequency shift $\delta$ is given in Eq. (5). It is important that we present here the function $T\left(z,t\right)$ and variable $y$ in the special form as

$$T(z,t)=t+P(z),~{}~{}~{}~{}y=t-[qZ(z)-P(z)].$$

(43)

Moreover, we require that the wave function $\psi(z,t)$ in (41) coincides with the wave function $U(z,t)$ in (2) for $z=0$. This initial condition for the wave functions $\psi(z,t)$ and $U(z,t)$ yields the following initial conditions for the functions $Z(z)$ and $T(z,t)$: $Z(0)=0$ and $T(0,t)=t$. Hence, using the expressions in Eq. (43) we have the initial conditions for the functions $Z(z)$ and $P(z)$: $Z(0)=0$ and $P(0)=0$.

Substitution of the expressions (41) and (43) into Eq. (40) leads to the following equations:

$$(P^{\prime}-qZ^{\prime}+D_{1}+2\delta D_{2})\frac{du}{dy}=\left[3\rho+2f-2\delta\left(3\chi_{1}+\chi_{2}-\chi_{3}\right)\right]u^{2}\frac{du}{dy},$$

(44)

$$D_{2}\frac{d^{2}u}{dy^{2}}+(3\chi_{1}+\chi_{2}+\chi_{3})u^{2}\frac{d^{2}u}{dy^{2}}+6\chi_{1}u\left(\frac{du}{dy}\right)^{2}-\left[R-\rho\delta+\delta^{2}\left(\chi_{1}+\chi_{2}+\chi_{3}\right)\right]u^{3}+(\kappa Z^{\prime}-\delta P^{\prime}-\delta D_{1}-\delta^{2}D_{2})u=0,$$

(45)

where $P^{\prime}=dP/dz$ and $Z^{\prime}=dZ/dz$. Eq. (44) is satisfied when the following relations are accepted:

$$\frac{dP}{dz}-q\frac{dZ}{dz}+D_{1}+2\delta D_{2}=0,$$

(46)

$$3\rho+2f-2\delta\left(3\chi_{1}+\chi_{2}-\chi_{3}\right)=0.$$

(47)

Now we perform the important transformation in Eq. (45) using the following change of the variable: $y\mapsto x$. This mapping also yields the change of function in (45) as $u(y)\mapsto u(x)$. One can see that above mapping does not change the form of function $u$. Moreover, Eq. (45) is equivalent to Eq. (4) when the above mapping of variable is performed ($y\mapsto x$) and the following equations are satisfied:

$$\frac{3\chi_{1}+\chi_{2}+\chi_{3}}{D_{2}}=\frac{3\sigma_{1}+\sigma_{2}+\sigma_{3}}{\alpha_{2}},~{}~{}~{}~{}\frac{\chi_{1}}{D_{2}}=\frac{\sigma_{1}}{\alpha_{2}},$$

(48)

$$\frac{R-\rho\delta+\delta^{2}\left(\chi_{1}+\chi_{2}+\chi_{3}\right)}{D_{2}}=\frac{\gamma-\lambda\delta+\delta^{2}\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)}{\alpha_{2}},$$

(49)

$$\frac{1}{D_{2}}\left(\kappa\frac{dZ}{dz}-\delta\frac{dP}{dz}-\delta D_{1}-\delta^{2}D_{2}\right)=\frac{\kappa-\alpha_{1}\delta-\alpha_{2}\delta^{2}}{\alpha_{2}}.$$

(50)

By solving Eqs. (46-50) with the initial conditions $Z(0)=0$ and $P(0)=0$ self-consistently, we obtain the following results that define the mapping variables:

	$\displaystyle\left.P(z)=\frac{\alpha_{1}}{\alpha_{2}}\int_{0}^{z}D_{2}(s)ds-\int_{0}^{z}D_{1}(s)ds,\quad Z(z)=\frac{1}{\alpha_{2}}\int_{0}^{z}D_{2}(s)ds,\right.$		(51)
	$\displaystyle\left.R(z)=\frac{\gamma}{\alpha_{2}}D_{2}(z),\quad\rho(z)=\frac{\lambda}{\alpha_{2}}D_{2}(z),\quad f(z)=\frac{\epsilon}{\alpha_{2}}D_{2}(z),\right.$		(52)
	$\displaystyle\left.\chi_{1}(z)=\frac{\sigma_{1}}{\alpha_{2}}D_{2}(z),\quad\chi_{2}(z)=\frac{\sigma_{2}}{\alpha_{2}}D_{2}(z),\quad\chi_{3}(z)=\frac{\sigma_{3}}{\alpha_{2}}D_{2}(z),\right.$		(53)

Thus, all varying parameters in Eq. (40) completely defined by arbitrary function $D_{2}(z)$ via the constraints given in Eqs. (52) and (53). We also have found that the function $T(z,t)$ and variable $y$ have the form,

$$T(z,t)=t+\frac{\alpha_{1}}{\alpha_{2}}\int_{0}^{z}D_{2}(s)ds-\int_{0}^{z}D_{1}(s)ds,$$

