Dark Soliton Dynamics for the Resonant Nonlinear Schrödinger Equation with Third- and Fourth-Order Dispersions

Abstract

Optical solitons have emerged as a highly active research domain in nonlinear fiber optics, driving significant advancements and enabling a wide range of practical applications. This study investigates the dynamics of dark solitons in systems governed by the resonant nonlinear Schrödinger equation (RNLSE). For the RNLSE with third-order (3OD) and fourth-order (4OD) dispersions, the dark soliton solution of the equation in the (1+1)-dimensional case is derived using the F-expansion method, and the analytical study is extended to the (2+1)-dimensional case via the self-similar method. Subsequently, the nonlinear equation incorporating perturbation terms is further studied, with particular attention given to the dark soliton solutions in both one and two dimensions. The soliton dynamics are illustrated through graphical representations to elucidate their propagation characteristics. Finally, modulation instability analysis is conducted to evaluate the stability of the nonlinear system. These theoretical findings provide a solid foundation for experimental investigations of dark solitons within the systems governed by the RNLSE model.

1. Introduction

As an important class of nonlinear physical phenomena, solitons exhibit unique mathematical properties and physical behaviors. Solitons have been discovered and studied in many branches of physics, including optics [1]. Solitons have found widespread applications across various domains, including plasma physics [2], nonlinear optics [3], fluid dynamics [4], and fiber laser technologies [5]. In 1973, the use of solitons in optical fibers for optical communications was proposed by Hasegawa [6]. Their stable propagation characteristics make them well-suited for long-distance and high-capacity data transmission [7]. In particular, dark solitons, as an important type of soliton, have advanced properties that have made them increasingly prominent in the field of optical communication [8]. Additionally, soliton dynamics are increasingly being explored in emerging areas such as optical computing, quantum communication, and photonic neural networks [9].

The physical phenomena described by nonlinear partial differential equations (NLPDEs) are relevant to many areas of sciences [10,11], such as nonlinear fiber optics, plasma physics, engineering sciences, fluid mechanics, and geochemistry. The study of the solitons in NLPDEs has been the focus of considerable scientific attention. The nonlinear Schrödinger equations (NLSE) describe wave propagation in optical fibers with nonlinear impacts [12] and provide a theoretical basis for understanding the wave dynamic behavior in complex systems. Among various NLSEs, the resonant nonlinear Schrödinger equation (RNLSE) is formulated to account for resonance effects, which are essential for a more precise characterization of the nonlinear dynamics of solitons [13,14,15]. Unlike prior studies limited to the (1+1)-dimensional ((1+1)D) case [13,15] or unperturbed models [14], our research presents a systematic study of dark soliton dynamics in the (2+1)-dimensional RNLSE, incorporating higher-order dispersion and perturbation terms (PTs) and utilizing self-similarity for dimensional extension.

Typically, group velocity dispersion (GVD) and self-phase modulation balance each other, allowing optical solitons to propagate stably over long distances: for example, in optical fibers. However, in certain cases, GVD may be negligible, and the dispersion effects are primarily compensated by third-order (3OD) and fourth-order (4OD) dispersion terms. Therefore, the NLSE with 3OD and 4OD is also referred to as the cubic-quartic nonlinear Schrödinger equation (CQ-NLSE) [16,17]. Many researchers have made significant contributions to the development of the CQ-NLSE. Y. X. Li et al. employed a method based on the complete discrimination system for polynomials to derive solutions to the CQ-NLSE [18]. This research focuses on the wave structures and chaotic behavior of the standard CQ-NLSE, without considering the resonance term and perturbation effects. Emad H. M. Zahran et al. utilized the extended simple equation method to obtain soliton solutions for the CQ-NLSE [19]. However, their model does not include resonance terms and is limited to a specific nonlinear form. Salman A. Alqahtani et al. obtained soliton solutions for the perturbed CQ-NLSE with cubic–quintic–septic–nonic nonlinearities, using Kudryashov’s method and an innovative mapping technique [20]. However, they solved for various soliton forms only in a single dimension. By comparison, our work investigates the dark soliton dynamics in the (2+1)-dimension of the RNLSE for the first time. J. Ahmad et al. proposed an improved modified extended tanh-expansion method for solving the time-fractional coupled NLSE, which describes pulse propagation in dual-mode optical fibers [21]. However, their research did not cover PTs or stability analysis. To the best of our knowledge, there have been several studies on the NLSE with 3OD and 4OD dispersions. However, research on the RNLSE incorporating both 3OD and 4OD dispersions and PTs is very limited. Based on the proposed model, this paper investigates the dynamics of dark solitons in resonant perturbation systems, offering a novel analytical perspective for exploring dark soliton dynamics in complex systems.

