Gap solitons in moiré optical lattice with quartic dispersion

Abstract

This work investigates the theoretical properties and stability mechanisms of semi-infinite gap solitons in moiré optical lattices governed by quartic dispersion. Using the framework of Floquet-Bloch theory, we explore linear modes in commensurate and incommensurate lattice configurations, revealing distinct localization characteristics influenced by the lattice twist angle and dispersion coefficient. Numerical simulations confirm that gap solitons can emerge in a self-focusing Kerr medium, originating from nonlinear evolution of localized linear Bloch modes. Linear stability analysis indicates that solitons are stable under normal quartic dispersion but unstable under anomalous dispersion, corroborated by direct perturbation evolution methods. Additionally, the study highlights the influence of lattice twist strength and angle on soliton localization, with incommensurate lattices enabling thresholdless soliton excitation. These findings provide new insights into the dynamics of solitons in moiré systems, with implications for designing tunable photonic structures and advancing applications in optical communications and nonlinear photonics.

Introduction

Solitons are wave packets that arise from the balance between nonlinear effects and dispersion in a medium¹. Due to this balance, solitons can maintain their temporal and spectral profiles during propagation². As a result, solitons are widely used in information transmission, particularly in fields such as nonlinear optics^3,4 and Bose-Einstein condensates^5,6. Optical media provide an excellent platform for studying soliton dynamics under various conditions. It is worth noting that as early as the 1990s, the influence of higher-order dispersion on the propagation of traditional solitons in optical fibers was theoretically proposed, and since then, it has become an active area of research⁷. However, for a long time, higher-order dispersion was regarded as a disruptive factor^8,9. Recent studies have demonstrated that pure quartic solitons, generated through the interplay between self-phase modulation and negative quartic dispersion, exhibit unique characteristics that distinguish them from traditional solitons governed by second-order dispersion. These pure quartic solitons have shown promising applications in high-energy pulse lasers and optical communications^{10,11,12,13,14,15,16,17}. Experimentally, the existence of pure quartic solitons has been observed in systems such as photonic crystal waveguides and mode-locked lasers, confirming that quartic dispersion significantly affects the shape, energy scaling, and dynamical properties of solitons^10,11,12,13.

While second-order dispersion is typically the dominant effect in many nonlinear systems, there are specific scenarios where quartic dispersion becomes the primary consideration. For example, in photonic crystal waveguides, the dispersion properties can be engineered to suppress second-order dispersion, allowing quartic dispersion to dominate. This has enabled the experimental observation of pure quartic solitons, which exhibit unique characteristics compared to traditional solitons governed by second-order dispersion^10,11. In mode-locked lasers, the dispersion profile can be tailored to achieve a purely quartic dispersion regime. In this regime, second-order dispersion is negligible, enabling the formation of pure quartic solitons^11,12. Moreover, specialized dispersion-engineered fibers have been designed to suppress second-order dispersion, allowing quartic dispersion to dominate, as explored in the context of pure quartic solitons¹⁴. In the present study, we focus on systems where quartic dispersion is the primary effect, providing insights into the unique properties of solitons in such regimes.

Optical lattices provide a rich platform for studying light propagation in periodic systems, where the interplay between nonlinearity and periodicity gives rise to novel dynamical phenomena^18,19,20,21. The experimental realization of quartic dispersion solitons has generated significant interest in exploring the existence, stability, and dynamics of soliton structures in periodic systems^{22,23,24,25,26}. quartic dispersion plays a crucial role in determining the existence and stability of solitons. Within certain parameter ranges, it can extend the region of soliton existence and enhance their robustness²⁷. This indicates that quartic dispersion has potential applications in modulating and enhancing soliton properties. Moreover, the diverse configurations of optical lattices can lead to new physical phenomena. For instance, in parity-time ($\mathscr{P}\mathscr{T}$) symmetric periodic systems, quartic dispersion enables the study of unique features in non-Hermitian systems²⁸. In such systems, quartic dispersion influences the exceptional points of $\mathscr{P}\mathscr{T}$-symmetric Bloch bands, and phenomena including dipolar solitons and bandgap solitons in semi-infinite bandgaps have been observed^29,30.

