Onsite Bright Solitons in Lattice Models

• Onsite bright solitons are spatially localized, time-independent nonlinear wave excitations that form precisely at lattice sites or potential peaks while balancing dispersion and nonlinear effects.
• Mathematical models, including the discrete nonlinear Schrödinger equation and continuous NLSE variants with spatially modulated nonlinearity, rigorously establish their existence and stability conditions.
• Experimental implementations in ultracold atoms, nonlinear optics, and mechanical lattices demonstrate their robust dynamics and potential for advancing photonic and quantum systems.

Onsite bright solitons are spatially localized, time-independent nonlinear wave excitations whose center coincides precisely with a lattice site, high-symmetry point, or the peak of a trapping/nonlinear potential. They emerge in a variety of continuous and discrete physical systems, both in conservative and dissipative regimes, and are characterized by a delicate balance between dispersion (or kinetic energy) and nonlinear effects—often stabilized or shaped by lattice discreteness, spatially modulated interactions, or complex multicomponent couplings.

1. Mathematical Models and Existence Criteria

Onsite bright solitons arise as stationary solutions of nonlinear wave equations with a symmetry or discreteness that allows their center to be located "onsite." Two major classes are:

• Discrete lattices (e.g., the discrete nonlinear Schrödinger equation, DNLS)
• Continuous nonlinear Schrödinger equations (NLSE) with nonlinear or linear potentials (modulated nonlinearity, waveguides, or optical lattices)

In the DNLS with focusing nonlinearity, the prototypical equation is

$$i \frac{d\psi_n}{dt} + C(\psi_{n+1} + \psi_{n-1} - 2 \psi_n) + |\psi_n|^2 \psi_n = 0$$

where $n$ indexes lattice sites and $C$ is the coupling constant. In the strong-coupling regime, onsite (centered at $n_0 \in \mathbb{Z}$) and intersite ($n_0 = \tfrac{1}{2} + \mathbb{Z}$) solutions are uniquely selected by the analytic structure of the exponentially small tail corrections (Adriano et al., 18 Jul 2025). The leading-order continuum approximation for the onsite soliton is

$\phi_0(x) = \sqrt{2} \, \operatorname{sech}(x)$

with $x = \epsilon(n-n_0)$ and $\epsilon = \sqrt{2/C}$. The cancellation of exponentially growing tails imposes $\sin(2\pi n_0) = 0$, restricting solutions to onsite ($n_0=0$) or intersite ($n_0=1/2$) types.

In continuous systems with spatially modulated nonlinearity, for example,

$$i\partial_\xi q = -\frac{1}{2}\nabla^2 q + \sigma(r)|q|^2 q,$$

where $\sigma(r)$ increases rapidly with $|r|$, bright solitons of the form $q(r,\xi) = A\exp(ib\xi - a r^2/2)$ can emerge even for defocusing nonlinearity (1108.3673). Here, the onsite character refers to solitons localized at minima of the effective trapping potential.

2. Stability and Symmetry Selection

Stability of onsite bright solitons is a central concern and is governed by the system's symmetry, dimensionality, and nonlinear dispersion balance:

• In DNLS lattices, the onsite bright soliton ($n_0=0$) is rigorously proven to be linearly stable in the strong-coupling regime. The squared eigenvalue associated with translation becomes

$$\lambda^2 \sim -4\pi^2|\Lambda_1|\epsilon^{-5}e^{-\pi^2/\epsilon}$$

(for explicit $\Lambda_1$ in (Adriano et al., 18 Jul 2025)). The negative sign indicates that $\lambda$ is purely imaginary, confirming stability. Intersite solitons, in contrast, are always unstable due to a real eigenvalue pair.

• For the PT-symmetric dimer chain, the discrete onsite bright soliton can be stable for small inter-site coupling and balanced gain/loss (Kirikchi et al., 2016). However, as the coupling or gain/loss increases, instabilities may emerge, manifesting as drift (traveling solitons) or blow-up.
• In spatially inhomogeneous mean-field systems, the localization mechanism and the sign and profile of the nonlinearity dictate stability. For instance, spatially increasing defocusing nonlinearity suppresses spreading, ensuring soliton localization and linear stability in broad parameter regimes (1108.3673).

