Soliton Hopping: Discrete Nonlinear Transport

• Soliton hopping is a process where localized nonlinear excitations undergo discrete, protocol-controlled transitions while maintaining invariants like mid-gap states, Hopf indices, or Chern numbers.
• It encompasses mechanisms such as adiabatic shuttling in Josephson junction arrays, three-dimensional hops in chiral liquid crystals, fractional Thouless pumping in nonlinear lattices, and chaotic switching in microresonators.
• The discrete transport is achieved via controlled protocols exploiting topology, nonlinearity, and driven-dissipative dynamics to ensure precise and stable state transfer.

Searching arXiv for the cited papers and closely related "soliton hopping" work. Soliton hopping denotes a class of processes in which a localized nonlinear or topological excitation undergoes a discrete change of position, host, or dynamical state rather than a purely continuous drift. Across recent literature, the term has been used for adiabatic shuttling of topological mid-gap modes in a bosonic Su-Schrieffer-Heeger chain implemented in a Josephson-junction array, for electric-pulse-driven three-dimensional hops of Hopf solitons in chiral liquid crystals, for integer or fractional unit-cell transport of nonlinear lattice solitons under Thouless pumping, and for periodic or chaotic transfer phenomena involving dissipative Kerr solitons in coupled or injection-locked microresonators (Kuzmanovski et al., 2023, Tai et al., 2021, Tao et al., 10 Feb 2025, Deshmukh et al., 13 Aug 2025, Moille et al., 11 Sep 2025). In this broad usage, the common element is discrete, protocol-dependent transport of a localized object together with the retention of a defining invariant such as a mid-gap branch, a Hopf index, a Chern number of a self-consistent effective Hamiltonian, or a periodic-orbit structure.

1. Scope and taxonomy

In the literature considered here, “soliton hopping” is not a single universal mechanism. In one-dimensional bosonic SSH systems, it refers to controlled motion of a topological zero-mode between neighboring dimers by local electrostatic gates. In chiral liquid crystals, it refers to a net three-dimensional displacement acquired after a full electric-field on/off cycle because the heliknoton $\to$ hopfion and hopfion $\to$ heliknoton transformations are non-reciprocal. In nonlinear AAH lattices, it denotes quantized adiabatic transport over one or several pump periods. In microresonator photonics, it can denote either periodic transfer between coupled resonators or random transitions between synchronized repetition-rate states (Kuzmanovski et al., 2023, Tai et al., 2021, Tao et al., 10 Feb 2025, Deshmukh et al., 13 Aug 2025, Moille et al., 11 Sep 2025).

Platform	Localized object	Hopping signature
Josephson metamaterial	topological mid-gap soliton states	final fidelities $F_{s_1},F_{s_2}$ exceed $0.998$
Chiral liquid crystal	heliknoton or hopfion	$\Delta x\approx1\,\mu{\rm m}$, $\Delta z\approx0.5\,p_0$ per cycle
Nonlinear off-diagonal AAH chain	soliton	average shift $1$, $1/2$, $1/3$, or $1/4$ unit cells per cycle
Coupled or driven microresonators	dissipative Kerr soliton	periodic transfer between resonators or random repetition-rate jumps

A recurrent misconception is to equate hopping with ordinary translational motion of a continuum soliton. The surveyed works do not support that reduction. In several cases the process is instead branch-following in an instantaneous spectrum, a topology-preserving reconversion between distinct embedded textures, or a transition between coexisting attractors. This suggests that “hopping” functions as a phenomenological label for discrete transport of localized nonlinear states across markedly different dynamical settings.

2. Bosonic SSH soliton shuttling in a Josephson metamaterial

A particularly explicit formulation appears in the bosonic SSH realization derived from a Josephson-junction array. In units $\to$0, the circuit Hamiltonian is

$\to$1

with $\to$2 set by gate and coupling capacitances, $\to$3, and $\to$4. In the charging-dominated regime, linearizing Cooper-pair hopping yields a non-interacting bosonic tight-binding model,

$\to$5

which becomes an SSH chain under alternating couplings $\to$6 and $\to$7 (Kuzmanovski et al., 2023).