The advancement of twistronics has sparked significant research interest in manipulating twist angles and extending this concept to various systems^{31,32,33,34,35}. In photorefractive crystals, the interference of light can generate two periodically rotated or twisted square sublattices, leading to a shallow modulation of the refractive index in the $x$-$y$ plane and forming photonic moiré lattices. Recently, a novel approach has been proposed to dynamically generate a twisted double-layer lattice in a cold atom system by utilizing the interactions between atoms³⁶. These lattices provide a versatile platform for exploring moiré physics^37,38. Within moiré systems, phenomena including the formation of optical solitons³⁷, flat-band physics^39,40, metasurfaces^41,42, and topological structures⁴³ have been realized. Notably, competing nonlinearities-such as those arising in dipolar quantum droplets or photonic crystals-can stabilize complex structures like vortex solitons, where repulsive and attractive interactions counteract azimuthal instabilities^44,45. Although the current work focuses on gap solitons, analogous mechanisms may extend to vortex solitons in moiré systems under specific conditions. By overlapping two identical sublattices that rotate against each other, monolayer and bilayer moiré lattices can be formed. Depending on the twist angle, the moiré lattices can exhibit periodicity (commensurability) based on the symmetry of the underlying sublattices or become aperiodic (incommensurate). In contrast to real material systems, photonic moiré lattices have highly tunable parameters, facilitating the study of physical phenomena that occur during transitions between commensurate and incommensurate states, such as the phase transition of localization-delocalization of two-dimensional beams^38,46,47. In the presence of nonlinear effects, significant progress has been made in the study of threshold-free Kerr solitons³⁷, multifrequency solitons⁴⁸, vortex solitons^49,50, and gap solitons^51,52 within moiré lattices. Moreover, the formation and stabilization mechanisms of two-dimensional gap solitons have been theoretically investigated by combining parity-time symmetry⁵³.

Previous studies on moiré lattices and solitons have primarily focused on systems governed by second-order dispersion^{51,52,53,54,55,56} . While these works have provided valuable insights into the formation and stability of solitons, the role of higher-order dispersion, particularly quartic dispersion, remains underexplored. In contrast to previous studies, our work investigates the unique properties of solitons in moiré lattices under purely quartic dispersion, which has not been systematically studied before. This allows us to uncover new phenomena, such as the stabilization of solitons under normal quartic dispersion and the thresholdless excitation in incommensurate lattices, which were not observed in systems dominated by second-order dispersion.

This paper investigates the existence and stability of semi-infinite gap solitons in moiré lattices with quartic dispersion. First, within the framework of Floquet-Bloch theory, we study the linear modes supported by the system, distinguishing between localized and delocalized cases. We further explore the influence of lattice twist strength and twist angle on the localization of these linear modes. Based on the linear modes, we identify the gap solitons supported by the system. We then evaluate their stability using a combination of linear stability analysis and direct perturbation evolution. Our results show that gap solitons exhibit robust propagation under normal quartic dispersion, while those under anomalous dispersion are uniformly unstable. Lastly, we examine the impact of quartic dispersion on the power threshold required for soliton formation.

Model

Starting from the generalized nonlinear Schrödinger equation (GNLSE) model as presented by Blanco et al.¹⁰, we proceed by neglecting the loss terms, including linear loss and two-photon absorption, and also omitting the second-order dispersion and third-order dispersion. Furthermore, we incorporate a periodic potential field $\mathscr {P}(\varvec{r})$ arising from moiré lattices. The resulting simplified equation is:

$$\begin{aligned} \begin{aligned} i\frac{\partial \Psi }{\partial z}=\frac{\beta }{24} \frac{\partial ^4\Psi }{\partial \varvec{r}^4}+\mathscr {P}(\varvec{r})\Psi + g|\Psi |^2\Psi , \end{aligned} \end{aligned}$$

(1)

here $\varvec{r} = (x, y)$ is the transverse coordinates, z is the propagation distance, $\beta$ is the adjustable quartic dispersion coefficient, and g is the nonlinear coefficient generated by medium polarization. The sign of $\beta$ determines the nature of the quartic dispersion: $\beta > 0$ corresponds to normal quartic dispersion, where the system supports self-focusing linear effects, while $\beta < 0$ represents anomalous quartic dispersion, which induces wave packet broadening. In physical systems, the sign and magnitude of $\beta$ can be engineered by tailoring the dispersion properties of photonic structures, such as photonic crystal waveguides or dispersion-managed fibers, where higher-order dispersion terms dominate over lower-order contributions^10,14. In the following discussion, unless explicitly stated otherwise, we restrict our analysis to solitons with a nonlinear coefficient $g=-1$. The moiré lattices $\mathscr {P} (\varvec{r})$ are formed by the superposition of two periodic sublattices, and their mathematical form is as follows:

$$\begin{aligned} \mathscr {P}(\varvec{r})=v_1(\cos ^2x_1+\cos ^2y_1)+v_2(\cos ^2x_2+\cos ^2y_2). \end{aligned}$$

