Chirped periodic and localized waves in a weakly nonlocal media with cubic-quintic nonlinearity

Propagations of spatial optical solitons through nonlocal nonlinear media have drawn considerable attention because of their experimental observations in liquid crystals Conti and lead glasses R1 in addition to several important theoretical predictions K1 ; G1 ; G2 . Nonlocal nonlinearity represents the fact that the refractive index change of a material at a particular location, is determined by the light intensity in a certain vicinity of this location K1 . Compared with the local nonlinear medium for which the response at a given point is only dependent on the light intensity at that point Kro , the response of the nonlocal nonlinear medium at a given point depends not only on the optical intensity at that point, but also on the intensity in its vicinity Kro ; Jia ; W1 . It is noteworthy that the nonlocality plays a vital role for the very narrow beam propagation in the system and thus the nonlocal contribution to the refractive index change has to be taken into account in this case Kro . It is worth mentioning that nonlinear media that feature the nonlocal nonlinearity also include nonlinear ion gas Sut , thermal nonlinear liquid Dreis , quadratic nonlinear media Nikolov , dipolar Bose-Einstein condensate Gries , and photorefractive crystal Segev ; Saffman .

It has been demonstrated that the propagation dynamics of beams and their localization is significantly influenced by nonlocality Bang . In this respect, important results have revealed that the nonlocal nonlinearity can suppress the modulational instability of plane waves W1 ; W2 and support novel soliton states such as ring vortex solitons Bri ; Kart , gap solitons Ras , multipole solitons Rot , spiraling solitons Sku ; Buc , soliton clusters Buc1 , and incoherent solitons Kro2 . Although soliton structures have been extensively studied in nonlocal Kerr-type media Kro ; Dre ; Kart1 ; Ge ; Kong1 ; Kong2 , their investigation in nonlocal non-Kerr systems has not been widespread. Some significant results have, however, been obtained, with previous theoretical studies considering nonlocality of nonlinear response and saturation Tsoy . Specifically, Tsoy studied the soliton solutions in an implicit form which propagate through a weakly nonlocal medium with cubic-quintic nonlinearity, and derived explicit solutions in bright and dark solitons in particular case Tsoy .

To the best of our knowledge, investigations discussing the formation and properties of periodic waves with nonlinear chirp in weakly nonlocal media exhibiting cubic-quintic nonlinearity are not available to date. Moreover, the control of chirped self-similar beams in a nonlocal nonlinear system with distributed diffraction, cubic-quintic nonlinearity, weak nonlocality, and gain or loss has been absent. The objective of the present work is to study the existence and propagation properties of periodic and localized waves with a nonlinear chirp in a weakly nonlocal cubic-quintic medium. Such chirping property is of practical interest in achieving effective beam compression or amplification. Additionally, the problem of self-similar light beam propagation through a weakly nonlocal medium in the presence of distributed cubic-quintic nonlinearity, diffraction, and gain (or loss) are investigated too.

The paper is organized as follows. In Sec. II, we present the cubic-quintic nonlinear Schrödinger equation (NLSE) with weak nonlocality describing optical beam propagation in a nonlocal medium with a saturation of the nonlinear response. We also present here the nonlinear equation that governs the evolution of the light beam intensity in the system and the general traveling-wave solutions of the mentioned equation. In Sec. III, we present results of novel chirped periodic wave solutions of the model equation and the nonlinear chirp accompanying these nonlinear waveforms. Considering the long-wave limit of the analytically determined periodic solutions, we find chirped solitary beam solutions which include gray, bright and dark solitons in Sec. IV. In Sec. V, the similarity transformation method is employed to construct exact self-similar periodic and localized wave solutions of the generalized NLS model with varied coefficients governing the beam evolution in presence of the inhomogeneities of nonlocal media. We also determine here the self-similar variables and constraints satisfied by the distributed coefficients in the inhomogeneous NLS model. We further investigate the propagation dynamics of the obtained self-similar localized waves (or “similaritons”) in a specified soliton control system. Finally, we give some conclusions in Sec. VI.

