In this section we will apply two methods to solve Eqs. (26) and (27). These methods are the Csch function method, and the Tanh–Coth method.

The solutions of many nonlinear equations can be expressed in the form [32]:

$$U\left(\xi \right)=A ^\left(\mu \xi \right),$$

(29)

$$V\left(\xi \right)=B ^\left(\mu \xi \right),$$

(30)

$$\left(\xi \right)=- A \tau \mu ^\left(\mu \xi \right) coth\left(\mu \xi \right),$$

(31)

$$^}\left(\xi \right)= A \tau ^ \lceil\tau ^\left(\mu \xi \right)+(\tau +1) ^\left(\mu \xi \right)\rceil,$$

(32)

where \(A ,\) \(\mu\), and \(\tau\) are parameters to be determined. Also, \(\mu\) is the wave width. Substituting Eqs. (29)-(32) into the reduced Eqs. (26) and (27), we get

$$a\left(M ^ ^^ \left(^\left(\mu \xi \right)+^\left(\mu \xi \right)\right)+^ ^\left(\mu \xi \right) \tau ^ \lceil\tau ^\left(\mu \xi \right)+(\tau +1) ^\left(\mu \xi \right)\rceil\right)-n\left(\omega +a^\right)^ ^\left(\mu \xi \right)+n_^ ^\left(\mu \xi \right)+n_ ^^ ^\left(\mu \xi \right)=0,$$

(33)

and

$$a\left(M^ ^^ \left(^\left(\mu \xi \right)+^\left(\mu \xi \right)\right) +^ ^\left(\mu \xi \right) \tau ^ \lceil\tau ^\left(\mu \xi \right)+(\tau +1) ^\left(\mu \xi \right)\rceil\right)-n\left(\omega +a^\right) ^ ^\left(\mu \xi \right)+n_^ ^\left(\mu \xi \right)+n_ ^^ ^\left(\mu \xi \right)=0.$$

(34)

Balancing the terms of the Csch functions, \(4\tau =2\tau +2\), we arrive at \(\tau =1\). Thus, Eqs. (33) and (34) simplify to

$$aM^ \left(^\left(\mu \xi \right)+^\left(\mu \xi \right)\right)+a ^\lceil ^\left(\mu \xi \right)+2^\left(\mu \xi \right)\rceil-n\left(\omega +a^\right) ^\left(\mu \xi \right)+n_^ ^\left(\mu \xi \right)+n_ ^ ^\left(\mu \xi \right)=0,$$

(35)

and

$$aM^ \left(^\left(\mu \xi \right)+^\left(\mu \xi \right)\right)+a ^\lceil ^\left(\mu \xi \right)+2^\left(\mu \xi \right)\rceil-n\left(\omega +a^\right) ^\left(\mu \xi \right)+n_^ ^\left(\mu \xi \right)+n_^ ^\left(\mu \xi \right)=0.$$

(36)

As a result, a set of algebraic equations is obtained:

$$\begin aM \mu^ + 2a \mu^ + n\sigma_ A^ + n\sigma_ B^ = 0, \hfill \\ aM \mu^ + 2a \mu^ + n\delta_ B^ + n\delta_ A^ = 0, \hfill \\ aM \mu^ + 2a \mu^ - n\left( } \right) = 0 \hfill \\ \end$$

(37)

Solving the system of equations in (37), we get:

$$A=B =\mp \sqrt^}_+_}}, \mu =\mp \sqrt^\right)}}$$

Therefore, the singular soliton solutions are extracted as

$$_\left(x,t\right)=_\left(x,t\right)=^}_+_}} Csch\left(\sqrt^\right)}}\left(x-\gamma t\right)\right) \right]}^\frac\text[i\left(-\kappa x+\omega t+_\right).$$

(38)

Figure 1 illustrates the behavior of a singular soliton, represented by the complex-valued solution (38). Figure 2 depicts a dark-singular straddled soliton, described by the complex-valued solution (49). In both figures, the evolution of the solitons is examined with the parameters set to: \(a=1\), \(k=1\), \(c=1\), \(_=1\), \(_=1\), and \(\omega =1\).