This article is organized as follows. Section 2 introduces the basic equation model and provides an overview of the F-expansion method [22,23] applied to the RNLSE. Section 3 presents a detailed derivation of the dark soliton solution for the (1+1)D RNLSE incorporating 3OD and 4OD dispersions using the F-expansion method. This analysis is then extended to the (2+1)-dimensional ((2+1)D) case via the self-similar method [24], with graphical illustrations provided to depict the soliton dynamics. Section 4 investigates the RNLSE incorporating 3OD and 4OD dispersions with PTs, obtaining the dark soliton solutions in both (1+1) and (2+1) dimensions. Section 5 conducts the modulation instability analysis [25] for the (1+1)D equations. The final section provides concluding remarks.

2. RNLSE with 3OD and 4OD Dispersions and the F-Expansion Method

The RNLSE with 3OD and 4OD dispersions is formulated as [26]:

$$i{\psi }_t+i{\alpha }_1{\nabla }^3\psi +{\alpha }_2{\nabla }^4\psi +\beta |\psi {|}^2\psi +\gamma {\displaystyle \frac{{\nabla }^2|\psi |}{|\psi |}}\psi =0$$

(1)

where $\psi$ denotes the complex-valued wave function, and $i{\psi }_t$ corresponds to the linear evolution term. ${\alpha }_1$ and ${\alpha }_2$ are the coefficients of 3OD and 4OD dispersion, respectively. $\beta$ is the Kerr nonlinear coefficient, arising from the dependence of light intensity on the refractive index of the medium. $\gamma$ represents the coefficient of the resonant term (also known as the Bohm potential). This term plays a significant role in the study of chiral solitons, particularly in terms of the quantum Hall effect, where it contributes to the nonlinear dynamics and the formation of soliton structures in quantum systems. The coefficients ${\alpha }_1$, ${\alpha }_2$, $\beta$ and $\gamma$ are real constants $({\alpha }_1,{\alpha }_2,\beta ,\gamma \ne 0)$.

The RNLSE with 3OD and 4OD dispersions in presence of PTs extends to:

$$i{\psi }_t+i{\alpha }_1{\nabla }^3\psi +{\alpha }_2{\nabla }^4\psi +\beta |\psi {|}^2\psi +\gamma {\displaystyle \frac{{\nabla }^2|\psi |}{|\psi |}}\psi =i(\delta {\psi }_x+\lambda {(|\psi {|}^{2m}\psi )}_x+\mu {(|\psi |)}_x^{2m}\psi )$$

(2)

where δ denotes the inter-modal dispersion, λ represents the self-steepening perturbation coefficient for short pulses, and μ corresponds to the higher-order dispersion coefficient. m represents the order of nonlinearity, which describes the complete nonlinearity strength of the system. The typical value of m is m ≥ 1.

To solve the above equations, the F-expansion method is applied and the following differential equation [27] is examined:

$$N(u,u_t,u_x,u_{xx},\dots )$$

(3)

where $N$ represents a polynomial involving $u(x,t)$ and its partial derivatives of various orders, while $u(x,t)$ denotes the unknown function to be solved, utilizing the wave transformation $u(x,t)=u(\xi )$ with $\xi =px+qt$, where $p$ and $q$ are real constants. It is assumed that $u(\xi )$ can be represented as a finite power series of $G(\xi )$ in the following form:

$$u(\xi )={\displaystyle \sum _{i=0}^n{h}_i{G}^i(\xi )},\hspace{1em}{h}_i\ne 0$$

(4)

where $h_i$ is a constant, and $n$ represents the highest-order power index. The power index $n$ of the highest-order term can be determined by balancing the highest-order nonlinear term with the highest-order derivative term in the ordinary differential Equation. $G(\xi )$ is defined as:

$${({\displaystyle \frac{dG(\xi )}{d\xi }})}^2=a_l{G}^l+a_{l-1}{G}^{l-1}+\dots +a_2{G}^2+a_{1}G+a_0$$

(5)

where the coefficients $p$,$q$,$a_l$,$a_{l-1}$, … $a_2$,$a_1$,$a_0$ constitute undetermined parameters, the exact values of which will be ascertained in subsequent computational stages of the solution process. The polynomial expression for $G(\xi )$ is obtained by substituting the Equations (4) and (5) into Equation (3). The power index $n$ in Equation (4) is determined by setting all coefficients corresponding to different powers of $G(\xi )$ in the polynomial to zero. These coefficients are then determined by solving the resulting system of equations. The exact solution $u(\xi )$ of Equation (3) is ultimately obtained by substituting both the solution for $G(\xi )$ and the determined coefficients from Equation (4).

