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

To solve Equation (4), by the *m* + *<sup>G</sup>*- *G* expansion method, we consider wave transformation Equation (17). Replacing Equation (17) into Equation (4) and separating the outcome equation into real and imaginary parts, we can write the real part as follows:

$$\left(\kappa^3 \left(\mathfrak{a} - \beta\mathfrak{k}\right) + \omega\right) \mathcal{U} - c\_1 \mathcal{U}^3 - c\_2 \mathcal{U}^5 - \left(c\_3 + 3\kappa \left(\mathfrak{a} - 2\beta\mathfrak{k}\right)\right) \mathcal{U}'' - \beta \mathcal{U}^{(4)} = 0,\tag{27}$$

and the imaginary part can be written as:

$$\left(3a\kappa^2 - 4\beta\kappa^3 + \nu\right)\mathcal{U}' - \left(a - 4\beta\kappa\right)\mathcal{U}'^{\prime\prime} = 0.\tag{28}$$

From Equation (28) *U*- = 0 and *U*---= 0, then:

$$\nu = 4\beta \kappa^3 - 3a\kappa^2, \qquad \qquad a = 4\beta \kappa. \tag{29}$$

Hence, Equation (27) can be rewritten as:

$$\left(3\beta\kappa^4 + \omega\right)\mathcal{U} - c\_1\mathcal{U}^3 - c\_2\mathcal{U}^5 - \left(c\_3 + 6\beta\kappa^2\right)\mathcal{U}'' - \beta\mathcal{U}^{(4)} = 0. \tag{30}$$

Multiplying both sides of Equation (30) by *U*and integrating it once with respect to *ξ*, we get:

$$\left(36\beta\kappa^4 + 12\omega\right)ll^2 - 6c\_1ll^4 - 4c\_2ll^6 - \left(12c\_3 + 72\beta\kappa^2\right)\left(ll'\right)^2 + \beta\left(12\left(ll'\right)^2 - 24ll'll'\right) = 0.\tag{31}$$

Finding the balance, we gain *n* = 1. Putting this value into Equation (8), we get the same result of Equation (23). Substituting Equation (23) with Equation (9) into Equation (4), we get the following solutions:

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

Solution 1. In the case Δ > 0, we have an exponential function solution:

$$\begin{aligned} \mathbf{u} \cdot (\mathbf{x}, t) &= \mathbf{e}^{\mathbf{i}\left(-\kappa x + \frac{1}{2}\beta \left(-6\kappa^{4} + \left(\left(2m + \lambda\right)^{2} - 4\mu\right)^{2}\right)t\right)} \\\\ &\left(a\_{0} - \frac{2\left(m \left(m + \lambda\right) - \mu\right)a\_{0}}{\left(m + \frac{1}{2}\left(-2m + \left(1 - \frac{2A\_{1}}{A\_{1} + A\_{2}\mathbf{e}^{\sqrt{\Delta}\left(x + 8\mathbf{i}\mathbf{k}^{T}\right)}}\right)\sqrt{\Delta} - \lambda\right)}\right)}\right) . \end{aligned} \tag{32}$$

Considering some values of parameters, Figure 4 shows singular soliton solution.

**Figure 4.** 3D figure of Equation (32), which is a singular soliton solution plotted when *A*<sup>1</sup> = 1, *A*<sup>2</sup> = 3, *λ* = 1, *m* = 1, *μ* = −1, *β* = 0.2, *a*<sup>0</sup> = 0.2, *κ* = 0.01, and *t* = 2 for 2D.

Solution 2. In the case Δ < 0, we have a trigonometric function solution:

$$\begin{aligned} \mathbf{u} \cdot (\mathbf{x}, t) &= \mathbf{e}^{-\mathrm{i}\mathbf{x} + \frac{1}{2}\beta \left( -\delta \mathbf{x}^{4} + \left( (2m + \lambda)^{2} - 4\mu \right)^{2} t \right)} \\ & \quad \left( a\_{0} + \frac{4a\_{0} \left( m^{2} + m\lambda - \mu \right) \left( A\_{2} \cos \left( \alpha \right) + A\_{1} \sin \left( \alpha \right) \right)}{\lambda \left( \left( -A\_{1} \sqrt{-\Delta} + A\_{2} \lambda \right) \cos \left( \alpha \right) + \left( A\_{2} \sqrt{-\Delta} + A\_{1} \lambda \right) \sin \left( \alpha \right) \right)} \right), \end{aligned} \tag{33}$$

which is *α* = <sup>1</sup> 2 √−<sup>Δ</sup> *x* + 8*βκ*3*t* .