(54)

$$y=t-\int_{0}^{z}D_{1}(s)ds-2\delta\int_{0}^{z}D_{2}(s)ds.$$

(55)

Hence, the dynamics of solitons depends on two arbitrary functions as $D_{1}(z)$ and $D_{2}(z)$. Incorporating these results into Eq. (41), we obtain the general self-similar wave solutions to the generalized NLSE with distributed coefficients (40) as

$$\psi(z,t)=u(t-\zeta(z))\exp[i(\kappa Z(z)-\delta T(z,t)+\theta)],$$

(56)

where

$$\zeta(z)=\int_{0}^{z}D_{1}(s)ds+2\delta\int_{0}^{z}D_{2}(s)ds,$$

(57)

and parameter $\delta$ is given here in Eq. (5). Thus, one can construct exact self-similar solutions to Eq. (40) by using the exact solutions of Eq. (1) via the transformation (56). It is worth noting that the existence of general self-similar solution (56) depends on the specific nonlinear and dispersive features of the medium, which have to satisfy the parametric conditions (52) and (53). These conditions present the exact balances among dispersive and nonlinear processes of different nature.

The equation for velocity of solitons follows from Eq. (56) which yields the relation $dt=(d\zeta/dz)dz$. Thus, the velocity of solitons $V(z)=dz/dt$ has the form,

$$V(z)=\left(\frac{d\zeta(z)}{dz}\right)^{-1}=\frac{1}{D_{1}(z)+2\delta D_{2}(z)}.$$

(58)

We emphasize that the solutions described by Eqs. (56), (57) and (51-(55) are satisfied to appropriate boundary conditions. The substitution into Eqs. (51-53) the limiting boundary conditions $D_{1}(z)=\alpha_{1}$ and $D_{2}(z)=\alpha_{2}$ yield the following relations: $P(z)=0$, $Z(z)=z$, $R(z)=\gamma$, $\rho(z)=\lambda$, $f(z)=\epsilon$, $\chi_{1}(z)=\sigma_{1}$, $\chi_{2}(z)=\sigma_{2}$, $\chi_{3}(z)=\sigma_{3}$ which transform NLSE (40) into generalized NLSE (1). Moreover, in this case we have $Z(z)=z$, $T(z,t)=t$ and $\zeta(z)=(\alpha_{1}+2\alpha_{2}\delta)z=qz$ which leads the wave function in (56) to the form,

$$\psi(z,t)=u(t-qz)\exp[i(\kappa z-\delta t+\theta)].$$

(59)

Hence, in this limiting case we have $\psi(z,t)=U(z,t)$ where the wave function $U(z,t)$ is defined in (2), and the velocity in (58) is $V(z)=1/q=v$. Note that from this limiting case we have the relation $\psi(0,t)=U(0,t)$ which is connected with our initial conditions as $Z(0)=0$ and $P(0)=0$.

We also note that the developed method of solving the generalized NLSE with variable coefficients is applied here to Eq. (40). However, this general method can be used for solving an arbitrary NLSE with variable coefficients. We emphasize that this method for solving the generalized NLSE with variable coefficients is significantly differ from the mapping given in the previous papers Kruglov3 ; Kruglov4 ; Dai2 ; Krug3 .

Having obtained the general self-similar solution (56) of the physically relevant model (40), we now analyze the problem of wave-speed management of soliton pulses by considering important spatial modulation of metamaterial parameters. In particular, we will study the deceleration (i.e., the slowing) and acceleration motions of the dipole soliton pulse as it propagates through the optical metamaterial.

To start with, we first construct the exact self-similar soliton solutions of the generalized NLSE with distributed coefficients (40). Substitution of the solution (31) into Eq. (56) leads to a family of exact self-similar dipole soliton solution for Eq. (40) of the form,

$$\psi(z,t)=\pm R_{0}~{}\mathrm{sech}\left(w_{0}\tau(z,t)\right)\mathrm{th}\left(w_{0}\tau(z,t)\right)\exp[i(\kappa Z(z)-\delta T(z,t)+\theta)],$$

(60)

where $\tau(z,t)=t-\zeta(z)-\eta$. Here the amplitude $R_{0}$ and inverse width $w_{0}$ are given in (32) and the wave number $\kappa$ in (22) with the parametric condition as $\sigma_{1}=-(\sigma_{2}+\sigma_{3})/7$.

Expression (60) shows that the amplitude and inverse width of dipole solitons remain constants as they propagate through the metamaterial while their velocity $V(z)$, given by Eq. (58), is affected by the varying dispersion coefficients $D_{1}(z)$ and $D_{2}(z).$ This may lead to the decelerating or accelerating soliton motions by suitable variations of these parameters. Below, we analyze the deceleration and acceleration processes of dipole solitons by managing the distributed parameters $D_{1}(z)$ and $D_{2}(z)$.