1 INTRODUCTION

Optical solitons are studied in a variety of contexts. These include fiber Bragg gratings, highly dispersive solitons, cubic–quartic (CQ) solitons and many others [1–15]. One of the latest innovations is the Kudryashov’s model for self-phase modulation that was first proposed during 2019 [5]. Subsequently, this model of refractive index was applied to a variety of situations such as in Bragg gratings, birefringent fibers and others. Today’s work will be addressing, for the first time, CQ solitons with Kudryashov’s form of refractive index. This is the situation when the usual chromatic dispersion runs low and therefore dispersive effect is introduced from third-order (3OD) and fourth-order dispersions (4OD). Thus, the necessary delicate balance between dispersion and nonlinearity is sustained hence the existence of soliton solutions is guaranteed. The current paper addresses CQ solitons for Kudryashov’s form of self-phase modulation by the aid of extended trial function approach. The solutions are first retrieved in terms of Jacobi’s elliptic functions and finally soliton solutions are revealed when limiting values of the corresponding modulus of ellipticity is approached. Finally, the conservation laws are recovered by multiplier approach and the corresponding conserved quantities are enlisted.

1.1 Governing Model

The dimensionless form of Kudryashov’s equation with 3OD and 4OD but without GVD reads as [4–8]

$$i{{q}_{t}} + ia{{q}_{{xxx}}} + b{{q}_{{xxxx}}} + \left( {{{c}_{1}}{{{\left| q \right|}}^{{ - 2n}}} + {{c}_{2}}{{{\left| q \right|}}^{{ - n}}} + {{c}_{3}}{{{\left| q \right|}}^{n}} + {{c}_{4}}{{{\left| q \right|}}^{{2n}}}} \right)q = 0.$$
(1)

In (1), the complex-valued function \(q(x,t)\) is the wave profile where \(x\) and \(t\) are respectively independent spatial and temporal variables. The first term stands for linear temporal evolution while the coefficients \(a\) and \(b\) are real parameters that independently controls 3OD and 4OD respectively. The next four terms, as introduced by Kudryashov, are nonlinear and stem from the law of refractive index of an optical fiber and gives self–phase modulation effect to the model.

2 PRELIMINARIES

To start off, the following structural form is picked:

$$q(x,t) = g(s){{e}^{{i\phi (x,t)}}},$$
(2)

where

$$s = x - {v}t,$$
(3)

and \({v}\) stands for soliton speed. The phase \(\phi \) has the split as

$$\phi = - \kappa x + \omega t + \theta ,$$
(4)

where \(\kappa \) is the frequency, \(\omega \) is the wave number and \(\theta \) is the phase center. Next, insert (2) into (1). Then, the real and imaginary parts give respectively:

$${{c}_{1}}g + {{c}_{2}}{{g}^{{1 + n}}} - ({{\kappa }^{3}}(a - b\kappa ) + \omega ){{g}^{{1 + 2n}}} + {{c}_{3}}{{g}^{{1 + 3n}}} + {{c}_{4}}{{g}^{{1 + 4n}}} + 3\kappa (a - 2b\kappa ){{g}^{{2n}}}g{\kern 1pt} '' + b{{g}^{{2n}}}{{g}^{{(4)}}} = 0,$$
(5)

and

$$({v} + 3a{{\kappa }^{2}} - 4b{{\kappa }^{3}})g{\kern 1pt} ' - (a - 4b\kappa ){{g}^{{(3)}}} = 0.$$
(6)

Now, differentiating (6) yields

$${{g}^{{(4)}}} = \frac{{({v} + 3a{{\kappa }^{2}} - 4b{{\kappa }^{3}})g{\kern 1pt} ''}}{{a - 4b\kappa }},$$
(7)

and then (5) modifies to

$$\begin{gathered} {{c}_{1}}(a - 4b\kappa )g + {{c}_{2}}(a - 4b\kappa ){{g}^{{1 + n}}} - (a - 4b\kappa )(a{{\kappa }^{3}} - b{{\kappa }^{4}} + \omega ){{g}^{{1 + 2n}}} + {{c}_{3}}(a - 4b\kappa ){{g}^{{1 + 3n}}} \\ + \;{{c}_{4}}(a - 4b\kappa ){{g}^{{1 + 4n}}} + (3{{a}^{2}}\kappa + 20{{b}^{2}}{{\kappa }^{3}} + b({v} - 15a{{\kappa }^{2}})){{g}^{{2n}}}g{\kern 1pt} '' = 0. \\ \end{gathered} $$
(8)