The paraxial wave equation governing one-dimensional beam propagation in a nonlinear medium is given by Kro :

For weakly nonlocal media with cubic-quintic nonlinearity, one finds Tsoy $\Delta n\left(I\right)=\gamma I+\sigma I^{2}+\mu\partial_{x}^{2}I$ with $\mu$ being the nonlocality parameter which is defined through $\mu=\frac{1}{2}\int_{-\infty}^{+\infty}x^{2}R(x)dx$. The accordingly equation governing the beam propagation in such nonlinear media is given by the NLS equation with weak nonlocality presented in Tsoy . For our studies, this NLS equation model is expressed as

where $\gamma,$ $\sigma$ and $\mu$  are real parameters related to the cubic nonlinearity, quintic nonlinearity, and weak nonlocality, respectively, while the coefficient $\beta$ accounts for the diffraction in the transverse plane.

In the absence of quintic nonlinearity (i.e., $\sigma=0$), Eq. (3) becomes the modified NLS equation which applies to the description of optical beam propagation in nonlocal nonlinear Kerr-type media Kro . Moreover, in the limit of vanishing weak nonlocality (i.e., $\mu=0$), Eq. (3) is reduced to the cubic-quintic NLS equation which governs the evolution of the optical beam in a local medium exhibiting third- and fifth-order nonlinearities. As previously mentioned, the soliton solutions in an implicit form of the model (3) with $\beta=1,$ which are expressed in terms of the elliptic integrals have been presented in Ref. Tsoy . But here we are concerned with explicit periodic wave and soliton solutions which are characterized by a nonlinear chirp. Our results introduce for the first time an efficient transformation which allows the derivation of periodic and localized solutions of Eq. (3) in an explicit form.

To obtain exact traveling wave solutions to the cubic-quintic NLS equation with weak nonlocality (3), we assume a solution given by the expression,

where both $u(\xi)$ and $\phi(\xi)$ are real functions of the traveling coordinate $\xi=x-vz$, with $v$ being the transverse velocity of the wave. Also $\kappa$ and $\delta$ represent the propagation constant and frequency shift, respectively. Substitution of Eq. (4) into Eq. (3) and separation of the real and imaginary parts of the equation yields the following two coupled ordinary differential equations,

The multiplication of Eq. (5) by the function $u(\xi)$ and integration of the resulting equation leads to the following equation,

where $J$ is the integration constant. Then Eq. (7) yields the following expression,

The accompanying chirp $\Delta\omega$ defined as $\Delta\omega=-\partial\left[\kappa z-\delta x+\phi(x)\right]/\partial x$ is given by

Further insertion of the result (8) into (6) gives to the following nonlinear ordinary differential equation,

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

where

Equation (13) presents one of the main results of our analysis, describing the evolution of beam intensity in a weakly nonlocal medium with cubic-quintic nonlinearity. This equation allow us to know whether the traveling-wave solutions exist in the nonlocal medium and in what parametric conditions they are formed. In general, this nonlinear differential equation with coexisting $f\left(df/d\xi\right)^{2}$ and $f^{4}$ terms is difficult to handle analytically. However, by introducing a special transformation in this paper, Eq. (13) is solved analytically to obtain a rich variety of nonlinear waveforms for the model (3). Such a transformation, to our knowledge not used before testifies about the novelty of the solutions obtained.

Incorporating these results back into Eq. (4), we find that general form of traveling wave solutions to the cubic-quintic NLS equation with weak nonlocality (3) is

This result shows that the phase modification $\phi(\xi)$ involves a nonlinear contribution that is inversely proportional to light beam intensity $\left|\psi\left(x,z\right)\right|^{2}=\left|u(\xi)\right|^{2}$. Interestingly, the nontrivial nature of the phase leads to the formation of chirped beams in the system. In particular, when $J=0$, the phase $\phi(\xi)$ in (16) can be reduced to a simple linear form as $\phi\left(\xi\right)=\left(\delta+v\beta^{-1}\right)(\xi-\eta)+\phi_{0}.\$In what follows, we are interested in periodic and solitary pulse solutions to Eq. (13) in the most general case when $J\neq 0,$ which describe nonlinearly chirped structures to the model (3).

As previously noted, the general solutions to Eq. (3) which are implicit and are expressed in terms of the elliptic integrals have been found in Tsoy . In this section, we introduce an efficient transformation that enables one to obtain explicit solutions of the full underlying cubic-quintic NLS equation with weak nonlocality (3). Interestingly, exact chirped periodic solutions are found in the presence of all material parameters for the first time.