The time-dependent driven Hamiltonian is

$\to$8

with only two adjacent islands pulsed to move the soliton. For the static chain with $\to$9, the bulk bands obey

$F_{s_1},F_{s_2}$0

so the band gap is $F_{s_1},F_{s_2}$1. A domain wall or a strong detuning defect produces mid-gap states pinned near $F_{s_1},F_{s_2}$2. In the semi-infinite case, the zero-mode has support

$F_{s_1},F_{s_2}$3

while on a finite ring with two detuned sites there is a pair of nearly degenerate mid-gap wavefunctions localized with length $F_{s_1},F_{s_2}$4 (Kuzmanovski et al., 2023).

The shuttling protocol moves the soliton from dimer $F_{s_1},F_{s_2}$5 by a three-segment sequence: ramp down $F_{s_1},F_{s_2}$6 from $F_{s_1},F_{s_2}$7, hold near zero for a buffer time, then ramp up $F_{s_1},F_{s_2}$8 from $F_{s_1},F_{s_2}$9. A convenient choice uses linear ramps of duration $0.998$0 and a relative shift $0.998$1, with amplitudes $0.998$2 and $0.998$3 on the two sites of a dimer so that the soliton never sees two strong gates at once. The evolution is tracked in the instantaneous basis $0.998$4 and propagated by $0.998$5.

Adiabaticity is controlled by

$0.998$6

with the rough requirement $0.998$7. The fidelity of a mid-gap branch,

$0.998$8

satisfies

$0.998$9

and a Landau-Zener estimate gives $\Delta x\approx1\,\mu{\rm m}$0. Numerical benchmarks used a ring of $\Delta x\approx1\,\mu{\rm m}$1 dimers, $\Delta x\approx1\,\mu{\rm m}$2, $\Delta x\approx1\,\mu{\rm m}$3, $\Delta x\approx1\,\mu{\rm m}$4, small sub-dimer splitting $\Delta x\approx1\,\mu{\rm m}$5, and effective temperature $\Delta x\approx1\,\mu{\rm m}$6. With total single-hop time $\Delta x\approx1\,\mu{\rm m}$7, protocol $\Delta x\approx1\,\mu{\rm m}$8 keeps the mid-gap states away from bulk crossings and yields final fidelities exceeding $\Delta x\approx1\,\mu{\rm m}$9, whereas protocol $\Delta z\approx0.5\,p_0$0 without the proper time-shift fails with $\Delta z\approx0.5\,p_0$1 (Kuzmanovski et al., 2023).

3. Three-dimensional hopping of Hopf solitons

In chiral liquid crystals, soliton hopping is formulated for three-dimensional topological textures characterized by the Hopf invariant. The director field $\Delta z\approx0.5\,p_0$2, with $\Delta z\approx0.5\,p_0$3, is governed by a Frank-Oseen free energy

$\Delta z\approx0.5\,p_0$4

where

$\Delta z\approx0.5\,p_0$5

and

$\Delta z\approx0.5\,p_0$6

The Hopf charge is the linking number of preimage loops and may be written as

$\Delta z\approx0.5\,p_0$7

(Tai et al., 2021).

Two embedded textures are emphasized. In the uniform background with $\Delta z\approx0.5\,p_0$8, a localized Hopf texture is termed a hopfion. In a helical background winding around an axis $\Delta z\approx0.5\,p_0$9 with period $1$0, the corresponding embedded object is a heliknoton, with a background ansatz

$1$1

Stability depends strongly on elastic anisotropy. By mixing a bent-core liquid crystal with a rod-like liquid crystal, the ratio $1$2 can be tuned from $1$3 to $1$4, and numerical minimization shows that for $1$5 and zero field heliknotons are energetically metastable, and in certain $1$6-ranges even thermodynamically favored, without any applied field or cell confinement (Tai et al., 2021).