For extracting closed form solutions, the transformation

$$g = {{\psi }^{{1/n}}}$$
(9)

is employed in Eq. (8) and thus

$$\begin{gathered} {{c}_{1}}{{n}^{2}}(a - 4b\kappa ) + {{c}_{2}}{{n}^{2}}(a - 4b\kappa )\psi - {{n}^{2}}(a - 4b\kappa )(a{{\kappa }^{3}} - b{{\kappa }^{4}} + \omega ){{\psi }^{2}} + {{c}_{3}}{{n}^{2}}(a - 4b\kappa ){{\psi }^{3}} \\ + \;{{c}_{4}}{{n}^{2}}(a - 4b\kappa ){{\psi }^{4}} - (n - 1)(3{{a}^{2}}\kappa + 20{{b}^{2}}{{\kappa }^{3}} + b({v} - 15a{{\kappa }^{2}})){{(\psi {\kern 1pt} ')}^{2}} \\ + \;n(3{{a}^{2}}\kappa + 20{{b}^{2}}{{\kappa }^{3}} + b({v} - 15a{{\kappa }^{2}}))\psi \psi {\kern 1pt} '' = 0. \\ \end{gathered} $$
(10)

3 EXTENDED TRIAL FUNCTION

In order for securing soliton solutions to (10), the assumption is chosen as [3]

$$\psi = \sum\limits_{j = 0}^\varsigma \,{{\varrho }_{j}}{{\varphi }^{j}},$$
(11)

where

$${{(\varphi {\kern 1pt} ')}^{2}} = \Delta (\varphi ) = \frac{{\Gamma (\varphi )}}{{\Upsilon (\varphi )}} = \frac{{{{\mu }_{\sigma }}{{\varphi }^{\sigma }} + \ldots + {{\mu }_{1}}\varphi + {{\mu }_{0}}}}{{{{\chi }_{\rho }}{{\varphi }^{\rho }} + \ldots + {{\chi }_{1}}\varphi + {{\chi }_{0}}}}.$$
(12)

Here, \({{\varrho }_{0}},...,{{\varrho }_{\varsigma }}\); \({{\mu }_{0}},...,{{\mu }_{\sigma }}\) and \({{\chi }_{0}},...,{{\chi }_{\rho }}\) are coefficients that need to be designated, such that the constants \({{\varrho }_{\varsigma }}\), \({{\mu }_{\sigma }}\) and \({{\chi }_{\rho }}\) are nonzero. Then, from Eq. (12) is

$$ \pm (s - {{s}_{0}}) = \int \frac{{d\varphi }}{{\sqrt {\Delta (\varphi )} }} = \int \,\sqrt {\frac{{\Upsilon (\varphi )}}{{\Gamma (\varphi )}}} {\kern 1pt} d\varphi .$$
(13)

Balancing \({{(\psi {\kern 1pt} ')}^{2}}\) or \(\psi \psi {\kern 1pt} ''\) with \({{\psi }^{4}}\) in (10) leads to

$$\sigma = \rho + 2\varsigma + 2.$$
(14)

For \(\rho = 0\), \(\varsigma = 1\) and \(\sigma = 4\),

$$\psi = {{\varrho }_{0}} + {{\varrho }_{1}}\varphi .$$
(15)

Substituting (15) into (10), some coefficients which need to be obtained, come out as:

$${{\mu }_{2}} = {{\mu }_{2}},\quad {{\mu }_{3}} = {{\mu }_{3}},\quad {{\mu }_{4}} = {{\mu }_{4}},\quad {{\varrho }_{0}} = {{\varrho }_{0}},\quad {{\varrho }_{1}} = {{\varrho }_{1}},$$
$${{c}_{3}} = - \frac{{{{c}_{4}}(2 + n)\left( {4{{\mu }_{4}}{{\varrho }_{0}} - {{\mu }_{3}}{{\varrho }_{1}}} \right)}}{{2{{\mu }_{4}}(1 + n)}},\quad {v} = \frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}},$$
$${{\mu }_{0}} = \frac{{{{c}_{1}}{{\mu }_{4}}(2 + n - {{n}^{2}}) + {{\varrho }_{0}}(n - 1)({{c}_{4}}{{\varrho }_{0}}(n - 2)(3{{\mu }_{4}}\varrho _{0}^{2} - 2{{\mu }_{3}}{{\varrho }_{0}}{{\varrho }_{1}} + {{\mu }_{2}}\varrho _{1}^{2}) - 2{{c}_{2}}{{\mu }_{4}}(1 + n))}}{{{{c}_{4}}\varrho _{1}^{4}(n - 2)(n - 1)}},$$
(16)
$${{\mu }_{1}} = \frac{{{{c}_{4}}{{\varrho }_{0}}(n - 2)(4{{\mu }_{4}}\varrho _{0}^{2} + {{\varrho }_{1}}(2{{\mu }_{2}}{{\varrho }_{1}} - 3{{\mu }_{3}}{{\varrho }_{0}})) - 2{{c}_{2}}{{\mu }_{4}}(1 + n)}}{{{{c}_{4}}\varrho _{1}^{3}(n - 2)}},$$
$$\omega = \frac{{{{\mu }_{4}}({{\kappa }^{3}}(1 + n)(b\kappa - a) - 6{{c}_{4}}\varrho _{0}^{2}) + {{c}_{4}}{{\varrho }_{1}}(3{{\mu }_{3}}{{\varrho }_{0}} - {{\mu }_{2}}{{\varrho }_{1}})}}{{{{\mu }_{4}}(1 + n)}},$$