Applying the transformation (85) to Eq. (13), one obtains a modified nonlinear differential equation of the form [see Appendix A]:

with the new coefficients $\alpha_{i}$ $(i=1,2,3)$ that are found in the Appendix A as

Thus the coefficient $\alpha_{0}$ in Eq. (17) is a free parameter because $J$ is the integration constant. Note that in Eq. (13) there are two free coefficients as $\nu_{0}$ and $\nu_{1}$ because $J$ and $C$ are two independent integration constants. However, Eq. (17) has only one free parameter $\alpha_{0}$. Thus one can use different parameter $\alpha_{0}$ for different solutions of Eq. (17).

We now introduce a new function $y(\xi)$ as

Thus the equation for function $y(\xi)$ is

where the coefficients $c_{n}$ are given by

We also introduce the polynomial $P(y)=c_{0}+c_{1}y+c_{2}y^{2}-y^{3}$ which is given by the right side of Eq. (21). The roots of polynomial $P(y)$ are given by equation,

where the coefficient $c_{0}=\alpha_{0}\alpha_{3}^{2}$ is a free parameter because $\alpha_{0}=-4J^{2}/\beta^{2}$ and $J$ is integration constant.

The periodic bounded solution of Eq. (21) defined in the interval $y_{2}\leq y(\xi)\leq y_{3}$ is

where the roots are real and ordered ($y_{1}<y_{2}<y_{3}$), and $\mathrm{cn}(z,k)$ is elliptic Jacobi function. The parameters $w$ and $k$ in this solution are

where $0<k<1$. It is shown in the Appendix B that ordered real roots are

The parameter $w$ in these equations is given by

where it is assumed that $\alpha_{2}^{2}-3\alpha_{1}\alpha_{3}>0$. It is shown in the Appendix B that the integration constant $J$ is fixed by relation,

Hence, in general case the roots $y_{n}$ must satisfy the condition $y_{1}y_{2}y_{3}\leq 0$.

1. Family of chirped periodic bounded waves with $J\neq 0$.

The amplitude in Eq. (4) is $u(\xi)=\pm\sqrt{-y(\xi)/\alpha_{3}}$ which lead to a family of periodic bounded solutions of Eq. (3) (with $0<k<1$) as

where the parameters $A=-y_{2}/\alpha_{3}$ and $B=-(y_{3}-y_{2}/\alpha_{3}$ are

The parameter $w$ in this solution is given in Eq. (28). It follows from this solution that the parameters $A$ and $B$ should satisfy the conditions $A>0$ and $A+B\geq 0$. Moreover, the conditions $\alpha_{2}^{2}-3\alpha_{1}\alpha_{3}>0$ and $y_{1}y_{2}y_{3}<0$ are also necessary for this family of periodic solutions with integration constant $J\neq 0$.

We note that parameter $w$ is given by Eq. (28), however one can consider $w$ as independent variable parameter in Eqs. (30) and (31) because the parameter $\alpha_{1}$ depends on $b$ which is variable parameter (see Eqs. (11) and (18)). In this case using Eq. (28) and relation $\alpha_{1}=-4b/a-\alpha_{2}/a$ we obtain the equation for the wave number $\kappa$ as

This equation means that the variable parameters $w$, $v$, $\delta$ and $\kappa$ are connected by this relation for fixed parameter $k$ of Jacoby elliptic function $\mathrm{cn}(\zeta,k)$. We present below three particular cases of periodic solutions with the integration constant $J=0$.

2. Bounded periodic dn-waves for the condition $y_{1}=0$.

The solution in Eq. (30) for $y_{1}=0$ ($J=0$) and $0<k<1$ reduces to the periodic waves,

where the parameters $\Lambda$ and $w$ are

The necessary conditions for this solution are $\alpha_{2}>0$ and $\alpha_{3}<0$. The phase $\theta(x,z)$ in this periodic solution is

where $\theta_{0}=\phi_{0}-\eta(\delta+v/\beta)$. In this case ( $y_{1}=0$) the parameter $w$ is fixed by Eq. (34) and hence Eq. (32) has the form,

3. Bounded periodic cn-waves for the condition $y_{2}=0$.

The solution in Eq. (30) for $y_{2}=0$ ($J=0$) and $0<k<1$ reduces to the periodic waves,

where the parameters $\Lambda$ and $w$ are

The necessary conditions for this solution are $\alpha_{2}>0$ and $\alpha_{3}<0$ for $1/\sqrt{2}<k<1$; and $\alpha_{2}<0$ and $\alpha_{3}<0$ for $0<k<1/\sqrt{2}$. The phase $\theta(x,z)$ in this solution is given by Eq. (35). In this case ( $y_{2}=0$) the parameter $w$ is fixed by Eq. (38) and hence Eq. (32) has the form,