3. Dark Soliton Solution for RNLSE with 3OD and 4OD Dispersions in the F-Expansion Method

The (1+1)- and (2+1)D dark soliton solutions are calculated via the F-expansion method. First, the (1+1)D RNLSE incorporating 3OD and 4OD dispersions with Kerr nonlinearity is given by:

$$i{\psi }_t+i{\alpha }_1{\psi }_{xxx}+{\alpha }_2{\psi }_{xxxx}+\beta |\psi {|}^2\psi +\gamma {\displaystyle \frac{|\psi {|}_{xx}}{|\psi |}}\psi =0$$

(6)

The traveling wave ansatz is employed for the wave function $\psi (x,t)$, expressed as:

$$\psi (x,t)=\phi (\xi )e^{i(kx+wt)}$$

(7)

Equation (7) includes the phase term $e^{i(kx+wt)}$, which describes the linear propagation of the carrier in the optical medium. k is the wave number and w is the angular frequency. The envelope term $\phi (\xi )$ describes the shape of the pulse. The variable $\xi$ is defined as:

$$\xi =px+qt$$

(8)

which is the coordinate system representing motion at a speed of q/p. $k$, $w$, $p$, $q$ are constant parameters to be determined. The modulus of $\psi (x,t)$ is represented as $|\psi |=\phi (\xi )$. Using the above formulas, all partial derivatives in Equation (6) can be expressed as:

$${\psi }_t=(q\phi \prime +iw\phi )e^{i(kx+wt)}$$

(9a)

$${\psi }_x=(p\phi \prime +ik\phi )e^{i(kx+wt)}$$

(9b)

$${\psi }_{xx}=(p^2\phi ″+2ikp\phi \prime -k^2\phi )e^{i(kx+wt)}$$

(9c)

$${\psi }_{xxx}=(p^3\phi ‴+3ik{p}^2\phi ″-3k^{2}p\phi \prime -i{k}^3\phi )e^{i(kx+wt)}$$

(9d)

$${\psi }_{xxxx}=(p^4\phi \prime \prime \prime \prime +4ik{p}^3\phi ‴-6k^2{p}^2\phi ″-4i{k}^{3}p\phi \prime +k^4\phi )e^{i(kx+wt)}$$

(9e)

$$|\psi {|}_{xx}=p^2\phi ″$$

(9f)

By substituting Equation (9) into Equation (6) and separating the imaginary and real parts, we obtain the following system of Equations:

$$({\alpha }_1+4{\alpha }_{2}k)p^3\phi ‴+(q-3{\alpha }_1{k}^{2}p-4{\alpha }_2{k}^{3}p)\phi \prime =0$$

(10a)

$${\alpha }_2{p}^4\phi \prime \prime \prime \prime -(3{\alpha }_{1}k{p}^2+6{\alpha }_2{k}^2{p}^2-\gamma p^2)\phi ″+({\alpha }_2{k}^4+{\alpha }_1{k}^3-w)\phi +\beta {\phi }^3=0$$

(10b)

By integrating both sides of Equation (10a), as shown in Equation (11a), we can then applying the boundary condition, where $\xi \to \infty$, $\phi ″\left(\xi \right)\to 0$ and $\phi ‴\left(\xi \right)\to 0$. We set the integration constant C₁ = 0 in Equation (11b):

$${\displaystyle \int ({\alpha }_1+4{\alpha }_{2}k)p^3\phi ‴+(q-3{\alpha }_1{k}^{2}p-4{\alpha }_2{k}^{3}p)\phi }\prime d\phi =C_1$$

(11a)

$$({\alpha }_1+4{\alpha }_{2}k)p^3\phi ″+(q-3{\alpha }_1{k}^{2}p-4{\alpha }_2{k}^{3}p)\phi =0$$

(11b)

We can deduce and obtain:

$$q=3{\alpha }_1{k}^{2}p+4{\alpha }_2{k}^{3}p$$

(12a)

Then, by substituting the above Equation (12a) to Equation (11a), and considering that for any value of $\phi ‴$, the Equation (11a) is equal to zero, we can obtain:

$$k=-{\displaystyle \frac{{\alpha }_1}{4{\alpha }_2}}$$

(12b)

To solve for $\phi (\xi )$ using the F-expansion method, we employ the following equation:

$$\phi (\xi )=G(\xi )$$

(13)

The $G(\xi )$ is provided as [27]:

$${({\displaystyle \frac{dG(\xi )}{d\xi }})}^2=a_4{G}^4+a_2{G}^2+a_0$$

(14)

Then, the second derivatives of $\phi$ can be represented as [28]:

$${(\phi \prime )}^2=a_4{\phi }^4+a_2{\phi }^2+a_0$$

(15a)

Subsequently, the derivatives of $\phi$ are taken with respect to both sides of Equation (15a):

$$2\phi \prime \phi ″=4a_4{\phi }^3\phi \prime +2a_2\phi \phi \prime$$

(15b)

Eliminating the $\phi \prime$ on both sides of Equation (15b), $\phi ″$ allows us to obtain:

$$\phi ″=2a_4{\phi }^3+a_2\phi$$

(15c)

