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摘要
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Discrete symmetries govern the motion and real axis crossings of interference zeros in the
meromorphic continuation of a wave field, thereby determining the singular structure of
its logarithmic derivatives. We considered two counter-propagating second-order
rational pulses that satisfy the one-dimensional massless wave equation exactly. With 𝑞ൌ
𝑟expሺ𝑖𝜑ሻ denoting the relative complex amplitude, the field admits two antilinear
reflection symmetries: real 𝑞 preserves collision-centered spacetime inversion followed
by complex conjugation, whereas |𝑞| ൌ 1 preserves fixed time spatial reflection followed
by conjugation up to an overall phase. The balanced in-phase state, 𝑞ൌ 1 , is the nondegenerate intersection of these symmetry manifolds; 𝑞ൌ−1 produces global
cancellation on the collision slice. For 𝑞ൌ 1, the two interference zero branches lie on the
imaginary axis and cross the real axis at 𝑐𝑡 ൌ 𝑥₀ േ ℓ, on opposite sides of the pulse center
collision. Away from the symmetry manifolds, the zero trajectories deform continuously,
and the associated pairing constraints are lost, while the crossing conditions remain
available in closed form. A logarithmic complex action representation yields local
momentum, energy, transport ratio, and a derived second-order complex action
descriptor without altering the underlying wave dynamics. Near an isolated noncharacteristic moving zero, the leading simple pole factors cancel in the transport ratio,
whereas the second-order term develops a double pole. The leading real axis response
therefore scales as 𝑑
ିଶ . A reference finite-window fit yields an exponent of −1.885 (𝜌ൌ
−0.995), and the fitted exponent approaches −1.998 as the fitting interval is narrowed
toward the isolated-zero regime. These results provide an exact benchmark linking
antilinear symmetry, complex zero-pole geometry, and real axis differential amplification.
The second-order quantity 𝑄𝑐 is used only as a descriptor generated by the logarithmic
representation; it is neither an externally imposed potential nor an additional dynamical
term. The loss of reflection symmetry away from the two symmetry manifolds is explicit
and parameter-induced, not spontaneous. |
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