Modern Mathematical Physics

We investigate the non-linear dynamics of de Broglie gravitational waves within a post-Minkowskian framework. Starting from the perturbative solution proposed by Feoli and Scarpetta, we derive the second-order gravitational field generated by the wave’s self-interaction. Owing to the extremely high oscillation frequency of de Broglie gravitational waves, we employ the Isaacson/Misner–Thorne–Wheeler averaging procedure to obtain an effective coarse-grained geometry. The resulting averaged theory takes the form of an effective linear perturbation theory at second order in the wave amplitude. Near the symmetry axis, the resulting spacetime exhibits a non-vanishing static curvature generated entirely by non-linear effects. We show that geodesics in this effective geometry undergo bounded transverse oscillations governed by a harmonic equation. Possible implications for de Broglie’s wave–particle framework are also discussed.

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We explore the capability of a Large Language Model (LLM) to perform specific computations in mathematical physics: the task is to compute the coordinate Bethe Ansatz solution of selected integrable spin chain models. We select three integrable Hamiltonians for which the solutions were unpublished; two of the Hamiltonians are actually new. We observed that the LLM semi-autonomously solved the task in all cases, with a few mistakes along the way. These were corrected after the human researchers spotted them. The results of the LLM were checked against exact diagonalization (performed by separate programs), and the derivations were also checked by the authors. The Bethe Ansatz solutions are interesting in themselves. Our second model manifestly breaks left–right invariance, but it is PT-symmetric; therefore its solution could be interesting for applications in Generalized Hydrodynamics. And our third model is solved by a special form of the nested Bethe Ansatz, where the model is interacting, but the nesting level has a free fermionic structure lacking $U\left(1\right)$-invariance. This structure appears to be unique and it was found by the LLM. We used ChatGPT 5.2 Pro and 5.4 Pro by OpenAI.