Similarly, $\phi ‴$ and $\phi \prime \prime \prime \prime$ can be obtained as follows:

$$\phi ‴=6a_4{\phi }^2+a_2$$

(15d)

$$\phi \prime \prime \prime \prime =12a_4\phi$$

(15e)

Substituting Equation (15) into Equation (11b), extracting $\phi$ and ${\phi }^3$ and simplifying the equation, we obtain:

$$\begin{array}{l}\left[12{\alpha }_2{a}_4{p}^4-(3{\alpha }_1{a}_{2}k{p}^2+6{\alpha }_2{a}_2{k}^2{p}^2-a_2\gamma p^2)+({\alpha }_2{k}^4+{\alpha }_1{k}^3-w)\right]\phi \\ -\left(6{\alpha }_1{a}_{4}k{p}^2+12{\alpha }_2{a}_4{k}^2{p}^2-2\gamma a_4{p}^2+\beta \right){\phi }^3=0\end{array}$$

(16)

To ensure that the equation holds identically, the terms before $\phi$ and ${\phi }^3$ should be equal to zero, that is:

$${\phi }^3:\hspace{1em}-12{\alpha }_2{a}_4{k}^2{p}^2-6{\alpha }_1{a}_{4}k{p}^2-2\gamma a_4{p}^2+\beta =0$$

(17a)

$$\phi :\hspace{1em}12{\alpha }_2{a}_4{p}^4-(3{\alpha }_1{a}_{2}k{p}^2+6{\alpha }_2{a}_2{k}^2{p}^2-a_2\gamma p^2)+({\alpha }_2{k}^4+{\alpha }_1{k}^3-w)=0$$

(17b)

By solving Equation (17), the coefficients can be written as:

$$a_4={\displaystyle \frac{\beta }{12{\alpha }_2{k}^2{p}^2+6{\alpha }_{1}k{p}^2-2\gamma p^2}}$$

(18a)

$$a_2={\displaystyle \frac{1}{6}}({\displaystyle \frac{{\alpha }_1{k}^3+{\alpha }_2{k}^4-w}{{\alpha }_2{k}^2{p}^2+\frac{1}{2}{\alpha }_{1}k{p}^2-\frac{1}{6}\gamma p^2}}+{\displaystyle \frac{{\alpha }_2\beta }{{({\alpha }_2{k}^2+\frac{1}{2}{\alpha }_{1}k-\frac{1}{6}\gamma )}^2}})$$

(18b)

When $a_0=a_2^2/4a_4$, the right-hand side of Equation (14) becomes a perfect square and its exact analytical solution can be obtained. In this scenario, $\phi (\xi )$ possesses an exact analytical solution. Then, we can reach the analytic solution [29]:

$$\phi (\xi )=-i\sqrt{{\displaystyle \frac{a_2}{2a_4}}}\tanh (\pm \sqrt{{\displaystyle \frac{a_2}{2}}}\xi )$$

(19)

By substituting Equation (19) into Equation (7), the analytical solution of dark soliton can be derived as:

$$\psi (x,t)=-i\sqrt{{\displaystyle \frac{a_2}{2a_4}}}\tanh (\pm \sqrt{{\displaystyle \frac{a_2}{2}}}(px+qt))e^{i(kx+wt)}$$

(20)

Substituting the coefficients of Equations (18) and (12) into Equation (20), we obtain the exact solution of $\psi (x,t)$:

$$\begin{array}{l}\psi (x,t)=-2i\sqrt{-{\displaystyle \frac{\frac{3{\alpha }_1^4}{256{\alpha }_2^3}+w}{\beta }}-{\displaystyle \frac{{\alpha }_2{p}^2}{\frac{{\alpha }_1^2}{16{\alpha }_2}+\frac{1}{6}\gamma }}}\tanh [\pm {\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\frac{3{\alpha }_1^4}{256{\alpha }_2^3}+w}{3(\frac{{\alpha }_1^2}{16{\alpha }_2}+\frac{1}{6}\gamma )}}+{\displaystyle \frac{{\alpha }_2\beta p^2}{3{(\frac{{\alpha }_1^2}{16{\alpha }_2}+\frac{1}{6}\gamma )}^2}}}\\ \times (x+{\displaystyle \frac{{\alpha }_1^3}{4{\alpha }_2^2}}t)]e^{i(-{\displaystyle \frac{{\alpha }_1}{4{\alpha }_2}}x+wt)}\end{array}$$

(21)

By applying Euler’s formula $e^{i(kx+wt)}=\cos (kx+wt)+i\sin (kx+wt)$, the real and imaginary parts of Equation (20) are obtained. Figure 1 shows the waveform plots of the dark soliton solution of Equation (21) with appropriate coefficients under static conditions t = 0, while Figure 2 shows the corresponding contour plots.