Analysis of the characteristics of a singular soliton, illustrated through its Surface, Contour, and 2D plots

Analysis of the characteristics of a dark-singular straddled soliton, illustrated through its Surface, Contour, and 2D plots

Assume \(\varpi =\varpi (\xi )\), by using the ansatz, [33, 34]:

$$Y=\text\left(\mu \xi \right),$$

(39)

that leads to the change of variables:

$$\frac=\mu \left(1-^\right)\frac,$$

(40)

and

$$\frac \varpi }} }} = \mu ^ \left[ } \right)\frac}} + \left( } \right)^ \frac \varpi }} }}} \right].$$

(41)

The solutions for Eqs. (26) and (27) are expressed in the form

$$U\left(Y\right)=\sum_^_ ^+\sum_^_ ^,$$

(42)

and

$$V\left(Y\right)=\sum_^_ ^+\sum_^_ ^.$$

(43)

By using the principle of the homogeneous balance method between the nonlinear term \(U ^}\) and the linear term \(^\) from Eq. (26), then \(N+N+2=4N\), which gives \(N=1\). Hence, Eq. (42) becomes:

$$U\left(Y\right)=\left(_+_Y+\frac_} \right).$$

(44)

Also by using balance method between the nonlinear term \(V ^}\) and the linear term \(^\) in Eq. (27), then \(M+M+2=4M\), which gives \(M=1\). Hence, Eq. (43) becomes

$$V\left(Y\right)=\left(_+_Y+\frac_} \right).$$

(45)

Here \(_, _, _ , _, _, and _\) are constants to be determined. Substituting Eqs. (44) and (45) along with their derivatives into Eqs. (26) and (27), we obtain:

$$aM_}^^\lceil\left(^+\frac^}\right)-4\left(^+\frac^}\right)+6\rceil+2a_^\lceil_\left(^+\frac^}\right)+ _\left(^+\frac^}\right)-_\left(Y+\frac\right)-2_ \rceil-n\left(\omega +a^\right) \lceil\left(_}^+2_}^\right)+2 __\left( Y+\frac \right)+ _}^\left(^+\frac^}\right)\rceil+n_\lceil_}^+2_}^\right)}^+4 __\left(_}^+2_}^\right)\left( Y+\frac \right)+_}^\left(_}^+2_}^\right)\left(^+\frac^}\right)+4_}^_}^\left( \left(^+\frac^}\right)+2\right)+4__}^\left( \left(^+\frac^}\right)+ \left(Y+\frac\right)\right)+_}^\left(_}^+2_}^\right)\left(^+\frac^}\right)+_}^\left(\left(^+\frac^}\right)+2\right)\rceil+n_\lceil\left(2\left(_}^+3 _}^\right)__+2\left(_}^+3_}^\right)__\right)\left( Y+\frac \right)+\left(\left(_}^+2_}^\right) _}^+\left(_}^+2_}^\right)_}^+2___\left(2_+_\right)\right)\left(^+\frac^}\right)+2__\left(__+__\right)\left(^+\frac^}\right)+8____\rceil=0,$$

(46)