The hopping mechanism is driven by electric pulses. With $1$7 on, the unwound background is favored and the soliton deforms into a hopfion in a uniform surround. When $1$8 is switched off, the helical background is restored and the texture becomes a heliknoton again. The critical point is that the director evolution is non-reciprocal: the path from helix to uniform to helix does not retrace itself. Each full on/off cycle therefore produces a net displacement while the Hopf index remains rigorously conserved at every moment. The evolution is modeled by overdamped relaxation,

$1$9

with the unit-length constraint and projected torques (Tai et al., 2021).

The reported transport is discrete and three-dimensional. Under periodic on/off pulses with period $1/2$0, the soliton drifts by $1/2$1, rotates its long axis by $1/2$2 per cycle, and exhibits a vertical hop $1/2$3 each cycle. The full heliknoton $1/2$4 hopfion $1/2$5 heliknoton cycle takes $1/2$6, and the two half-cycles are asymmetric, with hopfion $1/2$7 heliknoton taking $1/2$8 longer than the reverse. Simulations typically used a box of $1/2$9 in $1/3$0, height $1/3$1, a mesh of $1/3$2 grid points per $1/3$3, periodic boundaries in $1/3$4, strong Dirichlet boundaries in $1/3$5, and material parameters $1/3$6, $1/3$7, $1/3$8, $1/3$9, $1/4$0 (Tai et al., 2021).

4. Fractional Thouless pumping of nonlinear lattice solitons

A different use of soliton hopping arises in the nonlinear off-diagonal Aubry-André-Harper model, where transport is induced by adiabatic modulation of nearest-neighbor hoppings. The linear Hamiltonian is

$1/4$1

with

$1/4$2

After adding on-site Kerr nonlinearity, the amplitudes obey a discrete nonlinear Schrödinger equation,

$1/4$3

with conserved norm $1/4$4 (Tao et al., 10 Feb 2025).

The central mechanism is self-induced topology. In the instantaneous approximation, the stationary soliton profile $1/4$5 satisfies a nonlinear eigenproblem with eigenvalue $1/4$6. Its density produces an effective on-site shift

$1/4$7

so that the modified linear Hamiltonian

$1/4$8

can carry nontrivial topology even though the bands of $1/4$9 are topologically trivial. The pump parameter is ramped as

$\to$00

ensuring adiabatic evolution (Tao et al., 10 Feb 2025).

In a supercell of length $\to$01, one defines a Chern number

$\to$02

If after $\to$03 cycles the soliton has moved by exactly $\to$04 unit cells, the average displacement per cycle is

$\to$05

The reported cases realize one-cell transport in one, two, three, or four periods, corresponding to average shifts of $\to$06, $\to$07, $\to$08, or $\to$09 unit cells per cycle. The soliton center is tracked by

$\to$10

and exhibits perfectly quantized plateaux of $\to$11 per cycle for $\to$12 (Tao et al., 10 Feb 2025).

Representative parameter sets are explicitly given. For the integer pump, $\to$13, $\to$14, $\to$15, $\to$16, $\to$17, and $\to$18, and the soliton moves exactly one unit cell as $\to$19 goes from $\to$20 to $\to$21. For the half-cell pump, $\to$22, $\to$23, $\to$24, $\to$25, $\to$26, $\to$27, and one unit cell is traversed in two cycles. Analogous one-third-cell and one-quarter-cell pumps are reported for $\to$28 and $\to$29, respectively. The proposed implementation is a one-dimensional array of evanescently coupled nonlinear waveguides, with typical parameters $\to$30, modulation period $\to$31, and physical sample lengths of order $\to$32 per pump cycle (Tao et al., 10 Feb 2025).