where

$$\alpha = 15ab\kappa - 3{{a}^{2}} - 20{{b}^{2}}{{\kappa }^{2}},\quad \beta = {{c}_{4}}{{n}^{2}}\varrho _{1}^{2}{{\chi }_{0}}(a - 4b\kappa ).$$
(17)

Utilizing the results given in (16), the integral form (13) shapes up as:

$$ \pm (s - {{s}_{0}}) = \sqrt {\frac{{{{\chi }_{0}}}}{{{{\mu }_{4}}}}} {\kern 1pt} \int \frac{{d\varphi }}{{\sqrt {{{\varphi }^{4}} + \frac{{{{\mu }_{3}}}}{{{{\mu }_{4}}}}{{\varphi }^{3}} + \frac{{{{\mu }_{2}}}}{{{{\mu }_{4}}}}{{\varphi }^{2}} + \frac{{{{\mu }_{1}}}}{{{{\mu }_{4}}}}\varphi + \frac{{{{\mu }_{0}}}}{{{{\mu }_{4}}}}} }} = {{\vartheta }_{1}}\int \frac{{d\varphi }}{{\sqrt {\Delta (\varphi )} }}.$$
(18)

Thus, integrating (18) and setting \({{\bar {\varrho }}_{1}} = {{\varrho }_{0}} + {{\varrho }_{1}}{{\tau }_{1}}\), and \({{\bar {\varrho }}_{2}} = {{\varrho }_{0}} + {{\varrho }_{1}}{{\tau }_{2}}\), cubic–quartic solutions to the model are revealed as:

For \(\Delta (\varphi ) = (\varphi - {{\tau }_{1}}{{)}^{4}}\),

$$q(x,t) = {{\left\{ {{{{\bar {\varrho }}}_{1}} \pm \frac{{{{\varrho }_{1}}{{\vartheta }_{1}}}}{{x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t - {{s}_{0}}}}} \right\}}^{{1/n}}}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right].$$
(19)

If \(\Delta (\varphi ) = (\varphi - {{\tau }_{1}}{{)}^{3}}(\varphi - {{\tau }_{2}})\) and \({{\tau }_{2}} > {{\tau }_{1}}\),

$$q(x,t) = {{\left\{ {{{{\bar {\varrho }}}_{1}} + \frac{{4{{\varrho }_{1}}\vartheta _{1}^{2}({{\tau }_{2}} - {{\tau }_{1}})}}{{4\vartheta _{1}^{2} - {{{\left[ {({{\tau }_{1}} - {{\tau }_{2}})\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t - {{s}_{0}}} \right)} \right]}}^{2}}}}} \right\}}^{{1/n}}}{\kern 1pt} \exp \left[ {i( - \kappa x + \omega t + \theta )} \right].$$
(20)

However, when \(\Delta (\varphi ) = (\varphi - {{\tau }_{1}}{{)}^{2}}{{(\varphi - {{\tau }_{2}})}^{2}}\),

$$q(x,t) = {{\left\{ {{{{\bar {\varrho }}}_{2}} + \frac{{{{\varrho }_{1}}({{\tau }_{2}} - {{\tau }_{1}})}}{{\exp \left[ {\frac{{{{\tau }_{1}} - {{\tau }_{2}}}}{{{{\vartheta }_{1}}}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t - {{s}_{0}}} \right)} \right] - 1}}} \right\}}^{{1/n}}}{\kern 1pt} \exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(21)

and

$$q(x,t) = {{\left\{ {{{{\bar {\varrho }}}_{1}} + \frac{{{{\varrho }_{1}}({{\tau }_{1}} - {{\tau }_{2}})}}{{\exp \left[ {\frac{{{{\tau }_{1}} - {{\tau }_{2}}}}{{{{\vartheta }_{1}}}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t - {{s}_{0}}} \right)} \right] - 1}}} \right\}}^{{1/n}}}{\kern 1pt} \exp \left[ {i( - \kappa x + \omega t + \theta )} \right].$$
(22)