Periodic singular solution is plotted in Figure 5. Case 2. When *<sup>a</sup>*−<sup>1</sup> <sup>=</sup> 0, *<sup>a</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup>*a*<sup>0</sup> *<sup>λ</sup>* , *<sup>ω</sup>* <sup>=</sup> <sup>1</sup> 2 *β* −6*κ*<sup>4</sup> + - (2*m* + *λ*) <sup>2</sup> <sup>−</sup> <sup>4</sup>*<sup>μ</sup>* 2 , *<sup>c</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup>*βλ*<sup>2</sup>((2*m*+*λ*) <sup>2</sup>−4*μ*) *a*2 0 , *<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>3*βλ*<sup>4</sup> 2*a*<sup>4</sup> 0 , and *c*<sup>3</sup> = *β* - −6*κ*<sup>2</sup> + (2*<sup>m</sup>* + *<sup>λ</sup>*) <sup>2</sup> <sup>−</sup> <sup>4</sup>*<sup>μ</sup>* , we obtain the following solutions:

Solution 1. In the case Δ > 0, we get dark solution, as shown in Figure 6 :

$$\begin{split} \mathbf{u} \cdot (\mathbf{x}, y) &= \mathrm{e}^{\mathrm{i}\left(-\kappa x + \frac{1}{2}\beta \left(-6\alpha^{4} + \left((2m+\lambda)^{2} - 4\mu\right)^{2}\right)t\right)} \\ &\left(a\_{0} + \frac{2\left(m + \frac{1}{2}\left(-2m + \left(1 - \frac{2A\_{1}}{A\_{1} + A\_{2}\mathbf{e}^{\sqrt{\Delta}\left(x + 8\theta\alpha^{3}t\right)}}\right)\sqrt{\Delta} - \lambda\right)\right)a\_{0}}{\lambda}\right) . \end{split} \tag{34}$$

Figure 6 shows the dark structure this solution.

**Figure 5.** 3D surface of Equation (33), which is a periodic singular soliton solution plotted when *A*<sup>1</sup> = 1, *A*<sup>2</sup> = 2, *λ* = 1, *m* = <sup>1</sup> <sup>2</sup> , *μ* = 2, *β* = 0.1, *a*<sup>0</sup> = 2, *κ* = 0.1, and *t* = 2 for 2D.

**Figure 6.** 3D surface of Equation (34), which is a dark soliton solution plotted when *A*<sup>1</sup> = 1, *A*<sup>2</sup> = 3, *λ* = 1, *m* = 1, *μ* = −1, *β* = 0.2, *a*<sup>0</sup> = 0.2, *κ* = 0.01, and *t* = 2 for 2D.

Solution 2. In the case Δ < 0, we have a trigonometric function solution:

$$\begin{split} \mathbf{u} \cdot (\mathbf{x}, t) &= \mathbf{e}^{-\mathrm{i}\mathbf{x} + \frac{1}{2}\beta \left( -\delta \mathbf{x}^{4} + \left( (2m + \lambda)^{2} - 4\mu \right)^{2} \right)\_{\mathbf{f}}} \\ & \quad \left( \frac{\sqrt{-\Delta} \left( A\_{1} \cos \left( \frac{1}{2} \sqrt{-\Delta} \left( \mathbf{x} + 8\beta \mathbf{x}^{3} t \right) \right) - A\_{2} a\_{0} \sin \left( \frac{1}{2} \sqrt{-\Delta} \left( \mathbf{x} + 8\beta \mathbf{x}^{3} t \right) \right) \right)}{\lambda \left( A\_{2} \cos \left( \frac{1}{2} \sqrt{-\Delta} \left( \mathbf{x} + 8\beta \mathbf{x}^{3} t \right) \right) + A\_{1} \sin \left( \frac{1}{2} \sqrt{-\Delta} \left( \mathbf{x} + 8\beta \mathbf{x}^{3} t \right) \right) \right)} \right) . \end{split} \tag{35}$$

Periodic singular solution is plotted in Figure 7.

**Figure 7.** 3D figure of Equation (35), which is a periodic singular soliton solution plotted when *A*<sup>1</sup> = 1, *A*<sup>2</sup> = 2, *λ* = 1, *m* = <sup>1</sup> <sup>2</sup> , *μ* = 2, *β* = 0.1, *a*<sup>0</sup> = 0.2, *κ* = 0.1, and *t* = 2 for 2D.