Subsequently, RNLSE incorporating 3OD and 4OD dispersions with Kerr nonlinearity is investigated in the (2+1)D case. The equation is expressed as:

$$i{\psi }_t+i{\alpha }_1({\psi }_{xxx}+{\psi }_{yyy})+{\alpha }_2({\psi }_{xxxx}+{\psi }_{yyyy})+\beta |\psi {|}^2\psi +\gamma {\displaystyle \frac{|\psi {|}_{xx}+|\psi {|}_{yy}}{|\psi |}}\psi =0$$

(22)

By considering the self-similar method, the (2+1)D traveling wave ansatz can be expressed as:

$$\psi (x,y,t)=\phi (\xi )e^{i(k_{1}x+k_{2}y+wt)}$$

(23)

where the $\xi$ is expressed in the following (2+1)D form:

$$\xi =p_{1}x+p_{2}y+qt$$

(24)

where $k_1$, $k_2$, $w$, $p_1$, $p_2$ and $q$ are undetermined coefficients. We calculate the partial derivatives involved in Equation (22) and substitute them into Equation (22). Then, we separate the real and imaginary parts. The F-expansion method is reapplied to derive the expressions. Finally, by substituting the obtained expressions into Equation (20), the solution $\psi (x,y,t)$ can be formulated as:

$$\begin{array}{l}\psi (x,y,t)=-2i\sqrt{A}\tanh [\pm {\displaystyle \frac{1}{2}}\sqrt{B}(x+{\displaystyle \frac{p_2}{p_1}}y+((3{\alpha }_1(k_1^2+k_2^2)\\ +4{\alpha }_2(k_1^3+k_2^3))(1+{\displaystyle \frac{p_2}{p_1}}))t)]e^{i(k_{1}x+k_{2}y+wt)}\end{array}$$

(25)

where $A={\displaystyle \frac{{\alpha }_1(k_1^3+k_2^3)+{\alpha }_2(k_1^4+k_2^4)-w}{\beta }}+{\displaystyle \frac{{\alpha }_2\beta (p_1^4+p_2^4)}{({\alpha }_2(k_1^2+k_2^2)-\frac{{\alpha }_1^2}{8{\alpha }_2}-\frac{1}{6}\gamma )(p_1^2+p_2^2)}}$, $B={\displaystyle \frac{{\alpha }_1(k_1^3+k_2^3)+{\alpha }_2(k_1^4+k_2^4)-w}{3({\alpha }_2(k_1^2+k_2^2)-\frac{{\alpha }_1^2}{8{\alpha }_2}-\frac{1}{6}\gamma )(1+\frac{p_2^2}{p_1^2})}}+{\displaystyle \frac{{\alpha }_2\beta (p_1^4+p_2^4)}{3{(({\alpha }_2(k_1^2+k_2^2)-\frac{{\alpha }_1^2}{8{\alpha }_2}-\frac{1}{6}\gamma )(p_1+\frac{p_2^2}{p_1}))}^2}}$.

Figure 3 shows the waveform plots of the dark soliton solution of Equation (25) under appropriate coefficients and static conditions of $t=0$, while Figure 4 shows corresponding contour plots.

4. Dark Soliton Solution for RNLSE with 3OD and 4OD Dispersions with PTs

In order to understand the RNLSE with 3OD and 4OD dispersions in depth, the (1+1)D RNLSE incorporating 3OD and 4OD dispersions with PTs is investigated. The equation is formulated as:

$$i{\psi }_t+i{\alpha }_1{\psi }_{xxx}+{\alpha }_2{\psi }_{xxxx}+\beta |\psi {|}^2\psi +\gamma {\displaystyle \frac{|\psi {|}_{xx}}{|\psi |}}\psi =i(\delta {\psi }_x+\lambda {(|\psi {|}^{2m}\psi )}_x+\mu {(|\psi |)}_x^{2m}\psi )$$

(26)

The parameters δ, λ, m, and μ shown in Equation (26) have the same definitions as those in Equation (2). Like the process for deriving the soliton solutions of (1+1)D RNLSE incorporating 3OD and 4OD dispersions without PTs shown in Section 3, the dark soliton solution for the model with PTs can be obtained as:

$$\psi (x,t)=-2i\sqrt{C}\tanh [\pm {\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\frac{3{\alpha }_1^4}{256{\alpha }_2^3}+w+\frac{\delta {\alpha }_1}{4{\alpha }_2}}{3(\frac{{\alpha }_1^2}{16{\alpha }_2}+\frac{1}{6}\gamma )}}+{\displaystyle \frac{{\alpha }_2(\beta -\frac{\lambda {\alpha }_1}{4{\alpha }_2})p^2}{3{(\frac{{\alpha }_1^2}{16{\alpha }_2}+\frac{1}{6}\gamma )}^2}}}(x+({\displaystyle \frac{{\alpha }_1^3}{4{\alpha }_2^2}}+\delta )t)]e^{i(-{\displaystyle \frac{{\alpha }_1}{4{\alpha }_2}}x+wt)}$$

