*4.2. The Cubic-Quartic Resonant Nonlinear Schrödinger Equation*

To apply the exp (−*ϕ* (*ξ*)) expansion method on the cubic-quartic resonant nonlinear Schrödinger equation, Equation (4), we utilize the same wave transformation of Equation (17). As a result, we get Equation (31). Finding the balance, we gain *n* = 1. Via inserting the value of the balance into Equation (10), we get the same result of Equation (36). Substituting Equation (36) into Equation (31) and setting each summation of the coefficients of the exponential identities of the same power to be zero, we discuss the following cases of solutions.

Case 1. When *b*<sup>1</sup> = <sup>2</sup>*b*<sup>0</sup> *<sup>λ</sup>* , *<sup>c</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup>*βλ*<sup>2</sup>(*λ*2−4*μ*) *b*2 0 , *<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>3*βλ*<sup>4</sup> 2*b*<sup>4</sup> 0 , *κ* = √−*<sup>c</sup>*3+*β*(*λ*2−4*μ*) √6 <sup>√</sup>*<sup>β</sup>* , and *<sup>ω</sup>* <sup>=</sup> <sup>−</sup>*c*32+2*c*3*β*(*λ*2−4*μ*)+5*β*<sup>2</sup>(*λ*2−4*μ*) 2 <sup>12</sup>*<sup>β</sup>* , we get the following solutions:

Solution 1. When *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* > 0 and *<sup>μ</sup>* = 0, we get hyperbolic function solution:

$$\begin{cases} \text{u} \begin{pmatrix} \text{u} \\ \text{v} \end{pmatrix} = \text{e} \begin{pmatrix} \frac{-\sqrt{-\text{c}\_{3} + \beta \left( \lambda^{2} - 4\mu \right)}}{\sqrt{6} \sqrt{\beta}} \text{x} + \frac{\left( -\frac{2}{3} + 2\mu \beta \left( \lambda^{2} - 4\mu \right) + 5\beta^{2} \left( \lambda^{2} - 4\mu \right)^{2} \right)}{12\mu} \text{t} \end{pmatrix} \\\\ \begin{pmatrix} \text{u} \\ \text{b} \end{pmatrix} = \frac{4\mu \text{b}\_{0}}{\lambda \left( -\lambda - \sqrt{\lambda^{2} - 4\mu} \tanh \left[ \frac{1}{2} \left( \text{c} + \text{x} + \frac{2\sqrt{\frac{2}{3}} \text{t} \left( -\text{c} + \beta \left( \lambda^{2} - 4\mu \right) \right)^{3/2}}{3\sqrt{\beta}} \right)} \sqrt{\lambda^{2} - 4\mu} \right) \end{pmatrix}. \end{cases} \tag{41}$$

This is a dark soliton solution, as shown in Figure 12.

**Figure 12.** 3D surface of Equation (41), which is a dark soliton solution plotted when *b*<sup>0</sup> = 0.2, *β* = 0.2, *c*<sup>3</sup> = 1, *c* = 1, *λ* = 3, *μ* = 1, and *t* = 2 for 2D.

Solution 2. When *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* < 0 and *<sup>μ</sup>* = 0, we get trigonometric function solution:

$$\begin{aligned} \mathbf{u} \cdot \mathbf{u} \left( \mathbf{x}, t \right) &= \mathbf{e} \begin{pmatrix} \frac{x \sqrt{-c\_3 + \beta \left( \lambda^2 - 4\mu \right)}}{\sqrt{6} \sqrt{\beta}} + \frac{t \left( -\frac{c\_3}{3} + 2\mu \beta \left( \lambda^2 - 4\mu \right) + 5\mu^2 \left( \lambda^2 - 4\mu \right)^2 \right)}{12\mu} \\\\ b\_0 &= \frac{4\mu b b\_0}{\lambda^2 - \lambda \sqrt{-\lambda^2 + 4\mu} \tan \left[ \frac{1}{2} \left( c + \mathbf{x} + \frac{2\sqrt{\frac{2}{3}} (-c\_3 + \beta \left( \lambda^2 - 4\mu \right)^{3/2}}{3\sqrt{\beta}} t \right) \sqrt{-\lambda^2 + 4\mu} \right]} \end{pmatrix}. \end{aligned} \tag{42}$$

This is a periodic singular soliton solution, as shown in Figure 13.

**Figure 13.** 3D surface of Equation (42), which is a periodic singular soliton solution plotted when *b*<sup>0</sup> = 0.4, *β* = 0.1, *c*<sup>3</sup> = −5, *c* = 1, *λ* = 1, *μ* = 1, and *t* = 2 for 2D.

Solution 3. When *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* > 0 and *<sup>μ</sup>* = 0, we get hyperbolic function solution:

$$\begin{cases} \mathbf{u} \left( \mathbf{x}, t \right) = \mathbf{e} \begin{pmatrix} -\frac{\mathbf{x} \sqrt{-\varepsilon\_{3} + \beta \lambda^{2}}}{\sqrt{6}\sqrt{\beta}} + \frac{t \left( -\frac{2}{3} + 2\alpha \beta \lambda^{2} + 5\beta^{2} \lambda^{4} \right)}{12\beta} \\\\ b\_{0} + \frac{2b\_{0}}{\cosh\left[ \lambda \left( c + \mathbf{x} + \frac{2\sqrt{\frac{2}{3}t(-c\mathbf{y} + \beta \lambda^{2})^{3/2}}}{3\sqrt{\beta}} \right) \right] + \sinh\left[ \lambda \left( c + \mathbf{x} + \frac{2\sqrt{\frac{2}{3}t(-c\mathbf{y} + \beta \lambda^{2})^{3/2}}}{3\sqrt{\beta}} \right) \right] \end{cases} \tag{43}$$

This is a bright singular combo soliton solution, as shown in Figure 14.

