Optical Solitons with Cubic-Quintic-Septic-Nonic Nonlinearities and Quadrupled Power-Law Nonlinearity: An Observation

Abstract

The current paper considers the enhanced Kudryashov’s technique to retrieve solitons with a governing model having cubic-quintic-septic-nonic and quadrupled structures of self-phase modulation. The results prove that it is redundant to extend the self-phase modulation beyond cubic-quintic nonlinearity or dual-power law of nonlinearity.

1. Introduction

Optical soliton is one of the most important topics of study in nonlinear fiber optics during the present times [1,2,3,4,5]. The dynamics of such solitons is typically described by the nonlinear Schrödinger’s equation (NLSE) [6,7,8,9] with a singleton form of self-phase modulation (SPM) [10,11,12,13] that emerges from the nonlinear refractive index structure of an optical fiber [14,15,16,17,18,19,20]. This typically appears with cubic nonlinear structure AKA Kerr law of nonlinearity [21,22,23,24,25] and its generalization to power-law of nonlinear medium [26,27,28,29,30,31]. The third form of singleton SPM that leads to optical Gaussons, as opposed to optical solitons, is with logarithmic law of nonlinearity [32]. Apart from these three forms with single nonlinear term, the lesser known structures of SPM, sparingly visible, are saturable law and exponential form. The remaining forms of SPM typically contain two or more nonlinear structures that are applicable in various forms of materials such a ${\mathrm{LiNbO}}_3$ crystals. These are cubic-quintic nonlinearity, AKA parabolic form of SPM and its generalization to dual-power form of SPM. Several other forms of refractive index structures have emerged, such as quadratic-cubic (QC) form, generalized QC form, anti-cubic (AC) type, and generalized AC form of nonlinearity. The current paper draws attention to the possible extension of parabolic and dual power-laws of nonlinearity to cubic-quintic-septic-nonic (CQSN) form and its generalization to quadrupled power-law of nonlinearity (QPL) and beyond. Although the case of CQS law along with its generalization to triple power-law has been meaningfully addressed in the past [33,34], this paper carries out the analysis and proves that it is redundant to extend beyond CQS or triple-power law of nonlinear structure. This analysis has been carried out with chromatic dispersion (CD). The detailed analysis follows through with both forms of nonlinear refractive index structures.

• Governing Model

$$i{q}_t+a{q}_{xx}+F\left({\left|q\right|}^2\right)q=0,$$

(1)

where the first term stems from temporal evolution, where $i=\sqrt{-1}$, whilst F comes from SPM. x depicts spatial variable, whereas a describes CD. t imply to temporal variable, while $q(x,t)$ denotes the wave profile.

2. The Enhanced Kudryashov’s Technique

Consider a governing equation [35,36,37]

$$F(u,u_x,u_t,u_{xt},u_{xx},\dots )=0,$$

(2)

where $u=u(x,t)$ is dependent variable, whereas x and t are independent variables.

Step-1: Equation (2) reduces to

$$P(U,-kv{U}^{\prime },k{U}^{\prime },k^2{U}^{″},\dots )=0,$$

(3)

by using the restriction

$$\xi =k(x-vt),u(x,t)=U\left(\xi \right),$$

(4)

where v and k are constants.

Step-2: Equation (3) holds the solution structure

$$U\left(\xi \right)={\lambda }_0+\sum _{l=1}^N\sum _{i+j=l}{\lambda }_{ij}{Q}^i\left(\xi \right)R^j\left(\xi \right),$$

(5)

where N stems from the balancing procedure in Equation (3), while $R\left(\xi \right)$ and $Q\left(\xi \right)$ satisfy the ancillary equations

$${R^{\prime }\left(\xi \right)}^2={R\left(\xi \right)}^2(1-\chi {R\left(\xi \right)}^2),$$

(6)

and

$$Q^{\prime }\left(\xi \right)=Q\left(\xi \right)(\eta Q\left(\xi \right)-1),$$

(7)

along with the explicit solutions

$$R\left(\xi \right)=\frac{4c}{4c^2{e}^{\xi }+\chi e^{-\xi }},$$

(8)

and

$$Q\left(\xi \right)=\frac{1}{\eta +b{e}^{\xi }}.$$

(9)

Here $\chi$,${\lambda }_0$, $\eta$, ${\lambda }_{ij}(i,j=0,1,\dots ,N)$, a and b stand for constants.