Whenever \(\Delta (\varphi ) = (\varphi - {{\tau }_{1}}{{)}^{2}}(\varphi - {{\tau }_{2}})(\varphi - {{\tau }_{3}})\) and \({{\tau }_{1}} > {{\tau }_{2}} > {{\tau }_{3}}\),

$$q(x,t) = {{\left\{ {{{{\bar {\varrho }}}_{1}} - \frac{{2{{\varrho }_{1}}({{\tau }_{1}} - {{\tau }_{2}})({{\tau }_{1}} - {{\tau }_{3}})}}{{2{{\tau }_{1}} - {{\tau }_{2}} - {{\tau }_{3}} + ({{\tau }_{3}} - {{\tau }_{2}})\cosh \left[ {{{\mathcal{H}}_{1}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t} \right)} \right]}}} \right\}}^{{1/n}}}{\kern 1pt} \exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(23)

where

$${{\mathcal{H}}_{1}} = \frac{{\sqrt {({{\tau }_{1}} - {{\tau }_{2}})({{\tau }_{1}} - {{\tau }_{3}})} }}{{{{\vartheta }_{1}}}}.$$
(24)

Finally, if \(\Delta (\varphi ) = (\varphi - {{\tau }_{1}})(\varphi - {{\tau }_{2}})(\varphi - {{\tau }_{3}})(\varphi - {{\tau }_{4}})\) and \({{\tau }_{1}} > {{\tau }_{2}} > {{\tau }_{3}} > {{\tau }_{4}}\),

$$q(x,t) = {{\left\{ {{{{\bar {\varrho }}}_{2}} + \frac{{{{\varrho }_{1}}({{\tau }_{1}} - {{\tau }_{2}})({{\tau }_{4}} - {{\tau }_{2}})}}{{{{\tau }_{4}} - {{\tau }_{2}} + ({{\tau }_{1}} - {{\tau }_{4}}){\text{s}}{{{\text{n}}}^{2}}\left[ {{{\mathcal{H}}_{j}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t - {{s}_{0}}} \right),k} \right]}}} \right\}}^{{1/n}}}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(25)

where the modulus of elliptic integral is

$${{k}^{2}} = \frac{{({{\tau }_{2}} - {{\tau }_{3}})({{\tau }_{1}} - {{\tau }_{4}})}}{{({{\tau }_{1}} - {{\tau }_{3}})({{\tau }_{2}} - {{\tau }_{4}})}},$$
(26)

and \({{\mathcal{H}}_{j}}\) for \(j = 2,3\) are given by

$${{\mathcal{H}}_{j}} = \frac{{{{{( - 1)}}^{j}}\sqrt {({{\tau }_{1}} - {{\tau }_{3}})({{\tau }_{2}} - {{\tau }_{4}})} }}{{2{{\vartheta }_{1}}}}.$$
(27)

Here, the zeros of

$$\Delta (\varphi ) = 0$$
(28)

are \({{\tau }_{j}}\) for \(j = 1,...,4\). In the case of \({{\bar {\varrho }}_{1}} = 0\) and \({{s}_{0}} = 0\), the solutions (19)–(23) reduce to:

Rational function solutions are

$$q(x,t) = {{\left\{ { \pm \frac{{{{\varrho }_{1}}{{\vartheta }_{1}}}}{{x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t}}} \right\}}^{{1/n}}}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(29)
$$q(x,t) = {{\left\{ {\frac{{4{{\varrho }_{1}}\vartheta _{1}^{2}({{\tau }_{2}} - {{\tau }_{1}})}}{{4\vartheta _{1}^{2} - {{{\left[ {({{\tau }_{1}} - {{\tau }_{2}})\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t} \right)} \right]}}^{2}}}}} \right\}}^{{1/n}}}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right].$$
(30)

Singular solitons are

$$q(x,t) = {{\left\{ {\frac{{{{\varrho }_{1}}({{\tau }_{2}} - {{\tau }_{1}})}}{2}\left( {1 \mp \coth \left[ {\frac{{{{\tau }_{1}} - {{\tau }_{2}}}}{{2{{\vartheta }_{1}}}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t} \right)} \right]} \right)} \right\}}^{{1/n}}}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(31)

and bright soliton is

$$q(x,t) = \left\{ {\frac{{{{\mathcal{D}}_{1}}}}{{{{{\left( {{{\mathcal{F}}_{1}} + \cosh \left[ {{{\mathcal{H}}_{1}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t} \right)} \right]} \right)}}^{{1/n}}}}}} \right\}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(32)

where

$${{\mathcal{D}}_{1}} = {{\left( {\frac{{2{{\varrho }_{1}}({{\tau }_{1}} - {{\tau }_{2}})({{\tau }_{1}} - {{\tau }_{3}})}}{{{{\tau }_{3}} - {{\tau }_{2}}}}} \right)}^{{1/n}}},$$
(33)
$${{\mathcal{F}}_{1}} = \frac{{2{{\tau }_{1}} - {{\tau }_{2}} - {{\tau }_{3}}}}{{{{\tau }_{3}} - {{\tau }_{2}}}}.$$
(34)

Here, \({{\mathcal{D}}_{1}}\) is the soliton amplitude and \({{\mathcal{H}}_{1}}\) is its inverse width. The extracted solitons pose the restriction \({{\varrho }_{1}} < 0\).