(27)

where $C=-{\displaystyle \frac{\frac{3{\alpha }_1^4}{256{\alpha }_2^3}+w+\frac{\delta {\alpha }_1}{4{\alpha }_2}}{\beta -\frac{\lambda {\alpha }_1}{4{\alpha }_2}}}-{\displaystyle \frac{{\alpha }_2{p}^2}{\frac{{\alpha }_1^2}{16{\alpha }_2}+\frac{1}{6}\gamma }}$. The detailed derivation process of Equation (27) is given in Appendix A. Figure 5 presents the dark soliton solution from Equation (27) under static conditions and appropriate coefficients, along with its contour plot under time diffusion. Moreover, by comparing the two (1+1)D models, which are the Equation (27) with PTs and Equation (21) without PTs, the amplitude coefficient C in Equation (27) explicitly depends on the inter-modal dispersion δ and self-steepening coefficient λ of PTs. In Equation (27), the term $\beta -\lambda {\alpha }_1/\left(4{\alpha }_2\right)$ in the denominator in C replaces β in Equation (21), indicating that the self-steepening coefficient λ directly counteracts the Kerr nonlinearity β. On the other hand, the term $\left({\alpha }_1^3/\left(4{\alpha }_2^2\right)+\delta \right)$ before the variable t in Equation (27) gains an additional term δ compared with the Equation (21). This signifies that inter-modal dispersion δ directly alters the soliton propagation speed, independent of the dispersion coefficients α₁, α₂. This can also be clearly seen from the comparison of Figure 2a and Figure 5a; the deviation of the dark soliton in the x-direction is different within the same time range.

For the (2+1)D equation, the traveling wave ansatz from Equation (23) is applied and the F-expansion method is implemented following the standard procedure. Then, we derive the dark soliton solution for the (2+1)D RNLSE incorporating 3OD and 4OD dispersions and PTs, expressed as:

$$\begin{array}{ll}\psi (x,t)=& -2i\sqrt{D}\tanh [\pm {\displaystyle \frac{1}{2}}\sqrt{E}(x+{\displaystyle \frac{p_2}{p_1}}y+\\ & ((3{\alpha }_1(k_1^2+k_2^2)+4{\alpha }_2(k_1^3+k_2^3)+\delta )(1+{\displaystyle \frac{p_2}{p_1}}))t)]e^{i(k_{1}x+k_{2}y+wt)}\end{array}$$

(28)

where $D={\displaystyle \frac{{\alpha }_1(k_1^3+k_2^3)+{\alpha }_2(k_1^4+k_2^4)-w+\delta k_1}{\beta +\lambda k_1}}+{\displaystyle \frac{{\alpha }_2(\beta +\lambda k_1)(p_1^4+p_2^4)}{({\alpha }_2(k_1^2+k_2^2)-\frac{{\alpha }_1^2}{8{\alpha }_2}-\frac{1}{6}\gamma )(p_1^2+p_2^2)}}$, $E={\displaystyle \frac{{\alpha }_1(k_1^3+k_2^3)+{\alpha }_2(k_1^4+k_2^4)-w+\delta k_1}{3({\alpha }_2(k_1^2+k_2^2)-\frac{{\alpha }_1^2}{8{\alpha }_2}-\frac{1}{6}\gamma )(1+\frac{p_2^2}{p_1^2})}}+{\displaystyle \frac{{\alpha }_2(\beta +\lambda k_1)(p_1^4+p_2^4)}{3{(({\alpha }_2(k_1^2+k_2^2)-\frac{{\alpha }_1^2}{8{\alpha }_2}-\frac{1}{6}\gamma )(p_1+\frac{p_2^2}{p_1}))}^2}}$.

Figure 6 shows the dark soliton solution of Equation (28) under appropriate coefficients when t = 0. Combined with the comparison of the two (2+1)D models, which are Equation (28) with PTs and Equation (25) without PTs, this shows that, similar to the (1+1)D case, the amplitude coefficient D in Equation (25) incorporates PTs δ and λ. Moreover, the Kerr nonlinearity coefficient β is replaced by β+λk₁ in the denominator of Equation (28). This shows that the self-steepening coefficient λ interacts with the wave number k₁, modifying the effective nonlinearity governing the soliton’s amplitude. Additionally, the soliton propagation speed component gains an additive term δ. This inter-modal dispersion term δ directly increases the overall propagation speed in the (x+p₂/p₁y) direction, independent of the dispersion coefficients and wave numbers.