Step-3: Putting (5) together with (6) and (7) into (3) leaves us with a system of equations that enables us the much-needed constant parameters in (4)–(9).

3. Optical Solitons

The current section employs the integration tool to retrieve optical solitons to the model having CQSN and QPL nonlinearity structures of SPM.

3.1. CQSN Nonlinearity

In this case, the model shapes up as

$$i{q}_t+a{q}_{xx}+\left(b_1{\left|q\right|}^2+b_2{\left|q\right|}^4+b_3{\left|q\right|}^6+b_4{\left|q\right|}^8\right)q=0.$$

(10)

It must be noted that $b_j$$\left(1\le j\le 4\right)$ stem from ${\chi }^{\left(j\right)}$ for $\left(1\le j\le 4\right)$ nonlinearities. Although ${\chi }^{\left(1\right)}$ and ${\chi }^{\left(2\right)}$ are substantial for ${\mathrm{LiNbO}}_3$ crystals, ${\chi }^{\left(3\right)}$ and ${\chi }^{\left(4\right)}$ are negligibly small and miniscule. The current paper includes these nonlinearities to study the corresponding NLSE and check on its integrability aspect for the first time. The drawn conclusions will be interesting. It will be observed that these negligible nonlinear contributions must be set to zero for integrability purposes. This would lead to consistency between the Physics and Mathematics of the problem [38]. We consider the solution structure

$$q(x,t)=U\left(\xi \right)e^{i\varphi (x,t)},$$

(11)

with

$$\xi =k(x-vt),$$

(12)

and

$$\varphi (x,t)=-\kappa x+\omega t+\theta .$$

(13)

Here, $U\left(\xi \right)$ comes from the amplitude component, where $\xi$ is the wave variable and v is the velocity. Additionally, $\varphi (x,t)$ stems from the phase component, where $\theta$ is the phase constant, $\omega$ is the angular frequency and $\kappa$ is the wave number.

Putting (11) into (10) provides us the simplest equations

$$a{k}^2{U}^{″}-U\left(a{\kappa }^2+\omega \right)+b_4{U}^9+b_3{U}^7+b_2{U}^5+b_1{U}^3=0,$$

(14)

and

$$-k{U}^{\prime }(2a\kappa +v)=0.$$

(15)

Equation (15) enables us the soliton velocity

$$v=-2a\kappa .$$

(16)

Using the constraint

$$U\left(\xi \right)=V{\left(\xi \right)}^{\frac{1}{4}},$$

(17)

Equation (14) stands as

$$4a{k}^{2}V{V}^{″}-3a{k}^2{V^{\prime }}^2-16V^2\left(a{\kappa }^2+\omega \right)+16b_1{V}^{\frac{5}{2}}+16b_3{V}^{\frac{7}{2}}+16b_4{V}^4+16b_2{V}^3=0.$$

(18)

Setting $b_1=b_3=0$ reduces Equation (18) to

$$4a{k}^{2}V{V}^{″}-3a{k}^2{V^{\prime }}^2-16V^2\left(a{\kappa }^2+\omega \right)+16b_4{V}^4+16b_2{V}^3=0.$$

(19)

It must be noted that in Equation (18), $b_1$ and $b_3$ were set to zero simply for Equation (18) to be rendered integrable since these would Free (18) from all terms carrying fractional exponents of V. Thus, only $b_2$ and $b_4$ sustain to permit integrability of (18). This is equivalent to studying the governing model with only two non-zero terms, namely $b_2$ and $b_4$ terms. This is equivalent to saying that the governing NLSE is integrable with cubic—quintic nonlinear form of refractive index that is present in ${\mathrm{LiNbO}}_3$ crystals. Thus, extending the SPM beyond ${\chi }^{\left(5\right)}$ nonlinearity is redundant [14,38]. By the implementation of balancing procedure in Equation (19), the solution structure (5) stands as

$$V\left(\xi \right)={\lambda }_0+{\lambda }_{01}R\left(\xi \right)+{\lambda }_{10}Q\left(\xi \right).$$

