Two packets, with their mixed sextic interactions.
Compare the full H₄ + H₆ evolution with a control that removes mixed sextic terms, and with the quartic reference. Both packets share one physical clock. Each model starts from the same state.
Two wave packets occupy fixed Fourier sites. Their amplitudes move through carrier chains, supported by pairs of pump modes. This explorer shows how the sextic resonant interaction between the packets changes that motion.
The three evolutions begin from the same initial data: the complete quartic-plus-sextic model, a control with only mixed packet terms removed, and the quartic reference. Switch models at a fixed time to compare their carrier distributions.
Old packet
New packet
Old packet: carrier actions
Actual saved trajectory for the selected model, on the common physical clock. Actions are divided by this packet’s initial carrier mass.
New packet: carrier actions
The full time interval is shown; no cycle is repeated or stitched into this heatmap. The cursor matches both spatial panels.
Numerical values at the selected time
| Model | Old terminal fraction | Added terminal fraction | Full H⁸ gain | Old mean index / N | Added mean index / N |
|---|
Carrier fractions use each packet’s initial carrier mass. Positions are normalized to the chain length. This table provides the plotted values as text; the time slider works with the keyboard.
Full H⁸ norm, including all pumps
G₈(t) = ‖z(t)‖H⁸ / ‖z(0)‖H⁸. All three models remain visible. A difference from the quartic curve alone does not isolate the mixed interaction.
Where each carrier packet sits
Mean carrier index divided by chain length: 0 is the first carrier, 1 is the terminal carrier. Color identifies the packet; dashed curves remove mixed sextic terms.
Old packet: change caused by mixed terms
Signed carrier-action difference: full minus mixed-terms-removed control, divided by initial carrier mass. This comparison stays fixed when the display model changes.
New packet: change caused by mixed terms
Blue means less action in the full model; red means more. Both difference heatmaps use the same fixed symmetric color scale over the entire run.
Size of the interaction’s effect
Each packet’s weighted contribution
Selected model: Q₈,old / Q₈,total(0) + Q₈,new / Q₈,total(0) = G₈². These include carrier and both pump actions. Mass conservation is separate from weighted-norm change.
Static overview of the carrier motion
Model and numerical checks
The complete model is H₄ + Z₆, with the kinetic rotation and the conserved common-mass phase removed. The sextic term is Z₆ = ½ Πres{P₄, χ₄}, where {H₂, χ₄} = −P₄. Coefficients include unrestricted virtual intermediate frequencies before restriction to the support. No extra multiplier is applied to Z₆.
- Support
- 20 integer modes; 2 old links and 4 added links. Exact closure is checked through degree six.
- Initial amplitudes
- Old: 1; added: 0.4. Base pump parameter A = 30.
- Time window
- 0 ≤ t ≤ 1.30899694, with 4,001 saved samples.
- Sextic coefficients
- 1768 nonzero monomials; 1109 involve both packets. Conjugate monomials are counted separately.
- Integrator
- SciPy DOP853; relative/absolute tolerances (10⁻¹², 10⁻¹⁴), compared with (10⁻⁹, 10⁻¹¹).
- Carrier refinement
- Maximum normalized carrier-action discrepancy: 2.34211e-08.
- Invariant drift
- Maximum recorded normalized drift: 8.86513e-12. Checks include mass, momentum, kinetic energy, the model Hamiltonian, and chain invariants.
What the comparison measures
Full minus the mixed-terms-removed control isolates the effect of interaction between packets. Full minus quartic also includes each packet’s internal sextic correction. Each packet conserves its total and carrier masses on this support: coupling changes phases and internal transport without net mass transfer between packets.
The full H⁸ norm includes every carrier and both pump families. Gains use the combined initial norm. The interaction-distance plots show the sum of absolute carrier-action differences divided by the initial carrier mass, twice the total-variation distance.
| Evolution | Old terminal fraction | Added terminal fraction | Full H⁸ gain |
|---|---|---|---|
| Full sextic: packets interact | 0.999647 | 0.682268 | 6.64261 |
| Sextic control: mixed terms removed | 0.999893 | 0.9992 | 7.08038 |
| Quartic reference | 0.999811 | 0.999699 | 7.08492 |
Frequencies are fixed throughout the simulation. Site plots divide the physical integer coordinates by the displayed common scale only for presentation; the equations and Sobolev weights use the integer support. Reference peak buttons refer to the quartic seed periods, not a claim of sextic periodicity.
Mixed sextic Hamiltonian along the full trajectory
The mixed part H₆,mixed is a component of the Hamiltonian. It need not be conserved separately.
Model, numerical checks, and run metadata
The control keeps quartic dynamics and each packet’s own sextic terms while deleting mixed sextic terms. It is a comparison system, not a replacement for the complete normal form. These are finite-time trajectories from the saved initial data; sextic periodicity is not assumed. Fixed site coordinates are already divided by the one common scale specified in the metadata.
NLS Growth numerical study · 11 November 2026 · Dataset: sextic-two-packet-2fa4f2c4dd45. This page is self-contained, makes no network requests, and has no analytics. Use the model selector and time slider to inspect saved solver samples; playback starts only when requested.