(2)

In this case, both sublattices have a period of $\pi$. Experimentally, such sublattices can be realized via interference patterns of counter-propagating laser beams or spatially modulated refractive index profiles in photonic crystals^37,38. We fix the depth of the sublattice with $v_1$ and twist the sublattices with the depth of the lattices $v_2$ by an angle $\theta$ to generate the moiré lattices. The relative twist angle $\theta$ between the sublattices is a critical parameter that determines the moiré periodicity, which can be dynamically controlled in optical platforms through electro-optic modulation or mechanical rotation stages. The spatial coordinates between the two sublattices satisfy the following relationship:

$$\begin{aligned} \begin{pmatrix}x_2\\ y_2\end{pmatrix}=\begin{pmatrix}\cos (\theta ),& -\sin (\theta )\\ \sin (\theta ),& \cos (\theta )\end{pmatrix}\begin{pmatrix}x_1\\ y_1\end{pmatrix}. \end{aligned}$$

(3)

The twist angle $\theta$ plays a key role in determining the structure of the resulting moiré lattice. Specifically, when $\theta$ takes the form of $\arctan [(m^2 - n^2)/2mn]$, where $m> n > 0$ and $m, n \in \mathbb {N}$, the moiré lattice forms a quasi-periodic structure. This structure is referred to as commensurate moiré lattices and results from the superposition of two sublattices with a specific relationship, as described by the Pythagorean triple $(m^2 - n^2, 2mn, m^2 + n^2)$. For other values of $\theta$, however, the moiré lattice becomes incommensurate, lacking a periodic pattern. This difference is clearly illustrated in Fig. 1a,b, where the periodic and aperiodic patterns of commensurate and incommensurate moiré lattices, respectively, are depicted. In this study, we focus on the existence and stability of quartic solitary waves within these distinct moiré lattice structures, with a focus on twist angles of $\arctan (3/4)$ (commensurate) and $\pi /6$ (incommensurate).

To establish soliton solutions in the moiré lattice system, we start from the NLSE given by Eq. (1). By assuming a stationary solution of the form $\Psi (\varvec{r}, z) = \psi (\varvec{r}) \exp (i \mu z)$, where $\mu$ is the propagation constant and $\psi (\varvec{r})$ is the transverse field distribution, we obtain the nonlinear stationary equation (Eq. (4)):

$$\begin{aligned} \mu \psi =-\frac{\beta }{24} \frac{\partial ^4\psi }{\partial \varvec{r}^4} - \mathscr {P}(\varvec{r})\psi -g|\psi |^2\psi . \end{aligned}$$

(4)

This equation is solved numerically by combining the split-step Fourier method for the nonlinear part with the quartic Runge-Kutta method for the linear part. The resulting soliton solutions are subsequently analyzed for their stability through linear stability analysis and direct perturbation evolution techniques.

The soliton solutions are obtained by numerically solving Eq. (4). To better characterize the properties of this system, we introduce several physical quantities. Firstly, the power of the beam is defined as:

$$\begin{aligned} U=\int |\psi |^2 d\varvec{r}. \end{aligned}$$

(5)

Next, we define the shape factor $\chi$ to characterize the localization of the steady-state solution in the moiré lattice system with quartic dispersion. In this context, a higher value of $\chi$ indicates stronger localization.

$$\begin{aligned} \chi = U^{-1} \left( \int |\psi |^4d\varvec{r}\right) ^{1/2}. \end{aligned}$$

(6)

Once we have obtained the solitons of the system, their stability becomes a priority concern. In this work, we employ primarily two methods to assess the stability of the solitons. First, we used a linear stability analysis to determine the stability of the soliton by introducing a small perturbation to the soliton , expressed in the form of

$$\begin{aligned} \psi =[\phi {+}p\exp {(\lambda z)}{+}q^{*}\exp {(\lambda ^{*}z)}]\exp {(i \mu z)}, \end{aligned}$$