**Figure 14.** 3D surface of Equation (43), which is a singular soliton solution plotted when *b*<sup>0</sup> = 0.4, *β* = 0.2,*c*<sup>3</sup> = −1, *c* = 0.2, *λ* = 3, *μ* = 0, and *t* = 2 for 2D.

Case 2. When *b*<sup>0</sup> = √2 <sup>√</sup>*βλ*2(*λ*2−4*μ*) <sup>√</sup>*c*<sup>1</sup> , *<sup>b</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup> √2 <sup>√</sup>*βλ*2(*λ*2−4*μ*) <sup>√</sup>*c*1*<sup>λ</sup>* , *<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup> <sup>3</sup>*c*<sup>2</sup> 1 <sup>8</sup>*β*(*λ*2−4*μ*) <sup>2</sup> , *c*<sup>3</sup> = *β* −6*κ*<sup>2</sup> + *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* , and *ω* = <sup>1</sup> 2 *β* - −6*κ*<sup>4</sup> + *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* 2 , we get the following solutions:

Solution 1. When *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* > 0 and *<sup>μ</sup>* = 0, we get hyperbolic function solution:

$$u(x,t) = \sqrt{2\beta\lambda^2\left(\lambda^2 - 4\mu\right)} \operatorname{e}^{-i\mathbf{x}x + \frac{1}{2}i\beta\left(-6\kappa^4 + \left(\lambda^2 - 4\mu\right)^2\right)t}$$

$$\frac{\left(1 - \frac{4\mu}{\lambda^2 + \lambda\sqrt{\lambda^2 - 4\mu}\tanh\left[\frac{1}{2}(c + \mathbf{x} + 8\beta\kappa^3 t)\sqrt{\lambda^2 - 4\mu}\right]}{\sqrt{c\_1}}\right)}{\sqrt{c\_1}}.\tag{44}$$

This is a dark soliton solution, as shown in Figure 15.

**Figure 15.** 3D surface of Equation (44), which is a dark soliton solution plotted when *β* = 3, *c*<sup>1</sup> = 5, *c* = 0.03, *λ* = 1, *μ* = 0.1, *κ* = 0.1, and *t* = 2 for 2D.

Solution 2. When *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* < 0 and *<sup>μ</sup>* = 0, we get trigonometric function solution:

$$u(\mathbf{x},t) = \sqrt{2\beta\lambda^2\left(\lambda^2 - 4\mu\right)} \mathbf{e}^{-i\mathbf{x}\cdot\mathbf{x} + \frac{1}{2}i\beta\left(-6\mathbf{x}^4 + \left(\lambda^2 - 4\mu\right)^2\right)t}$$

$$\frac{\left(1 - \frac{4\mu}{\lambda^2 - \lambda\sqrt{-\lambda^2 + 4\mu}\tan\left[\frac{1}{2}\left(\mathbf{c} + \mathbf{x} + 8t\beta\mathbf{x}^3\right)\sqrt{-\lambda^2 + 4\mu}\right]}{\sqrt{c\_1}}\right)}{\sqrt{c\_1}}.\tag{45}$$

This is a periodic singular soliton solution, as shown in Figure 16.

**Figure 16.** 3D surface of Equation (45), which is a periodic singular soliton solution plotted when *β* = −3, *c*<sup>1</sup> = 5, *c* = 0.03, *λ* = 0.1, *μ* = 1, *κ* = 0.1, and *t* = 2 for 2D.

Solution 3. When *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* > 0 and *<sup>μ</sup>* = 0, we get the hyperbolic function solution:

$$u(\mathbf{x},t) = \frac{\sqrt{2}e^{-i\mathbf{x}\cdot\mathbf{x} + \frac{1}{2}i\beta\left(-6\kappa^{4} + \lambda^{4}\right)t}\sqrt{\beta\lambda^{4}}\coth\left(\frac{1}{2}\left(c+\mathbf{x}+8\beta\kappa^{3}t\right)\lambda\right)}{\sqrt{c\_{1}}},\tag{46}$$

which is a periodic singular solution, as shown in Figure 17.

**Figure 17.** 3D surface of Equation (46), which is a singular soliton solution plotted when *β* = 3, *c*<sup>1</sup> = 5, *c* = 0.03, *λ* = 0.1, *μ* = 0, *κ* = 0.1, and *t* = 2 for 2D.

#### **5. Conclusions**

In this research, the new dark, singular, bright singular combo soliton, and periodic singular solutions of the cubic-quantic nonlinear Schrödinger equation and the cubic-quantic resonant nonlinear Schrödinger equation were shown. Figures 1, 6, 8, 10, 12 and 15 are dark soliton solutions, Figures 2–4, 11, 14 and 17 are singular soliton solutions, Figures 5, 7, 13 and 16 are periodic singular solutions, and Figure 9 is bright singular combo soliton solution. The (*m* + *G*- /*G* ) expansion and exp(−*ϕ* (*ξ*)) expansion methods were utilized to study these two models with the parabolic law. The new solutions verified the main equations after we substituted them into Equations (3) and (4) for the existence of the equation.

Conte and Musette introduced that wave transformation, which we considered in this paper, protects the Painleve conditions and its properties [48]. Therefore, it can be seen that all results verified their physical properties and presented their estimated wave behaviors. Therefore, one can observe that the wave transformation considered in this paper in Equation (17) satisfies these conditions. We substituted all solutions to the main equations Equations (3) and (4), and they verified it; the constraint conditions Equations (20) and (29) were also used to verify this existence. The optical soliton solutions obtained in this research paper may be of concern and useful in many fields of science, such as mathematical physics, applied physics, nonlinear science, and engineering.

**Author Contributions:** All authors contributed equally.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