Moreover, in the case of \({{\bar {\varrho }}_{2}} = 0\) and \({{s}_{0}} = 0\), the solutions (25) reduce to:

$$q(x,t) = \left\{ {\frac{{{{\mathcal{D}}_{2}}}}{{{{{\left( {{{\mathcal{F}}_{2}} + {{{\operatorname{sn} }}^{2}}\left[ {{{\mathcal{H}}_{j}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t} \right),k} \right]} \right)}}^{{1/n}}}}}} \right\}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(35)

where

$${{\mathcal{D}}_{2}} = {{\left( {\frac{{{{\varrho }_{1}}({{\tau }_{1}} - {{\tau }_{2}})({{\tau }_{4}} - {{\tau }_{2}})}}{{{{\tau }_{1}} - {{\tau }_{4}}}}} \right)}^{{1/n}}},$$
(36)
$${{\mathcal{F}}_{2}} = \frac{{{{\tau }_{4}} - {{\tau }_{2}}}}{{{{\tau }_{1}} - {{\tau }_{4}}}}.$$
(37)

Remark 1. If \(k \to 1\), from (35), cubic–quartic singular solitons fall out as

$$q(x,t) = \left\{ {\frac{{{{\mathcal{D}}_{2}}}}{{{{{\left( {{{\mathcal{F}}_{2}} + {{{\tanh }}^{2}}\left[ {{{\mathcal{H}}_{j}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t} \right)} \right]} \right)}}^{{1/n}}}}}} \right\}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(38)

where \({{\tau }_{3}} = {{\tau }_{4}}\).

Remark 2. However, for \(k \to 0\), trigonometric function solutions are

$$q(x,t) = \left\{ {\frac{{{{\mathcal{D}}_{2}}}}{{{{{\left( {{{\mathcal{F}}_{2}} + {\text{si}}{{{\text{n}}}^{2}}\left[ {{{\mathcal{H}}_{j}}\left( {x - \left\{ {\frac{{\alpha \kappa {{\mu }_{4}}(1 + n) - \beta }}{{b{{\mu }_{4}}(1 + n)}}} \right\}t} \right)} \right]} \right)}}^{{1/n}}}}}} \right\}\exp \left[ {i( - \kappa x + \omega t + \theta )} \right],$$
(39)

where \({{\tau }_{2}} = {{\tau }_{3}}\).

4 CONSERVATION LAWS

In the system above, we let \(q = u + i{v}\) and \(n = 2m\) so that (1) decomposes into a system of two components:

$${{u}_{t}} + a{{u}_{{xxx}}} + b{{{v}}_{{xxxx}}} + {v}({{c}_{1}}{{({{u}^{2}} + {{{v}}^{2}})}^{{ - 2m}}} + {{c}_{2}}{{({{u}^{2}} + {{{v}}^{2}})}^{{ - m}}} + {{c}_{3}}{{({{u}^{2}} + {{{v}}^{2}})}^{m}} + {{c}_{4}}{{({{u}^{2}} + {{{v}}^{2}})}^{{2m}}}) = 0,$$
(40)
$$ - {{{v}}_{t}} - a{{{v}}_{{xxx}}} + b{{u}_{{xxxx}}} + u({{c}_{1}}{{({{u}^{2}} + {{{v}}^{2}})}^{{ - 2m}}} + {{c}_{2}}{{({{u}^{2}} + {{{v}}^{2}})}^{{ - m}}} + {{c}_{3}}{{({{u}^{2}} + {{{v}}^{2}})}^{m}} + {{c}_{4}}{{({{u}^{2}} + {{{v}}^{2}})}^{{2m}}}) = 0,$$
(41)

whose conserved flows \(({{T}^{t}},\;{{T}^{x}})\) are established employing the multiplier approach.

Fig. 1.
figure 1

Profile of bright solution (32) corresponding to the values \(n = 2\), \({v} = 3\), \({{\mathcal{D}}_{1}} = 3\), \({{\mathcal{H}}_{1}} = 0.3\) and \({{\mathcal{F}}_{1}} < 0\).