5. Modulation Instability for RNLSE with 3OD and 4OD Dispersions

Modulation instability (MI) is a fundamental phenomenon in nonlinear wave systems where a steady-state solution becomes unstable under small perturbations, leading to the exponential growth of sideband frequencies. MI analysis is implemented to investigate the (1+1)D RNLSE, incorporating 3OD and 4OD dispersions with Kerr nonlinearity and its PTs-included counterpart.

First, we give the steady-state solution of Equation (6). $\psi (x,t)$ is defined as:

$$\psi (x,t)=P_0{e}^{i(kx+wt)}$$

(29)

where $P_0$ represents the initial incident optical power. $k$ denotes the wave number and $w$ represents the frequency, all of which are real constants. By substituting Equation (29) into Equation (6), we obtain:

$$w={\alpha }_1{k}^3{P}_0+{\alpha }_2{k}^4{P}_0+\beta P_0^2$$

(30)

The perturbation $\tilde{P}$ is introduced and the perturbed solution is formulated as:

$$\psi (x,t)=(P_0+\tilde{P})e^{i(kx+wt)}$$

(31)

Then, the partial derivatives involved in Equation (31) can be formulated as:

$${\psi }_{xxx}=({\tilde{P}}_{xxx}+3ik{\tilde{P}}_{xx}-3k^2{\tilde{P}}_x-i{k}^3(P_0+\tilde{P}))e^{i(kx+wt)}$$

(32a)

$${\psi }_{xxxx}=({\tilde{P}}_{xxxx}+4ik{\tilde{P}}_{xxx}-6k^2{\tilde{P}}_{xx}-4i{k}^3{P}_x+k^4(P_0+\tilde{P}))e^{i(kx+wt)}$$

(32b)

By substituting Equations (31) and (32) into Equation (6) and applying linear stability analysis, we can derive:

$$\begin{array}{l}i{\tilde{P}}_t-w{P}_0-w\tilde{P}+i{\alpha }_1{\tilde{P}}_{xxx}-3{\alpha }_{1}k{\tilde{P}}_{xx}-3i{\alpha }_2{k}^2{\tilde{P}}_x+{\alpha }_1{k}^3(P_0+\tilde{P})+{\alpha }_2{\tilde{P}}_{xxxx}+4i{\alpha }_{2}k{\tilde{P}}_{xxx}\\ -6{\alpha }_2{k}^2{\tilde{P}}_{xx}-4i{\alpha }_2{k}^3{\tilde{P}}_x+{\alpha }_2{k}^3{\tilde{P}}_x+{\alpha }_2{k}^4(P_0+\tilde{P})+\beta P_0^2(\tilde{P}+{\tilde{P}}^{*})+\gamma {\tilde{P}}_{xx}=0\end{array}$$

(33)

where * denotes the complex conjugate. In addition, $\tilde{P}$ can be formulated as:

$$\tilde{P}(x,t)=P_1{e}^{i(\widehat{k}x+\widehat{w}t)}+P_2{e}^{-i(\widehat{k}x+\widehat{w}t)},$$

(34)

where $P_1$ and $P_2$ are perturbed power. $\widehat{k}$ and $\widehat{\omega }$ are the perturbed wave number and frequency, which are all real constants. By substituting Equation (34) into Equation (33) and organizing the coefficients $P_1$ and $P_2$, we obtain:

$$e^{i(\widehat{k}x+\widehat{w}t)}\left(M_{11}{P}_1+M_{12}{P}_2\right)=0$$

(35a)

$$e^{-i(\widehat{k}x+\widehat{w}t)}\left(M_{21}{P}_1+M_{22}{P}_2\right)=0$$

(35b)

The parameters in Equation (35) are calculated and expressed as:

$$M_{11}=(3{\alpha }_1{k}^2+4{\alpha }_2{k}^3)\widehat{k}+(3{\alpha }_{1}k+6{\alpha }_2{k}^2-\gamma ){\widehat{k}}^2+({\alpha }_1+4{\alpha }_{2}k){\widehat{k}}^3+{\alpha }_2{\widehat{k}}^4-\widehat{w}$$

(36a)

$$M_{12}=\beta P_0^2$$

(36b)

$$M_{21}=\beta P_0^2$$

(36c)

$$M_{22}=-(3{\alpha }_1{k}^2+4{\alpha }_2{k}^3)\widehat{k}+(3{\alpha }_{1}k+6{\alpha }_2{k}^2-\gamma ){\widehat{k}}^2-({\alpha }_1+4{\alpha }_{2}k){\widehat{k}}^3+{\alpha }_2{\widehat{k}}^4+\widehat{w}$$

(36d)

When the determinant of the coefficient matrix is equal to zero, $P_1$ and $P_2$ in Equation (36) are non-zero solutions, that is:

$$\left|\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right|=0$$

(37)