(20)

Substituting (20) along with (6) and (7) into (19) gives way to the results:

Result-1:

$${\lambda }_0=\frac{6\left(a{\kappa }^2+w\right)}{b_2},{\lambda }_{01}=0,{\lambda }_{10}=-\frac{6\eta \left(a{\kappa }^2+\omega \right)}{b_2},k=\pm 4\sqrt{\frac{a{\kappa }^2+\omega }{a}},b_4=-\frac{5b_2^2}{36\left(a{\kappa }^2+\omega \right)}.$$

(21)

Plugging (21) along with (9) into (20) provides us

$$q(x,t)={\left\{\frac{6(\omega +a{\kappa }^2)}{b_2}\frac{bexp\left[\pm 4\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]}{\eta +bexp\left[\pm 4\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]}\right\}}^{\frac{1}{4}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(22)

Setting $a(a{\kappa }^2+\omega )>0$ and $\eta =\pm b$ collapses Equation (22) to the dark and singular solitons

$$q(x,t)={\left\{\frac{3(\omega +a{\kappa }^2)}{b_2}\left(1\pm \tanh \left[2\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right)\right\}}^{\frac{1}{4}}{e}^{i(-\kappa x+\omega t+\theta )},$$

(23)

and

$$q(x,t)={\left\{\frac{3(\omega +a{\kappa }^2)}{b_2}\left(1\pm \coth \left[2\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right)\right\}}^{\frac{1}{4}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(24)

Result-2:

$${\lambda }_0={\lambda }_{01}=0,k=\pm 4\sqrt{\frac{a{\kappa }^2+\omega }{a}},{\lambda }_{10}=\frac{6\eta \left(a{\kappa }^2+\omega \right)}{b_2},b_4=-\frac{5b_2^2}{36\left(a{\kappa }^2+\omega \right)}.$$

(25)

Inserting (25) along with (9) into (20) enables us

$$q(x,t)={\left\{\frac{6(\omega +a{\kappa }^2)}{b_2}\frac{\eta }{\eta +bexp\left[\pm 4\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]}\right\}}^{\frac{1}{4}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(26)

Taking $a(a{\kappa }^2+\omega )>0$ and $\eta =\pm b$ turns Equation (26) into the dark and singular solitons

$$q(x,t)={\left\{\frac{3(\omega +a{\kappa }^2)}{b_2}\left(1\mp \tanh \left[2\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right)\right\}}^{\frac{1}{4}}{e}^{i(-\kappa x+\omega t+\theta )},$$

(27)

and

$$q(x,t)={\left\{\frac{3(\omega +a{\kappa }^2)}{b_2}\left(1\mp \coth \left[2\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right)\right\}}^{\frac{1}{4}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(28)

Result-3:

$${\lambda }_0=0,{\lambda }_{10}=0,k=\pm 4\sqrt{\frac{a{\kappa }^2+\omega }{a}},b_2=0,{\lambda }_{01}=\sqrt{\frac{5\chi \left(a{\kappa }^2+\omega \right)}{b_4}}.$$

(29)

Putting (29) along with (8) into (20) leaves us with

$$\begin{array}{ccc}& & q(x,t)={\left\{4\sqrt{5}\sqrt{\frac{a{\kappa }^2\chi +w\chi }{b_4}}\frac{c}{4c^{2}exp\left[\pm 4\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]+\chi exp\left[\mp 4\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]}\right\}}^{1/4}\hfill \\ & & \times e^{i(-\kappa x+\omega t+\theta )}.\hfill \end{array}$$

(30)

Setting $a(a{\kappa }^2+\omega )>0$ and $\chi =\pm 4c^2$ changes Equation (30) to the bright and singular solitons

$$q(x,t)={\left\{\pm \sqrt{\frac{5\left(\omega +a{\kappa }^2\right)}{b_4}}\mathrm{sech}\left[4\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right\}}^{\frac{1}{4}}{e}^{i(-\kappa x+\omega t+\theta )},$$

(31)

and

$$q(x,t)={\left\{\pm \sqrt{\frac{5\left(\omega +a{\kappa }^2\right)}{b_4}}\mathrm{csch}\left[4\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right\}}^{\frac{1}{4}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(32)