(7)

where $\phi$ is the unperturbed soliton solution obtained from Eq. (4), representing the steady-state profile of the soliton. Here, $p$ and $q$ are small perturbations added to the soliton, which can be complex functions of the spatial coordinates, and $\lambda$ is the eigenvalue associated with the perturbation modes, determining the growth or decay of the perturbations over the propagation distance $z$. The parameter $\mu$ is the propagation constant of the soliton, related to its phase evolution during propagation. Substituting such an expression into Eq. (1), we get the following eigenvalue problem:

$$\begin{aligned} \left. \left( \begin{array}{cc}L& g\phi ^2\\ -g\phi ^{*2}& -L \end{array}\right. \right) \left( \begin{array}{c}p\\ q \end{array}\right) =i\lambda \left( \begin{array}{c}p\\ q \end{array}\right) , \end{aligned}$$

(8)

with $L=\mu +\beta /24 \cdot \partial ^{4} /\partial \varvec{r}^{4} + 2g|\phi |^{2} + \mathscr {P}(\varvec{r})$. The stability of the soliton is determined by the eigenvalues $\lambda$: if all eigenvalues $\lambda$ have zero imaginary parts (i.e., $\lambda$ is purely real), the perturbations do not grow over time, and the soliton is considered stable; if any eigenvalue $\lambda$ has a non-zero imaginary part, the perturbations grow exponentially, leading to modulational instability, and the soliton is considered unstable. This formulation allows us to analyze the stability of the soliton by solving the corresponding eigenvalue problem derived from the linearized equations of motion.

Figure 1

Profiles of moiré lattices under commensurate and incommensurate conditions are presented in (a) and (b), respectively. The colors in these panels represent the potential well intensity of the moiré lattice. Specifically, panel (a) shows the periodic pattern formed by a Pythagorean angle ($\theta = \arctan (3/4)$), while panel (b) illustrates the non-periodic pattern resulting from an incommensurate twist angle ($\theta = \pi /6$). Characteristic profiles of the linear modes supported by the moiré lattices under varying parameters are showcased in panels (c–h), where the colors represent the squared modulus of the wavefunction. In panel (c), we have $\theta = \arctan (3/4)$, $v_2 = 2$, and $\beta = 0.2$. Linear modes for $\theta = \pi /6$, $v_2 = 0.5$, and $\beta = 0.2$ are illustrated in panel (d). Panels (e) and (f) are similar, both depicting $\theta = \pi /6$, $v_2 = 2$, and $\beta = 0.2$ with different propagation constants. Modes for $\theta = \pi /6$, $v_2 = 2$, and $\beta = 0.3$ are shown in panel (g), while panel (h) represents $\theta = \pi /6$, $v_2 = 2$, and $\beta = 0.2$. The propagation constants of the linear modes in (c-h) are: (0.98, 0.75, 1.06, 1.07, 1.12, 1.072), respectively, and the other parameter $v_1 = 2$.

Numerical results and discussion

Linear modes

We begin by investigating the linear regime, where the interaction coefficient $g = 0$, and focus on the linear modes supported by the moiré lattices. Specifically, we analyze two representative twist angles: the commensurate case ($\theta = \arctan (3/4)$) and the incommensurate case ($\theta = \pi /6$), both with a quartic dispersion coefficient of $\beta = 0.2$. As depicted in Fig. 1c–h, we present the characteristic profiles of the linear modes for various parameters, which are eigenstates of the linear Hamiltonian.

In the commensurate case, the moiré lattices exhibit a periodic spatial distribution. Under these conditions, all linear modes are confined to the lowest-energy sites of the moiré lattices, forming a nonlocal periodic pattern, as illustrated in Fig. 1c. In contrast, for the incommensurate case, the moiré lattices become aperiodic in space. Here, the linear modes are highly localized at specific positions, and their density distribution exhibits a strong dependence on the propagation constant. Different propagation constants support distinct localized linear modes, as shown in Fig. 1e–h, corresponding to propagation constants of 1.06, 1.07, and 1.072, respectively. Notably, the degree of localization increases with decreasing propagation constant, a characteristic feature of the moiré lattices. In the incommensurate case, the linear modes predominantly occupy the most localized ground states. Expanding these modes to neighboring lattice sites requires a larger propagation constant. Additionally, a transitional localized mode is observed between the states shown in Fig. 1e,h, as seen in Fig. 1f.

