In nature, many phenomena achieve stability through a delicate balance of opposing forces. As illustrated in Fig. 1, examples include patterned formations in vibrating sand piles, stable structures in magnetic fluids under certain conditions, water wave patterns, cloud patterns, as well as rotating, localized formations like hurricanes. In a similar way, light in nonlinear optical systems can form stable, localized structures in space and time. In the early 1960s, researchers observed optical solitons—self-stabilizing light waves—when lasers were focused on various materials. These observations showed that the balance between light spreading out (diffraction) and light concentrating (due to a nonlinear effect called the Kerr effect) could create stable structures. This behavior was captured by the mathematical model, the nonlinear Schrödinger equation (NLSE). In 1973, Hasegawa et al. demonstrated that the one-dimensional NLSE could also describe pulse propagation in optical fibers, predicting stable "temporal solitons" that could revolutionize optical communications. Although the technology to test these predictions didn't yet exist, fiber solitons were experimentally realized by 1980. This breakthrough sparked a wave of research that continues to shape optical science and technology today.

Figure 1 Examples of localized states observed in nature. From left to right: patterned formations in vibrating sand piles, stable structures in magnetic fluids, organized water wave patterns, cloud patterns, and hurricanes

When a laser pulse travels through an optical fiber, it naturally spreads out over time due to dispersion, where different frequency components within the pulse travel at different speeds. In an anomalous dispersion regime, high-frequency components move to the front of the pulse while low-frequency components lag behind, which causes the pulse to broaden, creating a "negative phase chirp". However, this spreading can be counteracted by the Kerr effect. The Kerr effect makes the fiber's refractive index vary with the laser intensity. This variation, known as "self-phase modulation" (SPM), introduces a "positive phase chirp" that counteracts the pulse's dispersive spreading. This balance enables the light pulse to maintain its shape as it propagates through the fiber, forming a stable structure known as a temporal soliton—a self-stabilizing pulse of light (see Fig. 2(a)). In a similar scenario, when dispersion is replaced by diffraction, the nonlinearity of the medium can similarly counteract the diffraction, allowing a laser beam to travel without expanding in size. This balance results in the formation of spatial solitons, where the beam remains confined in space as it propagates (see Fig. 2(b)).

Figure 2 (a) Temporal solitons propagate within a single mode fiber, forming due to the equilibrium between chromatic dispersion-induced temporal broadening and nonlinearity-induced pulse compression. (b) Spatial solitons manifest in Kerr nonlinear media, shaping a beam with a consistent beam waist as it travels, emerging from the interplay of spatial diffraction-induced beam divergence and nonlinear self-focusing. (c) Spatiotemporal multimode solitons propagate in graded-index multimode fibers, where the interaction between beam diffraction and fiber confinement creates spatial multimodal beams. MMS formation is more complex, necessitating a balance not only between chromatic dispersion broadening and nonlinear pulse compression, but also a balance between nonlinear beam trapping and modal walk-off due to the disparate group velocities of the fiber modes

Recent advancements have led to the exploration of spatiotemporal solitons—also known as "light bullets"—which need to take into account both space and time dimensions. The interaction of diffraction and spatial confinement in a multimode fiber (MMF) leads to the formation of multiple co-propagating spatial modes, each with distinct dispersive properties, such as different phase and group velocities, second and higher-order dispersions. The dispersion of these spatial modes can be balanced by Kerr nonlinearity, causing the modes to become locked together in time. As a result, the multimode field propagates as a single, shape-preserving "ball" of energy in both space and time. These equilibrium states are known as multimode solitons (MMS) (see Fig. 2(c)).

This paper reviews recent theoretical and experimental advances in the area of multimode solitons, focusing primarily on multimode fibers. The theoretical framework involves:

1. Basic concepts: We begin by introducing the foundational concepts of MMF and MMS, elucidating the phenomena associated with MMS in relation to MMF properties.
2. Generalized Coupled Mode Equations: Many of these phenomena, such as modal walk-off and MMS formation, can be effectively modeled by the generalized coupled mode equation, with simple examples.
3. Analytical Approaches: we review analytic approaches to MMS using the two coupled mode equations.
4. 3D+1 NLSE: we discuss extensions to the 3D+1 spatiotemporal NLSE.
5. Reduced 1D+1 NLSE Representation: To enhance understanding, the comprehensive 3D+1 model is simplified to a reduced 1D+1 NLSE representation, which is highly efficient for studying the self-imaging effect-induced phenomena in GIMFs.
6. Gaussian Quadrature Method: we introduce an innovative Gaussian quadrature approach tailored for numerical simulations of the generalized NLSE, potentially improving computational efficiency.
7. Impact of random mode mixing: we examine the propagation regime in space-division multiplexed (SDM) optical communication systems designed with MMFs, discussing the model using a generalized Jones formalism.

This model highlights the impact of linear coupling effects, primarily due to manufacturing imperfections and environmental perturbations. Furthermore, we focus on the relevant experimental studies involving the MMS phenomena. Experimentally, the focus is on observing MMS formation and behavior in both GRIN and step-index multimode fibers (SIMF). we overview recent experiments involving MMS, spanning various regimes such as the quasi-singlemode and fully multimode domains. These experiments encompass a diverse array of phenomena, including femtosecond walk-off solitons, MMS fission, and solitons in SIMF. Additionally, we explore the emergence of soliton attractors from long MMF, investigate soliton-soliton collision dynamics, and examine dispersive wave generation. Relevant research results were recently published in Photonics Research, Volume 12, Issue 11, 2024. [Yifan Sun, Pedro Parra-Rivas, Govind P. Agrawal, Tobias Hansson, Cristian Antonelli, Antonio Mecozzi, Fabio Mangini, Stefan Wabnitz, "Multimode solitons in optical fibers: a review," Photonics Res. 12, 2581 (2024)]

Prof. Stefan Wabnitz, the corresponding author, noted: "This work reviews recent advances in multimode solitons in fibers, covering fundamental concepts, theoretical models, numerical examples, analytical approaches, and experimental findings. This review paper is intended to help newcomers learn about multimode solitons and multimode fibers, while also providing experts with a comprehensive update on the latest theoretical and experimental developments."

Future research on MMS could address several open questions to deepen our understanding and control of MMS in fiber systems. Key areas of focus may include: Developing advanced methods to excite and control MMS. Investigating the dynamics of MMS in high-energy environments, such as hollow-core fibers. Exploring connections between MMS and other fields, including spatiotemporal mode-locked lasers, passive coherent driven cavities, and multimode fiber imaging applications like endoscopy. Building the potential connections with AI techniques, such as Physics-Informed Neural Networks, as powerful tools for advancing MMS research.