4. Bounded periodic sn-waves for the condition $y_{3}=0$.

The solution in Eq. (30) for $y_{3}=0$ ($J=0$) and $0<k<1$ reduces to the periodic waves,

where the parameters $\Lambda$ and $w$ are

The necessary conditions for this solution are $\alpha_{2}<0$ and $\alpha_{3}>0$. The phase $\theta(x,z)$ in this solution is given by Eq. (35). In this case ($y_{3}=0$) the parameter $w$ is fixed by Eq. (41) and hence Eq. (32) has the form,

We emphases that in Eqs. (36), (39) and (42) there are three variable parameters: $\delta$, $v$ and $\kappa$ for fixed parameter $k$. This means that one can fix any two of these variable parameters and then the third variable parameter follows from Eqs. (36), (39) and (42) respectively. For an example, if the variable parameters $\delta$ and $\kappa$ are fixed then the above equations yield the parameter $v$.

In this section, we present various nonlinearly chirped solitary wave solutions of the cubic-quintic NLS equation with weak nonlocality (3). At first we consider the solution in Eq. (30) in the limiting case $k=1$ for integration constant $J\neq 0$. In this case the periodic bounded solution in Eq. (30) reduces to solitary wave as

where the inverse width is $w_{0}=\frac{1}{2}(\alpha_{2}^{2}-3\alpha_{1}\alpha_{3})^{1/4}$. It is assumed here that the condition $\alpha_{2}^{2}-3\alpha_{1}\alpha_{3}>0$ is satisfied. The parameters $A$ and $B$ are

Note that in the case with $k=1$ the roots are

where $y_{1}=y_{2}$. The equation for variable parameters $w_{0}$, $v$, $\delta$ and $\kappa$ for this case ($k=1$) are connected by relation,

We consider here the case with integration constant $J\neq 0$, and hence Eq. (29) yields the condition $y_{3}<0$ because $y_{1}=y_{2}$ for $k=1$. It follows from Eq. (45) that the condition $y_{3}<0$ is satisfied for $\alpha_{2}<-8w_{0}^{2}$, and hence we have $\alpha_{2}<0$. Moreover, in this case ($J\neq 0$) we have $y_{n}\neq 0$, and hence $4w_{0}^{2}-\alpha_{2}\neq 0$ and $A\neq 0$. The parameter $A$ in solution (43) should be positive which is possible only in the case with $\alpha_{3}>0$ because we have $\alpha_{2}<0$ (see Eq. (44)). It also follows from Eq. (44) that $A+B=-(8w_{0}^{2}+\alpha_{2})/3\alpha_{3}>0$ because $y_{3}<0$ and $\alpha_{3}>0$. Thus we have found that the conditions $A>0$, $B<0$ and $A+B>0$ are satisfied for the solution given in Eq. (43). These conditions lead to gray soliton solution which is a dark soliton with nonzero minimum intensity.

5. Chirped gray soliton solution.

The solution given in Eq. (43) with $J\neq 0$ can also be written in the equivalent form as

where the inverse width of the gray soliton is $w_{0}=\frac{1}{2}(\alpha_{2}^{2}-3\alpha_{1}\alpha_{3})^{1/4}$ with the condition $\alpha_{2}^{2}>3\alpha_{1}\alpha_{3}$. The parameters $\Lambda=A+B$ and $D=-B$ are given by

We have shown that this solution occur only in the case when $\alpha_{2}<-8w_{0}^{2}$ and $\alpha_{3}>0$. These conditions also lead to inequalities $\Lambda>0$ and $D>0$. Hence, the minimum intensity of the pulse is $I=\Lambda\neq 0$ which typically for gray solitons. The equation for variable parameters $w_{0}$, $v$, $\delta$ and $\kappa$ for this case ($k=1$) are connected by Eq. (46).

We note that the condition $\Lambda D>0$ is satisfied for gray soliton solution. In this case the phase $\phi(\xi)$ in Eq. (47) follows by Eq. (16) as

where $R=\sqrt{\Lambda D}(\Lambda+D)$.

Typical intensity profiles of the chirped solitary wave (47) at $z=0$ are shown in Fig. 1 for different degrees of nonlocality $\mu$ as: $\mu=0.25,$ $\mu=0.27,$ $\mu=0.30$. The parameter values used are $\beta=1,$ $\sigma=-1,$ $\gamma=1,$ $v=0.1,$ $\delta=2.81,$ $\kappa=0.12,$ $\eta=0.$ An interesting observation in this figure is that the minimum intensity of chirped gray solitary wave decreases continuously with the increasing of the degree of nonlocality. We also find that the background intensity of the chirped gray solitary waves remains the same for different values of the nonlocality parameter $\mu$.