The potential strength of the lattices and the quartic dispersion coefficient also significantly influence the spatial distribution of the modes, as demonstrated in Fig. 1d,g. By tuning the lattice strength $v_2$, we observe that in weak lattices, the linear modes become delocalized due to the insufficient depth of the lattices to confine them, as shown in Fig. 1d. Moreover, increasing the quartic dispersion coefficient results in the delocalization of the linear modes. This is attributed to the flatter linear spectrum induced by the enhanced dispersion, which causes the modes to become more extended, as illustrated in Fig. 1g.

To characterize the influence of the twist angle and the lattice strength on the localization of linear modes, we fix $v_1 = 2$ and the quartic dispersion coefficient $\beta = 0.2$. To elucidate the effects of twist angle and lattice strength on the localization of linear modes, we fixed $v_1 = 2$ and the quartic dispersion coefficient $\beta = 0.2$, and present a comprehensive analysis in Fig. 2. Panel (a) of this figure depicts a contour plot of the shape factor $\chi$, which exhibits a pronounced symmetry with respect to the twist angle $\theta$, with the symmetry axis aligned at $\theta = \pi /4$. This symmetry arises from the intrinsic structural properties of the moiré lattice. It highlights the equivalent localization behavior of linear modes on either side of $\theta = \pi /4$, attributable to the balanced interference pattern characteristic of this specific twist angle. The shape factor $\chi$, a quantitative measure of mode localization, accordingly adopts similar values, indicative of the symmetric confinement properties of the moiré lattice around this point. This symmetric behavior is particularly pronounced in incommensurate lattices, such as at $\theta = \pi /4$, where enhanced localization is observed relative to commensurate conditions, as evidenced by $\theta = \arctan (3/4)$ and demarcated by the white dashed lines.

Figure 2

(a) presents a contour plot of the shape factor $\chi$ for the eigenmode with the largest propagation constant as a function of $\theta$ and $v_2$. The white dashed lines indicate specific Pythagorean angles. It exhibits symmetry around $\theta = \pi /4$, reflecting the structural symmetry of the moiré lattice. (b) illustrates the effect of the quartic dispersion coefficient $\beta$ on the shape factor $\chi$, indicating the degree of localization, for the maximum propagation constant in commensurate (dashed lines) and incommensurate (solid lines) moiré lattice systems. Red curves correspond to $\beta >0$ with $\theta = \arctan (3/4)$, and blue curves to $\beta <0$ with $\theta = \pi /6$, with $v_1=2$. The data reveal higher localization for $\beta >0$, a negative correlation between localization and $\vert \beta \vert$, and enhanced localization in incommensurate systems compared to commensurate ones.

The tunability of the quartic dispersion significantly enriches the degrees of freedom in the quartic dispersion moiré lattice system. To investigate this, we calculated the influence of the quartic dispersion coefficient $\beta$ on the shape factor for the maximum propagation constant, under both commensurate and incommensurate conditions. As shown in Fig. 2b, red curves correspond to $\beta > 0$, blue curves to $\beta < 0$, and dashed and solid lines represent the commensurate and incommensurate cases, respectively. We observe that, regardless of whether the system is commensurate or incommensurate, the degree of localization is consistently higher for $\beta > 0$ than for $\beta < 0$. This behavior can be attributed to the fact that $\beta < 0$ induces wave packet broadening, whereas $\beta > 0$ facilitates the formation of moiré-localized wave packets. In both cases, the localization is negatively correlated with $|\beta |$, as the band approaches flatness due to the quartic dispersion. When $|\beta |$ is smaller, the band flatness increases, leading to more localized linear modes. Thus, increasing the absolute value of the quartic dispersion coefficient invariably reduces the localization of linear modes. Comparing the commensurate and incommensurate cases, we find that the system exhibits enhanced localization under incommensurate conditions. Specifically, for $\beta < 0$, increasing the quartic dispersion coefficient within the commensurate framework only marginally enhances the system’s localization, without inducing a phase transition from delocalization to localization. In contrast, in the incommensurate regime, increasing the quartic dispersion coefficient can transition linear modes from a delocalized to a localized state. However, for $\beta > 0$, the quartic dispersion introduces a flat band, which inherently localizes the linear modes to a significant extent. As a result, variations in $\beta$ have a negligible impact on the overall localization of the system in this regime. We acknowledge that the shape factor ($\chi$) is a valuable indicator of localization within a specific system configuration, where higher values generally signify stronger localization. However, the precise value of ($\chi$) and the threshold for characterizing a ’localized’ state are influenced by multiple system parameters, including the quartic dispersion coefficient ($\beta$), the twist angle ($\theta$), and the lattice strengths ($v_1$) and ($v_2$). This multi-parameter dependence complicates the establishment of a universal ($\chi$) threshold applicable across all regimes. Consequently, the determination of localization in this system requires a nuanced approach that considers the interplay of these various parameters. This complexity highlights an important area for future research, as identifying a clear and consistent metric for localization remains a challenging yet intriguing problem.