and

$$aM_}^^\lceil\left(^+\frac^}\right)-4\left(^+\frac^}\right)+6\rceil+2a_^\lceil_\left(^+\frac^}\right)+ _\left(^+\frac^}\right)-_\left(Y+\frac\right)-2_ \rceil-n\left(\omega +a^\right) \lceil\left(_}^+2_}^\right)+2 __\left( Y+\frac \right)+ _}^\left(^+\frac^}\right)\rceil+n_\lceil_}^+2_}^\right)}^+4 __\left(_}^+2_}^\right)\left( Y+\frac \right)+_}^\left(_}^+2_}^\right)\left(^+\frac^}\right)+4_}^_}^\left( \left(^+\frac^}\right)+2\right)+4__}^\left( \left(^+\frac^}\right)+ \left(Y+\frac\right)\right)+_}^\left(_}^+2_}^\right)\left(^+\frac^}\right)+_}^\left(\left(^+\frac^}\right)+2\right)\rceil+n_\lceil\left(2\left(_}^+3 _}^\right)__+2\left(_}^+3_}^\right)__\right)\left( Y+\frac \right)+\left(\left(_}^+2_}^\right) _}^+\left(_}^+2_}^\right)_}^+2___\left(2_+_\right)\right)\left(^+\frac^}\right)+2__\left(__+__\right)\left(^+\frac^}\right)+8____\rceil=0.$$

(47)

Equations (46) and (47) yield the following set of algebraic equations:

$$\left(^+\frac^}\right): aM^+2a^+n__}^=0,$$

$$\left(^+\frac^}\right): aM^+2a^+n__}^=0,$$

$$\left(^+\frac^}\right): a_^+2__}^n_+_n_\left(__+__\right)=0,$$

$$\left(^+\frac^}\right): a_ ^+2__}^n_+_n_\left(__+__\right) =0,$$

$$\left(^+\frac^}\right): -4aM_}^^-n\left(\omega +a^\right)_}^ +n__}^\lceil\left(_}^+2_}^\right)+4_}^+\left(_}^+2_}^\right)\rceil+n_\left(\left(_}^+2_}^\right) _}^+\left(_}^+2_}^\right)_}^+2___\left(2_+_\right)\right)=0,$$

$$\left(^+\frac^}\right): -4 aM_}^^-n\left(\omega +a^\right)_}^ +n__}^\lceil\left(_}^+2_}^\right)+4_}^+\left(_}^+2_}^\right)\rceil+n_\left(\left(_}^+2_}^\right) _}^+\left(_}^+2_}^\right)_}^+2___\left(2_+_\right)\right)=0,$$

$$\left(Y+\frac\right): -2_a_^-2 __n\left(\omega +a^\right) +4 __n_\lceil\left(_}^+2_}^\right)+4__}^\rceil+2n_\left(\left(_}^+3 _}^\right)__+\left(_}^+3_}^\right)__\right)=0,$$

$$\left(Y+\frac\right): -2a_a}_^-2 __n\left(\omega +a^\right) +4 __n_\lceil\left(_}^+2_}^\right)+_}^\rceil+2 n_\lceil\left(_}^+3 _}^\right)__+\left(_}^+3_}^\right)__\rceil=0,$$

$$\left(^+\frac^}\right): 6aM_}^^-4_a_^-n\left(\omega +a^\right) \lceil\left(_}^+2_}^\right)\rceil+n_\lceil_}^+2_}^\right)}^+8_}^_}^+2_}^\rceil+8n_____=0,$$

$$\left(^+\frac^}\right): 6aM_}^^-4a_}^^-n\left(\omega +a^\right) \left(_}^+2_}^\right)+n_\lceil_}^+2_}^\right)}^+8_}^_}^+2_}^\rceil+8n _____=0.$$

(48)

Solving the set of algebraic Eqs. (48), we obtain:

Set 1

$$_=_=0, _=_= p, _=_=q.$$

Family 1

$$_=_=_=_=_=\mp \frac\sqrt^\right)\left(M+2\right)}}, _=\mp \frac\sqrt^\right)}}.$$

Thus, the dark-singular straddled solitons are presented as below:

$$_\left(x,t\right)=_\left(x,t\right)=_}^ }_\left(x-\gamma t\right)\right)+coth\left(_\left(x-\gamma t\right)\right) \right]}^\frac\times \text\left[i\left(-\kappa x+\omega t+_\right)\right].$$

(49)

Family 2

$$_=_=_=_=_=\mp \frac\sqrt^\right)\left(M+2\right)}}, _=\mp \frac\sqrt^\right)}}.$$