6. Bright soliton solution.

The solutions in Eqs. (33) and (37) for limiting case $k=1$ and $J=0$ reduce to chirped bright soliton solution,

where the necessary conditions for this solution are $\alpha_{2}>0$ and $\alpha_{3}<0$. In this bright soliton solution the phase $\theta(x,z)$ is given by Eq. (35). Note that we have $c_{0}=y_{2}^{2}y_{3}$ which leads to relation $c_{0}=\alpha_{0}\alpha_{3}^{2}=0$ because $y_{2}=0$. This yields $\alpha_{0}=0$ and integration constant $J=0$. Moreover, Eqs. (36) and (39) lead to equation for variable parameters $\delta$ and $\kappa$ and $v$ as

Note that in this case the inverse width is fixed as $w_{0}=\sqrt{\alpha_{2}}/2$.

Figure 2 depicts the intensity profiles of the chirped bright solitary wave (50) for different values of the nonlocality parameter $\mu.$ It can be observed that with the increasing of nonlocality, the intensity of the solitary wave gradually decreases, while the width increases leading to broadening of the light beam if the parameter $\mu$ is further increased.

7. Dark soliton solution.

The solution in Eq. (40) for limiting case $k=1$ and $J=0$ reduces to chirped dark soliton solution,

where the necessary conditions for this solution are $\alpha_{2}<0$ and $\alpha_{3}>0$. In this dark soliton solution the phase $\theta(x,z)$ is given by Eq. (35). Note that in this case we have $c_{0}=y_{2}^{2}y_{3}$ where $y_{3}=0$, and hence $c_{0}=\alpha_{0}\alpha_{3}^{2}=0$ which yields $\alpha_{0}=0$ and integration constant $J=0$. Moreover, Eq. (42) leads to equation for variable parameters $\delta$ and $\kappa$ and $v$ as

In this case the inverse width is fixed as $w_{0}=\sqrt{-\alpha_{2}/8}$. We note that in Eqs. (51) and (53) there are three variable parameters: $\delta$, $v$ and $\kappa$. This means that one can fix any two of these variable parameters and then the third parameter follows from above equations respectively.

The intensity profile of the chirped dark solitary wave (52) for different values of $\mu$ is displayed in Fig. 3. One can see that unlike chirped gray solitary waves, the background intensity of the chirped dark solitary waves increase as the degree of nonlocality grows.

In a real optical material including nonlocal nonlinear media, the physical parameters vary along with the propagation of light beams due to the presence of the inhomogeneities of media. The inclusion of the distributed coefficients into the NLS equations is currently an effective way to reflect the inhomogeneous effects of the optical beams LWang . In what follows, we analyze the beam propagation phenomena in a realistic weakly nonlocal cubic-quintic medium exhibiting varied physical parameters. An accurate description of the beam evolution in such inhomogeneous system can be achieved by means of variation with respect to the propagation distance of all the material parameters in Eq. (3), resulting in the generalized NLS model with distributed coefficients:

where $D(s),$ $R_{1}(s)$ and $R_{2}(s)$ represent the variable diffraction, cubic and quintic nonlinearity coefficients, respectively. Parameter $N(s)$ denotes the weak nonlocality coefficient, while $G(s)$ represents the loss ($G(s)<0$) or gain ($G(s)>0$) coefficient.

In order to connect solutions of Eq. (54) with those of Eq. (3), we will use the transformation Dai1 ; Dai4 ; Dai3 as

where $\rho(s),$ $z(s),$ $x(s,\chi),$ and $\varphi(s,\chi)$ are real functions to be determined. Substituting Eq. (55) into Eq. (54) leads to Eq. (3), but we must have the following set of parametric equations:

Here the parameters $\chi_{0}$, $W_{0}$, $c_{0}$ and $b_{0}$ are constants representing the initial values of central position of beam, width, chirp, and position of the wavefront, respectively. Further, the constraint conditions on the management parameters depicting weak nonlocality, quintic nonlinearity, and gain (or loss) are given by

We notice that the accumulated diffraction $d(s)$ influences not only the characteristics of the self-similar pulse such as the amplitude, width, center position, and phase but also the effective propagation distance.