Full size image

Corresponding to multiplier \(Q = ( - u, - {v})\) the conserved density is

$$T_{1}^{t} = \frac{1}{2}({{u}^{2}} + {{{v}}^{2}}),$$
(42)

so that a corresponding conserved density of the complex system is

$$\Phi _{1}^{t} = {\text{|}}{\kern 1pt} q{\kern 1pt} {{{\text{|}}}^{2}}.$$
(43)

Also, the multiplier \(({{{v}}_{x}},{{u}_{x}})\) leads to the density of linear momentum is

$$T_{2}^{t} = \frac{1}{2}{{{v}}_{x}}u - \frac{1}{2}{{u}_{x}}{v},$$
(44)

and the momentum density is

$$\Phi _{2}^{t} = \Im (q{\kern 1pt} {\text{*}}{{q}_{x}}).$$
(45)

The energy density is obtained via the multiplier \(({{{v}}_{t}},{{u}_{t}})\) given by \(\Phi _{3}^{t}\) below in which

$$\begin{gathered} \Phi _{a}^{t} = - {v}b{{{v}}_{{\{ xxxx\} }}} - {{u}^{2}}{{c}_{3}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{m}} + ua{{{v}}_{{\{ xxx\} }}} - 4{v}a{{u}_{{\left\{ {xxx} \right\}}}}{{m}^{4}} + 5{v}a{{u}_{{\{ xxx\} }}}{{m}^{2}} - 4{v}b{{{v}}_{{\{ xxxx\} }}}{{m}^{4}} + 5{v}b{{{v}}_{{\{ xxxx\} }}}{{m}^{2}} \\ + \;4ua{{{v}}_{{\{ xxx\} }}}{{m}^{4}} - 4ub{{u}_{{\{ xxxx\} }}}{{m}^{4}} - 5ua{{{v}}_{{\{ xxx\} }}}{{m}^{2}} + 5ub{{u}_{{\{ xxxx\} }}}{{m}^{2}} - ub{{u}_{{\{ xxxx\} }}} - {v}a{{u}_{{\{ xxx\} }}} - 2{{u}^{2}}{{c}_{1}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m}}}m \\ \end{gathered} $$
$$\begin{gathered} - \;{{u}^{2}}{{c}_{2}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m}}}m + 2{{u}^{2}}{{c}_{4}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m}}}m - 2{{{v}}^{2}}{{c}_{1}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m}}}m - {{{v}}^{2}}{{c}_{2}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m}}}m + {{{v}}^{2}}{{c}_{3}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{m}}m + 2{{{v}}^{2}}{{c}_{4}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m}}}m \\ + \;{{u}^{2}}{{c}_{3}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{m}}m - {{{v}}^{2}}{{c}_{1}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m}}} - {{u}^{2}}{{c}_{2}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m}}} - {{u}^{2}}{{c}_{1}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m}}} - {{{v}}^{2}}{{c}_{2}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m}}} - {{u}^{2}}{{c}_{4}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m}}} - {{{v}}^{2}}{{c}_{4}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m}}} \\ - \;{{{v}}^{2}}{{c}_{3}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{m}} - 4{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{m}}{{{v}}^{2}}{{c}_{3}}{{m}^{3}} + 4{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{m}}{{{v}}^{2}}{{c}_{3}}{{m}^{2}} + 4{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{m}}{{u}^{2}}{{c}_{3}}{{m}^{2}} - 4{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{m}}{{u}^{2}}{{c}_{3}}{{m}^{3}} + 2{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m}}}{{{v}}^{2}}{{c}_{1}}{{m}^{3}} \\ \end{gathered} $$
$$\begin{gathered} + \;{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m}}}{{{v}}^{2}}{{c}_{1}}{{m}^{2}} + 2{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m}}}{{u}^{2}}{{c}_{1}}{{m}^{3}} + {{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m}}}{{u}^{2}}{{c}_{1}}{{m}^{2}} + 4{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m}}}{{{v}}^{2}}{{c}_{2}}{{m}^{3}} + 4{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m}}}{{{v}}^{2}}{{c}_{2}}{{m}^{2}} \\ - \;2{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m}}}{{{v}}^{2}}{{c}_{4}}{{m}^{3}} + {{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m}}}{{{v}}^{2}}{{c}_{4}}{{m}^{2}} + 4{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m}}}{{u}^{2}}{{c}_{2}}{{m}^{3}} + 4{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m}}}{{u}^{2}}{{c}_{2}}{{m}^{2}} - 2{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m}}}{{u}^{2}}{{c}_{4}}{{m}^{3}} + {{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m}}}{{u}^{2}}{{c}_{4}}{{m}^{2}} \\ \end{gathered} $$
