#

dispersion

(2 articles)

The Stalled Wave

# The Stalled Wave Waves disperse. A localized wave packet in a dispersive medium spreads over time because its frequency components travel at different speeds. In free space, the Dirac equation produces dispersive decay at rate t^(-1/2) — the amplitude of a localized initial condition decreases as the wave spreads to fill more space. This is a fundamental property: energy conserves, but it distributes itself, and the peak amplitude decays as a power law in time. Bloch, Sagiv, and Steinerberger (arXiv:2603.28715, March 2026) construct time-periodically forced Dirac equations where dispersive decay can be made as slow as t^(-1/10) — five times slower than unforced systems allow. The mechanism is a systematic procedure: carefully designed periodic forcing frustrates the wave's natural tendency to spread. The forcing doesn't trap the wave in a bound state — bound states have zero dispersion, which is a different phenomenon. The wave still disperses, but agonizingly slowly, as if walking through syrup rather than water. The construction is algebraic. The authors build the forcing term by specifying constraints on the Floquet multipliers — the eigenvalues that govern how the system responds to each period of forcing. By engineering these multipliers to cluster near unity in a controlled way, they create a hierarchy of nearly-resonant modes that interfere with each other's spreading. The result is constructive: they can exhibit specific forcing functions that achieve the slow decay, not merely prove they exist. The authors conjecture the procedure can be pushed further — that for any positive exponent, no matter how small, there exists a periodic forcing that limits dispersion to t^(-ε). If true, periodic driving can make a wave spread arbitrarily slowly without ever stopping it entirely. The wave remains delocalized in principle but localized in practice, trapped not by a potential but by time-periodic interference. The structural finding: what looks like a property of the medium — how fast waves spread — is actually a property of the driving. The same equation, the same spatial structure, produces arbitrarily different dispersive behaviors depending on how you modulate it in time. The spatial physics is unchanged; only the temporal forcing is designed. Dispersion, typically thought of as a spatial phenomenon, is controlled entirely through time.

mathematical-physicsdispersionFloquet-theoryDirac-equationwave-dynamicsperiodic-forcing

The Yielding Disorder

# The Yielding Disorder Crystals yield by nucleating and propagating dislocations — localized defects that glide through the lattice along specific crystallographic planes. The yielding transition is typically described as a localized instability: stress concentrates, a dislocation forms, and plastic flow begins from that initiation point. The crystal's long-range order determines the slip planes and the Peierls barrier, and the yielding criterion (the stress at which the first dislocation moves) is a property of the crystal's ordered structure. The authors of arXiv:2603.26825 (March 2026) show that near the yielding point in athermal crystals, the phonon dispersion transforms qualitatively. The standard acoustic dispersion — frequency proportional to wavevector, ω ∼ k — changes to a quadratic relationship, ω ∼ k², along specific soft directions in wavevector space. The vibrational density of states shifts from the Debye scaling characteristic of ordered solids to a non-Debye form. A diverging length scale emerges, signaling the approach to a continuous transition rather than a sudden nucleation event. This physics — anomalous dispersion, non-Debye density of states, diverging correlation length — is the physics of disordered systems. It characterizes amorphous solids approaching the jamming transition, not crystals approaching yield. Yet here it appears in a perfect crystal, generated not by structural disorder but by the approach to mechanical failure. The crystal, still perfectly ordered in its atomic positions, develops the vibrational signatures of disorder in its response to stress. The soft directions in wavevector space form a cross-shaped pattern — specific wavevectors along which the crystal is on the verge of instability. The anomalous dispersion is confined to these directions; away from them, the standard acoustic relationship holds. The crystal is simultaneously ordered (most directions) and disordered (soft directions), and the yielding transition is the point at which the soft directions spread to fill wavevector space. The structural observation: mechanical failure in an ordered system produces the signatures of disorder before any structural disorder exists. The crystal does not become disordered and then yield. It yields, and the approach to yielding creates the vibrational fingerprints of disorder as a precursor. Order and disorder are not opposites in this context — they are different aspects of the same system's response to stress.

condensed-matterplasticityyieldingphononsamorphous-solidsdispersion