5. Dissipative Kerr-soliton hopping in microresonators

In coupled Kerr-nonlinear ring resonators, soliton hopping appears as a periodic orbit of coupled Lugiato-Lefever equations rather than as adiabatic motion in a static spectral gap. For a dimer or trimer of identical resonators, the normalized mean-field equations are

$\to$33

A stationary supermode soliton can lose stability when a pair of complex-conjugate eigenvalues crosses the imaginary axis, producing a periodic orbit in which the soliton periodically hops between resonators. For the dimer with $\to$34 and $\to$35, the stationary branch $\to$36 is stable in a window of $\to$37 and undergoes a subcritical Hopf at $\to$38, with $\to$39. The resulting branch $\to$40 initially emerges unstable and becomes stable only after saddle-node bifurcations of the orbit. For the trimer at the same $\to$41 and $\to$42, the stationary branch $\to$43 is already unstable to a real eigenvalue, and a supercritical Hopf at $\to$44 creates an unstable $\to$45 that stabilizes after a subsequent fold. These bifurcation differences explain why trimer hopping can be experimentally accessible at lower pump power than dimer hopping, and why the dimer exhibits hysteresis, multistability, and strong path dependence in laser scans (Deshmukh et al., 13 Aug 2025).

A distinct microresonator realization concerns chaotic group-velocity hopping of a single dissipative Kerr soliton under Kerr-induced synchronization to an externally injected weak reference laser. The underlying single-ring dynamics follow the Lugiato-Lefever equation

$\to$46

With a reference laser near comb mode $\to$47, synchronization fixes

$\to$48

and phase modulation $\to$49 allows locking to the carrier or sidebands. The reduced dynamics are described by a second-order Adler equation,

$\to$50

and, under modulation, by the autonomous three-dimensional system

$\to$51

Because the second-order term supplies effective inertia, the modulated system supports multiple coexisting attractors and strange attractors, yielding chaotic switching between repetition-rate states (Moille et al., 11 Sep 2025).

The experiment used a $\to$52 radius $\to$53 microring pumped in TE mode at $\to$54 with $\to$55 on chip, a counterpropagating TM cooler laser at $\to$56, and a weak TE reference at $\to$57 phase-modulated up to $\to$58. Detection employed EO-comb bridging of $\to$59 sidebands at $\to$60. Near $\to$61 for $\to$62 and $\to$63, the theory yields a positive Lyapunov exponent $\to$64 and the $\to$65 chaos test gives $\to$66. Experimentally the RF spectrum shows two sharp tones separated by

$\to$67

and time-resolved spectrograms reveal random jumps between them over a $\to$68-wide detuning range. Binary tests on the hopping sequence yield $\to$69-values all $\to$70 (Moille et al., 11 Sep 2025).

6. Related discrete transport problems and conceptual distinctions

A related but not identical framework is provided by one-dimensional mechanical linkage models. There the chain consists of rigid rotors with angles $\to$71, arm lengths $\to$72, pivot spacings $\to$73, and springlike constraints encoded by the discrete Lagrangian

$\to$74

with

$\to$75

Topological zero-energy solitons are defined as trajectories on the configuration-space manifold $\to$76, and scattering at a junction between chains with different parameters yields reflection and transmission amplitudes

$\to$77

with an effective impedance $\to$78. If both sides have the same topological index $\to$79, the soliton is fully transmitted; if $\to$80, the zero-energy cycle does not exist across the junction and the soliton is totally reflected (Sato et al., 2018).

This scattering problem clarifies an important distinction. Not every discrete change in a soliton’s position is a driven hop in the sense of the later SSH, Hopf-soliton, or microresonator literature. In the linkage model, the central issue is transmission or reflection at an interface; in the Josephson SSH model it is adiabatic branch-preserving shuttling; in chiral liquid crystals it is non-reciprocal cycle-to-cycle displacement with strictly conserved Hopf index; in the nonlinear AAH problem it is quantized topological pumping mediated by a self-induced potential; and in photonic microresonators it is either periodic resonator transfer or chaotic switching between synchronized states (Sato et al., 2018).

Taken together, these works indicate that the most stable cross-disciplinary meaning of soliton hopping is discrete transport of a localized nonlinear object under a protocol or bifurcation structure that suppresses ordinary dispersive delocalization. What is preserved differs from system to system: a mid-gap identity in bosonic SSH chains, a linking invariant $\to$81 in Hopf textures, a Chern-number-controlled displacement law in nonlinear pumping, or an invariant periodic orbit or synchronization branch in dissipative photonics. This suggests that the term is best understood not as a single mechanism but as a family resemblance across topological, nonlinear, and driven-dissipative transport phenomena.