3.2. QPL Nonlinearity

In this case, the model sticks out as

$$i{q}_t+a{q}_{xx}+\left(b_1{\left|q\right|}^{2n}+b_2{\left|q\right|}^{4n}+b_3{\left|q\right|}^{6n}+b_4{\left|q\right|}^{8n}\right)q=0,$$

(33)

where $b_j$$(j=1-4)$ come from QPL nonlinearity. Putting (11) into (33) paves way to the auxiliary equations

$$a{k}^2{U}^{″}-U\left(a{\kappa }^2+\omega \right)+b_4{U}^{8n+1}+b_3{U}^{6n+1}+b_2{U}^{4n+1}+b_1{U}^{2n+1}=0,$$

(34)

and

$$-k{U}^{\prime }(2a\kappa +v)=0.$$

(35)

Equation (35) leaves us with the soliton velocity

$$v=-2a\kappa .$$

(36)

Using the restriction

$$U\left(\xi \right)=V{\left(\xi \right)}^{\frac{1}{4n}},$$

(37)

Equation (34) reads as

$$4a{k}^{2}nV{V}^{″}+a{k}^2(1-4n){V^{\prime }}^2-16n^2{V}^2\left(a{\kappa }^2+\omega \right)+16b_1{n}^2{V}^{\frac{5}{2}}+16b_3{n}^2{V}^{\frac{7}{2}}+16b_4{n}^2{V}^4+16b_2{n}^2{V}^3=0.$$

(38)

Taking $b_1=b_3=0$ simplifies Equation (38) to

$$4a{k}^{2}nV{V}^{″}+a{k}^2(1-4n){V^{\prime }}^2-16n^2{V}^2\left(a{\kappa }^2+\omega \right)+16b_4{n}^2{V}^4+16b_2{n}^2{V}^3=0.$$

(39)

By the implementation of balancing technique in Equation (39), the formal solution (5) turns into

$$V\left(\xi \right)={\lambda }_0+{\lambda }_{01}R\left(\xi \right)+{\lambda }_{10}Q\left(\xi \right).$$

(40)

Substituting (40) along with (6) and (7) into (39) leaves us with the results:

Result-1:

$$\begin{array}{ccc}& & {\lambda }_0=\frac{2(2n+1)\left(a{\kappa }^2+w\right)}{b_2},{\lambda }_{01}=0,{\lambda }_{10}=-\frac{2(2n+1)\eta \left(a{\kappa }^2+\omega \right)}{b_2},\hfill \\ & & k=\pm 4n\sqrt{\frac{a{\kappa }^2+\omega }{a}},b_4=-\frac{(4n+1)b_2^2}{4(2n+{10}^2)\left(a{\kappa }^2+\omega \right)}.\hfill \end{array}$$

(41)

Plugging (41) along with (9) into (40) provides us

$$q(x,t)={\left\{\frac{2(1+2n)(\omega +a{\kappa }^2)}{b_2}\frac{bexp\left[\pm 4n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]}{\eta +bexp\left[\pm 4\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]}\right\}}^{\frac{1}{4n}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(42)

Taking $a(a{\kappa }^2+\omega )>0$ and $\eta =\pm b$, the dark and singular solitons stand as

$$q(x,t)={\left\{\frac{(1+2n)\left(\omega +a{\kappa }^2\right)}{b_2}\left(1\pm \tanh \left[2n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right)\right\}}^{\frac{1}{4n}}{e}^{i(-\kappa x+\omega t+\theta )},$$

(43)

and

$$q(x,t)={\left\{\frac{(1+2n)\left(\omega +a{\kappa }^2\right)}{b_2}\left(1\pm \coth \left[2n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right)\right\}}^{\frac{1}{4n}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(44)

Result-2:

$${\lambda }_0={\lambda }_{01}=0,k=\pm 4n\sqrt{\frac{a{\kappa }^2+\omega }{a}},{\lambda }_{10}=\frac{2(2n+1)\eta \left(a{\kappa }^2+\omega \right)}{b_2},b_4=-\frac{(4n+1)b_2^2}{4{(2n+1)}^2\left(a{\kappa }^2+\omega \right)}.$$