Figure 3

The soliton power $U$ (red curve) and shape factor $\chi$ (blue curve) are displayed as functions of the propagation constant in moiré lattices for cases (a,b) with $\beta =- 0.2$ and (c,d) with $\beta = 0.2$. Solid lines denote stable solitons, while dashed lines denote unstable solitons. The shaded regions indicate the band gaps of the commensurate lattices in the linear regime, with vertical gray lines marking the propagation constants of the incommensurate lattices. The black dots, labeled with corresponding letters, represent solitons whose profiles are illustrated in panels (e,f). Here, the parameters are set to $v_1 = 2$ and $v_2 = 2$.

Nonlinear localized modes

In this subsection, we investigate the impact of nonlinearity in Eq. (4) on the quartic dispersion moiré lattice system, with a particular focus on solitons residing within the system’s semi-infinite band gap. The stability and properties of these band gap solitons are intrinsically linked to the linear eigenmodes of the moiré lattices. Consequently, it is crucial to examine the solitons supported by the moiré lattices that emerge from different linear eigenmodes. We explore two values of the quartic dispersion coefficient, $\beta = 0.2$ and $\beta = -0.2$, and analyze both commensurate ($\theta = \arctan (3/4)$) and incommensurate ($\theta = \pi /6$) twist angles.

The soliton solutions are obtained by numerically solving the nonlinear stationary equation (Eq. (4)) for various values of the propagation constant $\mu$. To ensure accurate and stable numerical results, we employ a combination of the split-step Fourier method and the quartic Runge-Kutta method. The stability of the solitons is evaluated by introducing small perturbations to the soliton solutions and examining their evolution over the propagation distance $z$. This approach allows us to systematically assess the stability properties of the solitons.

Figure 3 presents the fundamental soliton within the semi-infinite band gap across four distinct scenarios. Panels (a–d) illustrate the dependence of the power $U$ (blue lines) and the shape factor $\chi$ (red lines) on the propagation constant $\mu$. Solid and dashed lines represent the results of the linear stability analysis, where solid lines indicate stable solitons and dashed lines signify modulational instability, indicating an unstable regime. In panels (a) and (c), the gray regions denote the linear band gap for the commensurate case, while in panels (b) and (d), the gray vertical lines correspond to the propagation constants obtained through numerical calculations for the linear case. From panels (a-d), it is evident that, regardless of whether $\beta = -0.2$ or $\beta = 0.2$, the linear eigenstates of the system are non-localized under commensurate conditions, as shown in Fig. 1c. In the nonlinear regime, the corresponding solitons emerge from the bifurcation of non-localized linear eigenstates. For $\beta = -0.2$, as indicated by the shape factor (red line) in panel (a), the localization of solitons decreases rapidly near the linear band gap, approaching the non-localized eigenstates, which results in a rapid increase in power (blue line) near the band gap. This phenomenon leads to a minimum power, termed the threshold power, below which solitons cannot exist. In contrast, for $\beta = 0.2$, the influence of quartic dispersion results in a higher degree of localization for the linear eigenstates, with a smooth transition in the shape factor between the linear and nonlinear cases, as depicted by the red dashed line in Fig. 2b. As the propagation constant approaches the linear band gap, the shape factor monotonically decreases. Here, the power variation is smooth, yet a minimum power (threshold power) persists, below which solitons cannot exist. In the incommensurate case, whether $\beta = 0.2$ or $\beta = -0.2$, the linear eigenstates exhibit high localization, remaining localized throughout the region, as shown by the solid line in Fig. 2b. Similar to the commensurate case with $\beta = 0.2$, the shape factor transitions smoothly between the linear and nonlinear regimes. As the propagation constant decreases, the shape factor also decreases monotonically, as indicated by the red lines in panels (b) and (d). Correspondingly, the power decreases with the propagation constant, but unlike the commensurate case, in the incommensurate scenario, the power approaches zero in the linear limit. This implies that solitons in the incommensurate case are generated by thresholdless excitations from the linear band gap, leading to the power approaching zero in the linear limit. This phenomenon is attributed to the local eigenstates caused by the incommensurate angle in the moiré lattices. For these four cases, we plot the characteristic profiles of the solitons in Fig. 2e–h, with the propagation constants corresponding to the annotations in panels (a-d). A clear comparison between panels (e-f) and (g-h) reveals that the solitons exhibit a higher degree of localization for $\beta = 0.2$.

