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---
## Known starting point
The matter-wave effective equation already has a concrete calculation from a selected extended gapped channel to a slow envelope. Equation (5.1.22) specifies its gap frequency and second-order propagation tensor. Equations (5.1.30)–(5.1.33) substitute the fast carrier into the original second-order equation: the carrier terms cancel, leaving an exact envelope relation that retains the second slow-time derivative.
With the energy–carrier matching of (5.1.34), long wavelengths, low speeds, a weak background and slow variation, (5.1.35) gives the leading effective equation. The exact substitution, the discarded time derivative and the low-wavenumber expansion of constant-coefficient dispersion are explicit. The remaining task is to calculate that branch and its coefficients from a complete core–field system and bound errors for a specified initial–boundary problem, rather than search again for a formally similar equation.
## Mathematical and physical task
Separate three levels: full discrete-connection dispersion, the long-wave continuum equation and its slow envelope. The first reduction needs $ka_0\ll1$; the second also needs $|\Omega|/\omega_0\ll1$, $ck/\omega_0\ll1$ and weak background modulation. The second-order dispersion holds within the selected continuum model and does not contain every lattice correction throughout the Brillouin zone.
Calculate the extended gapped branch, propagation tensor $B^{ij}$, normalization and $\omega_0$ from a specific action and stable background, and check intrinsic-energy matching. For band-limited initial data and explicit boundaries, control fast-branch excitation and estimate the retained exact remainder, accumulated time error and failure scales. Keep coefficient derivatives in divergence form for variable backgrounds. Treat interfaces, turning regions and suddenly switched boundaries separately; the earliest causal front is still governed by the original second-order equation. Use spectral or suitable norm conditions instead of dividing by the amplitude at envelope nodes.
## Reviewable deliverable
A first result can cover one family of band-limited initial data in a uniform background: provide the second-order and envelope solutions, error norms with the same normalization, a time interval, parameter scales and convergence plots. Then add one specified slowly varying coefficient or boundary problem. Separate the established carrier-substitution identity from approximate error bounds, and identify which coefficients were calculated from the core and which were adopted as inputs.
## What does not complete the task
Schrödinger form does not independently establish the microscopic branch, mass matching or Born event statistics. Conversely, unfinished microscopic closure does not remove the existing carrier substitution and effective limit. Treating full lattice dispersion, second-order continuum dispersion and the slow envelope as one exact equation hides distinct errors that require control.
## Research dependencies
- [The effective wave equation and exact remainder](/en/theory/quantum/effective-wave-equation)
- [Continuum variables, sources and boundaries](/en/theory/space-dynamics/continuum-dynamics)
[Quantum phenomena overview](/en/theory/quantum) · [Phenomenological requirements](/en/tests/quantum)