(45)

Inserting (45) along with (9) into (40) enables us

$$q(x,t)={\left\{\frac{2(1+2n)(\omega +a{\kappa }^2)}{b_2}\frac{\eta }{\eta +bexp\left[\pm 4n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]}\right\}}^{\frac{1}{4n}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(46)

Setting $a(a{\kappa }^2+\omega )>0$ and $\eta =\pm b$, the dark and singular solitons stick out as

$$q(x,t)={\left\{\frac{(1+2n)\left(\omega +a{\kappa }^2\right)}{b_2}\left(1\mp \tanh \left[2n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right)\right\}}^{\frac{1}{4n}}{e}^{i(-\kappa x+\omega t+\theta )},$$

(47)

and

$$q(x,t)={\left\{\frac{(1+2n)\left(\omega +a{\kappa }^2\right)}{b_2}\left(1\mp \coth \left[2n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right)\right\}}^{\frac{1}{4n}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(48)

Result-3:

$${\lambda }_0=0,{\lambda }_{10}=0,k=\pm 4n\sqrt{\frac{a{\kappa }^2+\omega }{a}},b_2=0,{\lambda }_{01}=\sqrt{\frac{(4n+1)\chi \left(a{\kappa }^2+\omega \right)}{b_4}}.$$

(49)

Putting (49) along with (8) into (40) paves way to

$$\begin{array}{ccc}& & q(x,t)={\left\{\sqrt{\frac{a{\kappa }^2\chi +\omega \chi }{b_4}}\frac{4\sqrt{(4n+1)}c}{4c^{2}exp\left[\pm 4n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]+\chi exp\left[\mp 4n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]}\right\}}^{\frac{1}{4n}}\hfill \\ & & \times e^{i(-\kappa x+\omega t+\theta )}.\hfill \end{array}$$

(50)

Setting $a(a{\kappa }^2+\omega )>0$ and $\chi =\pm 4c^2$, the bright and singular solitons evolve as

$$q(x,t)={\left\{\pm \sqrt{\frac{(1+4n)\left(\omega +a{\kappa }^2\right)}{b_4}}\mathrm{sech}\left[4n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right\}}^{\frac{1}{4n}}{e}^{i(-\kappa x+\omega t+\theta )},$$

(51)

and

$$q(x,t)={\left\{\pm \sqrt{\frac{(1+4n)\left(\omega +a{\kappa }^2\right)}{b_4}}\mathrm{csch}\left[4n\sqrt{\frac{\omega +a{\kappa }^2}{a}}(x-vt)\right]\right\}}^{\frac{1}{4n}}{e}^{i(-\kappa x+\omega t+\theta )}.$$

(52)

4. An Observation

This paper simply shows that the NLSE with CD for CQSN or QPL nonlinearity, it is redundant to extend the nonlinear structure of SPM beyond the quintic form or its corresponding generalization in the QPL nonlinear structure. The results fall back to the case of QC or dual-power law of nonlinearity structure, respectively. In both forms of SPM structures, one is compelled to choose $b_1=b_3=0$ thus collapsing the NLSE given by (10) or (33) to the form of parabolic law of nonlinearity or dual-power law of nonlinearity respectively. The respective exponents of the coefficients of $b_2$ and $b_4$ can be renamed from $(4,8)$ and $(4n,8n)$ to $(2,4)$ and $(2n,4n)$, respectively, so that the results for the soliton structure collapse and conform to the pre-existing results known earlier [39]. The extension to CQS and triple-power forms of SPM is also studied in [40].

5. Conclusions

The current paper derives 1-soliton solutions to the model with CD having CQSN and QPL nonlinearity structures of SPM. In both cases it was established that the extension beyond septic form of nonlinearity and its generalized form is redundant. It is only with dual-power and parabolic forms of nonlinear refractive index structure the model would make sense. Any extension that is beyond septic or its generalized form would collapse to parabolic dual-power laws. This true with CD being the source of dispersion terms. Additional form(s) of dispersion sources have not been examined yet. This is, thus, an open problem and will be later investigated. The results are yet to be released and are currently awaited. This would subsequently lead to a very interesting structure of the results.