Figure 4

The evolution of solitons with different propagation constants under weak perturbations is shown. Panel (a) plots the maximum amplitude of solitons during propagation, where the red line corresponds to solitons with $\beta = 0.2$ and the blue line to solitons with $\beta = -0.2$. Panels (b) through (e) provide snapshots of solitons at different propagation distances ($z = 100$ for panels (b) and (c), $z = 250$ for panels (d) and (e) for the four cases with $\beta = 0.2$ and $\beta = -0.2$ under commensurate ($\theta = \arctan (3/4)$) and incommensurate ($\theta = \pi /6$) conditions. The figures show that stable solitons retain their shape and peak amplitude during propagation, while unstable solitons break up, forming multiple peaks and failing to preserve their initial profiles.

We employed a standard linear stability analysis to assess the stability of all solitons depicted in Fig. 3a–d. This involved numerically solving the eigenvalue problem of Eq. (8) to derive stability criteria for the solitons. The results indicated that at $\beta = -0.2$, the solitons, whether commensurate or incommensurate, exhibited instability across the entire parameter range. In contrast, at $\beta = 0.2$, the solitons remained stable throughout the parameter space, as distinguished by the solid and dashed lines in Fig. 3a–d.To further validate the stability, we examined the nonlinear propagation of the solitons under weak perturbations. We applied 10% random noise to four selected solitons from Fig. 3e–h as initial states and simulated their propagation using Eq. (1). The simulation employed a split-step Fourier method for the nonlinear component and a quartic Runge-Kutta method for temporal integration.According to the stability analysis, stable solitons maintained a constant maximum amplitude during propagation, indicating robustness against perturbations. Conversely, modulationally unstable solitons exhibited rapid amplitude oscillations and failed to retain their original profiles, as illustrated in Fig. 4a. The propagation simulation results were consistent with those from the linear stability analysis. Figure 4b,c presents snapshots of unstable solitons at a propagation distance of $z = 100$, while Fig. 4d,e shows stable solitons at $z = 250$. It is evident that stable solitons preserved their shape and peak amplitude, whereas unstable gap solitons diverged, forming multiple side peaks and losing their initial characteristics.

Conclusion

In this study, we investigated the existence and stability of solitons in a quartic dispersion moiré lattice system, considering both commensurate and incommensurate configurations. Through linear stability analysis and nonlinear propagation simulations, we found that solitons are stable when the quartic dispersion coefficient $\beta$ is 0.2, regardless of commensurability, while they are unstable at $\beta = -0.2$. Simulations confirmed that stable solitons maintain their shape and amplitude, whereas unstable ones exhibit amplitude fluctuations and profile distortions. Additionally, we observed that commensurate solitons require a threshold power for existence, whereas incommensurate solitons can be excited without such a threshold, highlighting the significant influence of the twist angle on soliton properties. This work demonstrates robust gap soliton formation in quartic-dispersion moiré lattices, with stability governed by lattice geometry and nonlinear confinement. Our study introduces several novel aspects compared to previous research. We demonstrate that quartic dispersion plays a pivotal role in modulating the stability and formation of solitons in moiré lattices, a feature absent in systems governed by second-order dispersion. Additionally, we elucidate the distinct dynamics of solitons under normal and anomalous quartic dispersion, thereby enhancing our understanding of higher-order dispersion effects in nonlinear systems. Furthermore, by extending the concept of twistronics to photonic moiré lattices, we provide a new experimental platform for investigating soliton behavior in tunable periodic potentials. Future directions include exploring vortex solitons in anisotropic moiré systems, leveraging insights from quantum droplet studies where competing nonlinearities stabilize topological structures^44,57. Additionally, incorporating symmetry or cubic-quintic nonlinear terms, as demonstrated in photonic crystals²⁶, could further enhance soliton versatility and functionality. These findings contribute essential theoretical insights into soliton dynamics within nonlinear optical systems and underscore the importance of higher-order dispersion in photonic structures, paving the way for future explorations in this domain.