3. Experimental Implementations and Observations

Onsite bright solitons have been observed in a range of platforms:

• Ultracold Atoms in Optical Lattices: Discrete bright matter-wave solitons with attractive interactions are generated by loading a Bose–Einstein condensate (BEC) of cesium atoms into an optical lattice, followed by an interaction-quench protocol (Cruickshank et al., 15 Apr 2025). By preparing the density profile to favor occupation of a single lattice site, and by rapidly switching to attractive interactions, single-site (onsite) bright solitons are created. Multi-site (but still soliton-like) structures can also be formed.
• Electrical and Mechanical Lattices: Bright discrete breathers have been realized in diatomic-like electrical circuits driven at a frequency within the bandgap of the linear spectrum (Palmero et al., 2018). As the driving frequency is tuned, the system transitions from dark breathers near the band edge to highly localized onsite bright breathers deep in the gap, with stability confirmed both experimentally and through Floquet analysis.
• Nonlinear Optical Media: Onsite bright solitons occur in arrays of coupled waveguides or microresonators, as well as in hybrid light–matter systems (e.g., exciton–polariton waveguides), where the nonlinearity, dispersion, and gain/loss are judiciously balanced (Yulin et al., 2022, Tinkler et al., 2014).

4. Theoretical and Computational Tools

A suite of analytical and numerical methods has been developed for describing onsite bright solitons:

• Exponential asymptotics: This method captures the impact of exponentially small lattice corrections and uniquely selects the allowed spatial centers for localized solitons. It explains the emergence of onsite and intersite soliton families in discrete lattices and provides analytical predictions for their stability spectra that match high-precision numerics (Adriano et al., 18 Jul 2025).
• Variational methods: By adopting a Gaussian or hyperbolic secant trial function, one can map the energy landscape as a function of wavepacket width and interaction strength. This approach clarifies the conditions for soliton stability, collapse, and dispersion in lattices and provides critical thresholds for the existence of onsite localized states (Cruickshank et al., 15 Apr 2025).
• Bilinearization and integrability techniques: In integrable multicomponent and high-dimensional systems (such as 2+1-dimensional long-wave–short-wave resonance systems), Gram determinant representations enable the explicit construction and classification of onsite bright solitons (Sakkaravarthi et al., 2014, Kirane et al., 2022).

5. Role of Nonlinearity, Discreteness, and External Potentials

Onsite bright solitons are fundamentally shaped by the interplay between nonlinearity and the "onsite" attributes of the underlying structure:

• Inhomogeneous Interactions: In Bose-Hubbard and nonlinear Schrödinger lattice models, spatially dependent onsite repulsive interactions that increase away from the center can create an effective trapping for bosons, leading to "quantum bright solitons" (Barbiero et al., 2014). The stability and localization depend on the exponent of the spatial variation and the overall strength of interactions.
• Lattice Discreteness: Lattice potentials support both the existence and pinned nature of onsite bright solitons, introducing effective Peierls–Nabarro barriers that impede the free movement of solitons and can result in energetic and dynamic pinning.
• Spatial Modulation: In continuous systems, spatially structured nonlinearities (e.g., $\sigma(r) \propto \exp(a r^2)$) induce a robust nonlinear trapping mechanism, enabling bright soliton formation even for globally defocusing media (1108.3673).

6. Dynamical Properties and Applications

• Oscillations and Mobility: Onsite bright solitons can behave as quasi-particles oscillating in an effective potential (e.g., in BECs with spatially modulated nonlinearity) or traveling robustly in lattice potentials. Their dynamics under perturbation, including breathing oscillations and quasi-elastic collisions, have been extensively characterized in numerical simulations and experiments (Cruickshank et al., 15 Apr 2025, Palmero et al., 2018, Sakkaravarthi et al., 2014).
• Energy Barriers and Collapse: The stability and dynamical lifetime of onsite matter-wave solitons are governed by the relative heights of energy barriers (against collapse or dispersion), which depend nonlinearly on interaction strength, lattice depth, and wavepacket shape (Cruickshank et al., 15 Apr 2025).
• Stability under Dissipation: Dissipative effects, especially in real physical realizations, can profoundly affect soliton integrity. Experiments in mid-infrared laser chips demonstrate that dissipative bright solitons, stabilized by fast bistability and gain saturation, can remain robustly pinned onsite for hours (Kazakov et al., 30 Jan 2024).
• Quantum Fluctuations and Transitions: In quantum lattice models, quantum bright solitons formed by a localized density peak surrounded by a residual background can undergo a transition to a Mott insulator as interactions intensify (Barbiero et al., 2014). Quantum droplets in multi-component BECs highlight further complexity, where beyond-mean-field effects mediate transitions between onsite bright solitons and denser droplet states (Cheiney et al., 2017).

7. Outlook and Research Directions

Onsite bright solitons continue to serve as testbeds for nonlinear dynamics in discrete and continuous systems. Future research will address:

• The interplay of discreteness and quantum fluctuations in many-body models and their signatures in the dynamics of localized excitations.
• Nonequilibrium phenomena, including bistability, quantum–classical crossover, and transitions between soliton, droplet, and breather states in driven-dissipative systems.
• Application-driven exploration of robust, onsite-localized bright solitons in photonic, optoelectronic, and quantum platforms, particularly in the context of frequency comb generation, nonlinear switching, and coherent information processing.

Their universality and robustness across disparate physical scales underscore their enduring significance in modern nonlinear science and technology.