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## Physical setting
Work with the dimensionless complex superconducting order parameter
$\psi(x,y,t)$. Its inertial dynamics obey
$$
\frac{1}{2}\frac{\partial^2\psi}{\partial t^2}
+\gamma\frac{\partial\psi}{\partial t}
=\psi-|\psi|^2\psi+(\boldsymbol{\nabla}-i\mathbf A)^2\psi .
$$
Time is measured in inverse Higgs-mode frequency and distance in coherence
lengths. The system starts in the uniform equilibrium state,
$$
\psi(x,y,0)=1,\qquad \partial_t\psi(x,y,0)=0,
$$
and approaches that equilibrium state far from the illuminated region.
Use a continuous resonant, linearly polarized Gaussian field,
$$
\mathbf A(x,y,t)=\hat{\mathbf x}\cos t
\sum_j A_j
\exp\!\left[-\frac{(x-x_j)^2+y^2}{w_0^2}\right].
$$
Here $A_j$ is the peak dimensionless vector-potential amplitude of one beam.
Investigate the following two physical cases:
| case | beam centres $x_j$ | $A_j$ | $w_0$ | $\gamma$ |
|---|---|---:|---:|---:|
| S | $0$ | $1.0$ | $10$ | $0.05$ |
| P | $-40,+40$ | $2.0$ | $20$ | $0.10$ |
The second-order inertial equation and the two field configurations define the
physical problem. You may choose a stable spatial representation, boundary
realization, time propagator, sampling strategy, transform convention and
evidence layout, provided that your choices preserve this problem and do not
materially contaminate the claims.
## Research problem
Determine the physical origin and classification of the collective response
generated by the two specified optical configurations. Establish which
conclusions about the response are supported across the S and P cases, and what
material alternatives remain compatible with the computed dynamics.
Base the argument on auditable calculations of the complex order parameter and
student-chosen observables with enough discriminating force to support the
classification. State clearly which claims are established by these two cases
and which broader claims are not.
The solver, grid, box, boundary implementation, sampling interval, spectral
window, momentum convention, plotting range, normalization and number or order
of figures are not prescribed. Declare the conventions that materially affect
your interpretation.
## Submission and resources
Submit one self-contained scientific report and an auditable offline
calculation package containing the task-specific derivation, code, numerical
data underlying every material claim, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise
run and environment information. No fixed file tree, filename scheme, panel
count or complete exploration history is required.
Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary are binding. No separate aggregate CPU-hour limit applies. Greater computational expense
receives no additional credit.
## Reference and integrity policy
This examination is conducted without network access or live literature
retrieval. You may rely on your existing general scientific knowledge and, if any are
explicitly supplied with the examination, on those offline materials or
software documentation. Record every such source actually consulted. For formulas,
methods, constants, or conventions used from general knowledge rather than a
consulted source, state enough of the method basis, definition, value, units,
and convention to make the calculation auditable; do not invent a citation.
You are not required to identify or prove a publication date or to retrieve
literature during the examination.
The hidden target article and its identifying metadata, figures or captions,
supplementary material, author code or data, target-specific reproduction
code, solution or scoring material, online solutions, and worked
reproductions of this case must not be used. All submitted scientific figures
and numerical data must be generated by your own implementation of the stated
problem.
See [submission requirements](../../submission_requirements.md) for the common submission entry points.
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