$$\begin{gathered} = (1 + 4{{m}^{4}} - 5{{m}^{2}})(a\Im (q{\kern 1pt} {\text{*}}{{q}_{{xxx}}}) - b\Re (qq_{{xxxx}}^{*}))\; + \;{\text{|}}q{\kern 1pt} {{|}^{2}}\{ {{c}_{3}}{\text{|}}q{\kern 1pt} {{|}^{{2m}}}( - 1 - m + 4{{m}^{2}} - 4{{m}^{3}}) + {{c}_{4}}{\text{|}}q{\kern 1pt} {{|}^{{4m}}}( - 1 + 2m \\ + \;{{m}^{2}} - 2{{m}^{3}}) + {{c}_{1}}{\text{|}}q{\kern 1pt} {{|}^{{ - 4m}}}( - 1 - 2m + {{m}^{2}} + 2{{m}^{3}}) + {{c}_{2}}{\text{|}}q{\kern 1pt} {{|}^{{ - 2m}}}( - 1 - m + 4{{m}^{2}} + 4{{m}^{3}})\} , \\ \end{gathered} $$
$$\begin{gathered} \Phi _{b}^{t} = \mathop {\lim }\limits_{\epsilon \to {{0}^{ + }}} \text{[} - {{c}_{2}}{{\epsilon }^{{ - 2m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m + 1}}} - {{c}_{1}}{{\epsilon }^{{ - 4m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m + 1}}} - {{c}_{4}}{{\epsilon }^{{4m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m + 1}}} - {{c}_{3}}{{\epsilon }^{{2m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{m + 1}}} - {{\epsilon }^{2}}{v}b{{{v}}_{{\{ xxxx\} }}} \\ - \;{{\epsilon }^{2}}{v}a{{u}_{{\{ xxx\} }}} - 2{{c}_{4}}{{\epsilon }^{{4m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m + 1}}}{{m}^{3}} + {{\epsilon }^{2}}ua{{{v}}_{{\{ xxx\} }}} + 4{{c}_{3}}{{\epsilon }^{{2m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{m + 1}}}{{m}^{2}} + {{c}_{1}}{{\epsilon }^{{ - 4m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m + 1}}}{{m}^{2}} \\ + \;4{{c}_{2}}{{\epsilon }^{{ - 2m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m + 1}}}{{m}^{3}} + 4{{c}_{2}}{{\epsilon }^{{ - 2m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m + 1}}}{{m}^{2}} - {{c}_{2}}{{\epsilon }^{{ - 2m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - m + 1}}}m - 4{{c}_{3}}{{\epsilon }^{{2m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{m + 1}}}{{m}^{3}} \\ \end{gathered} $$
$$\begin{gathered} + \;{{c}_{3}}{{\epsilon }^{{2m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{m + 1}}}m + 2{{c}_{1}}{{\epsilon }^{{ - 4m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m + 1}}}{{m}^{3}} - 2{{c}_{1}}{{\epsilon }^{{ - 4m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{ - 2m + 1}}}m + {{c}_{4}}{{\epsilon }^{{4m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m + 1}}}{{m}^{2}} - {{\epsilon }^{2}}ub{{u}_{{\{ xxxx\} }}} \\ + \;2{{c}_{4}}{{\epsilon }^{{4m + 2}}}{{({\text{|}}q{\kern 1pt} {{{\text{|}}}^{2}})}^{{2m + 1}}}m - 4ub{{u}_{{\{ xxxx\} }}}{{\epsilon }^{2}}{{m}^{4}} + 5ub{{u}_{{\{ xxxx\} }}}{{\epsilon }^{2}}{{m}^{2}} - 5ua{{{v}}_{{\{ xxx\} }}}{{\epsilon }^{2}}{{m}^{2}} - 4{v}a{{u}_{{\{ xxx\} }}}{{\epsilon }^{2}}{{m}^{4}} \\ + \;5{v}a{{u}_{{\{ xxx\} }}}{{\epsilon }^{2}}{{m}^{2}} - 4{v}b{{{v}}_{{\{ xxxx\} }}}{{\epsilon }^{2}}{{m}^{4}} + 5{v}b{{{v}}_{{\{ xxxx\} }}}{{\epsilon }^{2}}{{m}^{2}} + 4ua{{{v}}_{{\{ xxx\} }}}{{\epsilon }^{2}}{{m}^{4}}] \\ \end{gathered} $$
$$\begin{gathered} = {{\epsilon }^{2}}\mathop {\lim }\limits_{\epsilon \to {{0}^{ + }}} \text{[}(1 + 4{{m}^{4}} - 5{{m}^{2}})(a\Im (q{\kern 1pt} {\text{*}}{{q}_{{xxx}}}) - b\Re (qq_{{xxxx}}^{*}))\; + \;{\text{|}}q{\kern 1pt} {{|}^{2}}\{ {{c}_{3}}{{\epsilon }^{{2m}}}{\text{|}}q{\kern 1pt} {{|}^{{2m}}}( - 1 + m + 4{{m}^{2}} - 4{{m}^{3}}) \\ + \;{{c}_{4}}{{\epsilon }^{{4m}}}{\text{|}}q{\kern 1pt} {{|}^{{4m}}}( - 1 + 2m + {{m}^{2}} - 2{{m}^{3}}) + {{c}_{1}}{{\epsilon }^{{ - 4m}}}{\text{|}}q{\kern 1pt} {{|}^{{ - 4m}}}( - 1 - 2m + {{m}^{2}} + 2{{m}^{3}}) \\ + \;{{c}_{2}}{{\epsilon }^{{ - 2m}}}{\text{|}}q{\kern 1pt} {{|}^{{ - 2m}}}( - 1 - m + 4{{m}^{2}} + 4{{m}^{3}})\} ] \\ \end{gathered} $$

