*4.1. The Cubic-Quartic Nonlinear Schrödinger Equation*

To apply this method on the cubic-quartic nonlinear Schrödinger equation, Equation (3), we utilize the same wave transformation of Equation (17). As a result, we get Equation (22). Finding the balance, we gain *n* = 1. By inserting the value of the balance into Equation (10), we get:

$$\mathcal{U}I\left(\xi\right) = b\_0 + b\_1 \mathfrak{e}^{-\varphi\left(\xi\right)}.\tag{36}$$

Substituting Equation (40) into Equation (22) and setting each summation of the coefficients of the exponential identities of the same power to be zero, we discuss the following cases of the solutions.

Case 1. When *b*<sup>0</sup> = *<sup>λ</sup>b*<sup>1</sup> <sup>2</sup> , *<sup>c</sup>*<sup>1</sup> <sup>=</sup> <sup>8</sup>*a*2(*λ*2−4*μ*) *b*2 1 , *<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>24*a*<sup>2</sup> *b*4 1 , *κ* = − <sup>√</sup>*λ*2−4*<sup>μ</sup>* <sup>√</sup><sup>6</sup> , and *<sup>ω</sup>* <sup>=</sup> <sup>5</sup> <sup>12</sup> *a*<sup>2</sup> *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* 2 , we get the following solutions:

Solution 1. In the case *<sup>λ</sup>*<sup>2</sup> − <sup>4</sup>*<sup>μ</sup>* > 0 and *<sup>μ</sup>* = 0, we have a hyperbolic function solution:

$$\begin{aligned} \mu \left( \mathbf{x}, t \right) &= \mathbf{e} \begin{pmatrix} \frac{\left( \sqrt{\lambda^2 - 4\mu} \right)}{\sqrt{6}} \mathbf{x} + \frac{\mathbf{s}}{12} a\_2 \left( \lambda^2 - 4\mu \right)^2 t \end{pmatrix} \\\\ \left( \frac{\lambda b\_1}{2} + \frac{2\mu b\_1}{-\lambda - \sqrt{\lambda^2 - 4\mu} \tanh\left[ \frac{1}{2} \left( c + x - \frac{2}{3} \sqrt{\frac{2}{3}} a\_2 (\lambda^2 - 4\mu)^{3/2} t \right) \sqrt{\lambda^2 - 4\mu} \right]} \right) . \end{aligned} \tag{37}$$

This is a dark soliton solution, as shown in Figure 8.

**Figure 8.** 3D surface of Equation (37), which is a bright singular combo soliton solution plotted when *b*<sup>1</sup> = 0.2, *β* = 0.2, *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, *<sup>μ</sup>* = 0, and *<sup>λ</sup>* = 0, we have hyperbolic function solutions:

$$\begin{aligned} \text{Im}\,(\mathbf{x},t) &= \mathbf{e}^{\mathbf{i}\left(\frac{3}{2\pi}\beta\lambda^{4}t + \frac{\sqrt{2}}{\sqrt{6}}\mathbf{x}\right)} \\ &\quad \left(\frac{\lambda b\_{1}}{2} + \frac{\lambda b\_{1}}{-1 + \cosh\left(\lambda\left(c + \mathbf{x} - \frac{2}{3}\sqrt{\frac{2}{3}}\beta(\lambda^{2})^{3/2}t\right)\right) + \sinh\left(\lambda\left(c + \mathbf{x} - \frac{2}{3}\sqrt{\frac{2}{3}}\beta(\lambda^{2})^{3/2}t\right)\right)}\right) . \end{aligned} \tag{38}$$

This is a bright singular combo soliton solution, as shown in Figure 9.

**Figure 9.** 3D surface of Equation (38), which is a bright singular combo soliton solution plotted when *b*<sup>1</sup> = 0.04, *β* = 0.2, *c* = 0.2, *λ* = 1, *μ* = 0, and *t* = 2 for 2D.

Case 2. When *b*<sup>0</sup> = √3 <sup>√</sup>*c*1*<sup>λ</sup>* 2 √−*<sup>c</sup>*2(*λ*2−4*μ*) , *b*<sup>1</sup> = √3 <sup>√</sup>*<sup>c</sup>* <sup>√</sup> <sup>1</sup> −*c*2(*λ*2−4*μ*) , *κ* = − <sup>√</sup>*λ*2−4*<sup>μ</sup>* <sup>√</sup><sup>6</sup> , *<sup>ω</sup>* <sup>=</sup> <sup>−</sup>5*c*<sup>1</sup> 2 32*c*<sup>2</sup> , and *<sup>β</sup>* <sup>=</sup> <sup>−</sup> <sup>3</sup>*c*<sup>2</sup> 1 <sup>8</sup>*c*2(*λ*2−4*μ*) 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 a dark solution, as shown in Figure 10:

$$\begin{cases} \mathbf{u} \left( \mathbf{x}, t \right) \stackrel{\text{i} \left( -\frac{5z\_1^2}{32\xi\_1} t + \frac{\sqrt{\lambda^2 - 4\mu}}{\sqrt{6}} \mathbf{x} \right)} \\\\ \mathbf{u} \left( \frac{\sqrt{3}\sqrt{c\_1} \left( \lambda^2 - 4\mu + \lambda\sqrt{\lambda^2 - 4\mu} \tanh\left( \frac{c\_1^2 t}{4\sqrt{bc\_2}} + \frac{1}{2} \left( c + \mathbf{x} \right) \sqrt{\lambda^2 - 4\mu} \right) \right)}{2\sqrt{-c\_2 \left( \lambda^2 - 4\mu \right)} \left( \lambda + \sqrt{\lambda^2 - 4\mu} \tanh\left( \frac{c\_1^2 t}{4\sqrt{bc\_2}} + \frac{1}{2} \left( c + \mathbf{x} \right) \sqrt{\lambda^2 - 4\mu} \right) \right)} \right) \end{cases} \tag{39}$$

**Figure 10.** 3D surface of Equation (39), which is a dark soliton solution plotted when *b*<sup>1</sup> = 0.2, *β* = 0.2, *c*<sup>1</sup> = 0.1, *c*<sup>2</sup> = −0.1, *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 hyperbolic function solution:

$$\mathbf{u}\left(\mathbf{x},t\right) = \mathbf{e}^{\mathbf{i}\left(-\frac{5c\_{1}^{2}}{32c\_{2}}t + \frac{\sqrt{\lambda^{2}-4\kappa}}{\sqrt{6}}\mathbf{x}\right)} \left(\frac{\sqrt{3}\sqrt{c\_{1}}\lambda}{2}\mathbf{\hat{i}\left(12\left(c+\mathbf{x}\right)+\frac{\sqrt{6c\_{1}^{2}}}{c\_{2}\sqrt{\lambda^{2}}}t\right)}\right) \,\mathrm{.}\tag{40}$$

This is a singular soliton solution, as shown in Figure 11.

**Figure 11.** 3D figure of Equation (40), which is a singular soliton solution plotted when *b*<sup>1</sup> = 4, *β* = 0.2, *c*<sup>1</sup> = 0.1, *c*<sup>2</sup> = −0.1, *c* = 0.2, *λ* = 1, *μ* = 0, and *t* = 2 for 2D.