Consequently, the dark-singular straddled solitons are recovered as:

$$_\left(x,t\right)=_\left(x,t\right)=_}^ }_\left(x-\gamma t\right)\right)+coth\left( _\left(x-\gamma t\right)\right) \right]}^\frac\times \text[i\left(-\kappa x+\omega t+_\right).$$

(50)

Set 2

Family 1

$$_=_= _=_=_=_=_=\mp \frac \sqrt}_, j=\text,3,$$

$$_=\mp \sqrt^\right)}},$$

$$_=\mp \sqrt^\right)}},$$

$$_=\mp \sqrt^\right)}_\left(19_+8_\right)\left(M+2\right)\right)}}.$$

In this case, the dark-singular straddled solitons are structured as:

$$_\left(x,t\right)=_\left(x,t\right)=_}^\frac_\left(x-\gamma t\right)\right)+coth\left( _\left(x-\gamma t\right)\right) \right]}^\frac\times \text[i\left(-\kappa x+\omega t+_\right).$$

(51)

Family 2

$$_=_= _=_=_=_=_=\mp \sqrt}_, j=\text,3,$$

$$_=\mp \sqrt^\right)}},$$

$$_=\mp \sqrt^\right)},}$$

$$_=\mp \sqrt^\right)}.}$$

Hence, the dark-singular straddled solitons are extracted as:

$$_\left(x,t\right)=_\left(x,t\right)=_}^\frac_\left(x-\gamma t\right)\right)+coth\left( _\left(x-\gamma t\right)\right) \right]}^\frac\times \text[i\left(-\kappa x+\omega t+_\right).$$

(52)

Family 3

$$_=_= _=_=_=_=_=\mp \sqrt_-_\right) n}}_, j=\text,3,$$

$$_=\mp \sqrt^\right)}},$$

$$_=\mp \sqrt^\right)}},$$

$$_=\mp \sqrt^\right)}}.$$

Therefore, the dark-singular straddled solitons are introduced as below:

$$_\left(x,t\right)=_\left(x,t\right)=_}^\frac_\left(x-\gamma t\right)\right)+coth\left( _\left(x-\gamma t\right)\right) \right]}^\frac\times \text[i\left(-\kappa x+\omega t+_\right).$$

(53)

Family 4

$$_=_= _=_=_=_=_=\mp \sqrt_+16_\right) n}}_, j=\text,3,$$

$$_=\mp \sqrt_+16_\right)\left(\omega +a^\right)}_+4_\right)2M +\left(30_+36_\right)\right)}},$$

$$_ =\mp \sqrt_+16_\right)\left(\omega +a^\right)}_+8_\right)+\left(42 _+24_\right)2M\right)}},$$

$$_=\mp \sqrt^\right)}.}$$

Thus, the dark-singular straddled solitons are indicated below:

$$_\left(x,t\right)=_\left(x,t\right)=_}^\frac_\left(x-\gamma t\right)\right)+coth\left( _\left(x-\gamma t\right)\right) \right]}^\frac\times \text[i\left(-\kappa x+\omega t+_\right).$$

(54)

Family 5

$$_=_= _=_=_=_=_=\mp \sqrt_+28_\right)}}_, j=\text,3,$$

$$_=\mp \sqrt_+28_\right)\left(\omega +a^\right)}_+12_\right)9-2\right)2M}},$$

$$_=\mp \sqrt_+28_\right)\left(\omega +a^\right)}_+28_\right)-\left(14 _+8_\right)18M\right)}},$$

$$_=\mp \sqrt^\right)}}.$$

As a result, the dark-singular straddled solitons come out as:

$$_\left(x,t\right)=_\left(x,t\right)=_}^\frac_\left(x-\gamma t\right)\right)+coth\left( _\left(x-\gamma t\right)\right) \right]}^\frac\times \text[i\left(-\kappa x+\omega t+_\right).$$

(55)