and

$$\Phi _{3}^{t} = \frac{1}{{4{{m}^{4}} - 5{{m}^{2}} + 1}}(\Phi _{a}^{t} - \Phi _{b}^{t}).$$
(46)

From (46), it is clear there exists two integrals of motion, namely the power (\(P\)) and linear momentum (\(M\)). The Hamiltonian does not exist since the integrals in (46) are rendered divergent. Thus, the two conserved quantities are:

$$P = \int\limits_{ - \infty }^\infty {{{{\left| q \right|}}^{2}}dx} = \frac{{2{{A}^{2}}}}{{{{2}^{{1/n}}}B}}F\left( {\frac{1}{n},\frac{1}{n};\frac{1}{n} + \frac{1}{2};\frac{{1 - D}}{2}} \right)\frac{{\Gamma \left( {\frac{1}{n}} \right)\Gamma \left( {\frac{1}{2}} \right)}}{{\Gamma \left( {\frac{1}{n} + \frac{1}{2}} \right)}},$$
(47)

and

$$M = ia\int\limits_{ - \infty }^\infty {(qq_{x}^{ * } - q{\kern 1pt} {\text{*}}{{q}_{x}})dx} = \frac{{2a\kappa {{A}^{2}}}}{{{{2}^{{1/n}}}B}}F\left( {\frac{1}{n},\frac{1}{n};\frac{1}{n} + \frac{1}{2};\frac{{1 - D}}{2}} \right)\frac{{\Gamma \left( {\frac{1}{n}} \right)\Gamma \left( {\frac{1}{2}} \right)}}{{\Gamma \left( {\frac{1}{n} + \frac{1}{2}} \right)}},$$
(48)

where Gauss’ hypergeometric function is listed as:

$$F(\alpha ,\beta ;\gamma ;z) = \sum\limits_{n = 0}^\infty \frac{{{{{(\alpha )}}_{n}}{{{(\beta )}}_{n}}}}{{{{{(\gamma )}}_{n}}}}\frac{{{{z}^{n}}}}{{n!}},$$
(49)

and the Pochhammer symbol is:

$${{(p)}_{n}} = \left\{ \begin{gathered} 1,\quad n = 0, \hfill \\ p(p + 1) \cdots (p + n - 1),\quad n > 0. \hfill \\ \end{gathered} \right.$$
(50)

Convergence of the series is guaranteed by the condition

$$\left| z \right| < 1,$$
(51)

which, for (9) and (10), gives rise to

$$ - 1 < D < 3.$$
(52)

Finally, Rabbe’s criteria of convergence implies

$$\gamma < \alpha + \beta .$$
(53)

Thus, condition (53) and the domain of definition of Euler’s gamma function together implies that the CQ solitons for Kudryashov’s law of refractive index would exist for

$$0 < n < 2.$$
(54)

5 CONCLUSIONS

This paper retrieved bright and singular CQ optical soliton solutions with Kudryashov’s proposed law of refractive index. The applied integration algorithm is extended trial function approach. These soliton solutions are the limiting values to Jacobi’s elliptic functions when the modulus of ellipticity approached unity. The multiplier method yielded the conserved quantities that are presented and are in terms of Gauss’ hypergeometric function. The results thus provide immense future scope with the model of study. One of the first avenues to address soliton perturbation theory to this model, since the two conserved quantities are recovered. Subsequently, this model needs to be extended to birefringent fibers and recover its soliton solutions followed by the retrieval of its corresponding conservation laws in birefringent fibers. One notable feature is that (54) gives the condition of existence for the solitons that emerge from the convergence criterion of the hypergeometric function, thus avoiding the study of Benjamin-Fier stability analysis for this model. Pretty slick!