The dispersion relation is obtained by solving the determinant and is expressed by:

$$\widehat{w}=({\alpha }_1+4{\alpha }_{2}k){\widehat{k}}^3+(3{\alpha }_1{k}^2+4{\alpha }_{2}k)\widehat{k}\pm \sqrt{{[(3{\alpha }_{1}k+6{\alpha }_2{k}^2-\gamma ){\widehat{k}}^2+{\alpha }_2{\widehat{k}}^4-\beta P_0^2]}^2-{(\beta P_0^2)}^2}$$

(38)

For Equation (38), ${[(3{\alpha }_{1}k+6{\alpha }_2{k}^2-\gamma ){\widehat{k}}^2+{\alpha }_2{\widehat{k}}^4-\beta P_0^2]}^2>{(\beta P_0^2)}^2$, $\widehat{w}$ is a real number, and Equation (38) is stable. Otherwise, when ${[(3{\alpha }_{1}k+6{\alpha }_2{k}^2-\gamma ){\widehat{k}}^2+{\alpha }_2{\widehat{k}}^4-\beta P_0^2]}^2<{(\beta P_0^2)}^2$, $\widehat{w}$ is an imaginary number, and Equation (38) is unstable. By using the same approach to Equation (25), we derive its dispersion relation as follows:

$$\widehat{w}=({\alpha }_1+4{\alpha }_{2}k){\widehat{k}}^3+(3{\alpha }_1{k}^2+4{\alpha }_{2}k)\widehat{k}\pm \sqrt{\Delta }$$

(39)

where the expansion of $\Delta$ is given by:

$$\begin{array}{ll}\Delta =& {\alpha }_2^2{\widehat{k}}^8+2{\alpha }_2(3{\alpha }_{1}k+6{\alpha }_2{k}^2-\gamma ){\widehat{k}}^6+(2{\alpha }_2(\beta +\lambda k)P_0^2+{(3{\alpha }_{1}k+6{\alpha }_2{k}^2-\gamma )}^2){\widehat{k}}^4\\ & +(2(\beta +\lambda k)(3{\alpha }_{1}k+6{\alpha }_2{k}^2-\gamma )P_0^2+(\lambda +\mu )P_0^2+{(\lambda +\mu )}^2{P}_0^4+\delta ){\widehat{k}}^4\end{array}$$

(40)

For Equation (39), if $\Delta >0$, $\widehat{w}$ is a real constant, and the Equation (39) is stable. If $\Delta <0$, $\widehat{w}$ is an imaginary number, and the Equation (39) becomes unstable. Besides, the MI gain spectrum quantifies the exponential growth rate imposed by the interplay of nonlinearity and dispersion on specific perturbation frequencies riding on an intense continuous-wave background. The MI gain spectrum quantifies the instability strength as a function of the perturbation wavenumber $\widehat{k}$ and can be derived as:

$$g(\widehat{k}) =2Im(\widehat{w})$$

(41)

Figure 7 shows the MI gain spectrum of the RNLSE with and without PTs. It can be seen that the shapes of the gain spectra are similar for the two cases. This is because, based on Equation (40), the product of the high power of $\widehat{k}$ and the dispersion coefficient ${\alpha }_1$ or ${\alpha }_2$, like ${{\alpha }_2}^2{\widehat{k}}^8$, ${{\alpha }_1\alpha }_2{\widehat{k}}^6$ terms, dominate at large $\widehat{k}$. It is suspected that these terms contribute to stabilizing the high-frequency perturbations so that the MI gain is small for the large absolute value of $\widehat{\mathrm{k}}$. Moreover, the peak of the MI gain spectrum for the RNLSE with PTs is lower than that without PTs, as shown in Figure 7. As shown in Equation (40), compared with the model without PTs, the value of $∆$ can be reduced depending on the different values of parameters δ, λ, μ, which will compress the unstable regions and lead to a decrease in the peak gain. Qualitatively, the generation of MI mainly depends on a certain balance between nonlinear effects and dispersion effect [30]. If the perturbed parameter μ related to the high-order dispersion is a negative number, this means that the total dispersion of the model will be reduced. In this case, the peak gain will also decrease in order to reduce the nonlinear effect to make a new balance with the reduced dispersion.

6. Conclusions

In this study, the RNLSE model incorporating 3OD and 4OD dispersions with Kerr nonlinearity are investigated. We derive dark soliton solutions for both (1+1)D and (2+1)D cases using the F-expansion and self-similar methods, extending the analysis to the models with PTs. By comparing the (1+1)D and (2+1)D models and graphical results, it is found that the parameters δ and λ in PTs alter the soliton amplitude by modifying the Kerr nonlinearity coefficient β. Moreover, inter-modal dispersion δ induces a direct shift in the soliton propagation speed. MI analysis reveals significant suppression of the MI growth rate induced by PTs. The theoretical results provide a foundation for the research on solitons in systems governed by the RNLSE model and further experimental investigation.