Matthew Dominique Mitchell

arXiv2026

The Small-Dispersion Limit of the Intermediate Long Wave Equation via Semiclassical Soliton Ensembles

We study the small dispersion limit of the intermediate long wave (ILW) equation, specifically on a class of well-behaved initial conditions $u_0$ where the number of solitons in the solution increases without bound. First, we conduct a formal WKB-style analysis on the ILW direct scattering problem, generating approximate eigenvalues and norming constants. We then use this to define a modified set of scattering data and rigorously analyze the associated inverse scattering problem. The main results include demonstrating $L^2$-convergence of the solution at $t = 0$ to the original initial condition $u_0$ and for $0 < t < t_\mathrm{c}$ to the associated solution of invicid Burgers' equation, where $t_\mathrm{c}$ is the time of gradient catastrophe.

math.AP↗

arXiv2021

On Extended Supersymmetry of 4d Galileons and 3-Brane Effective Actions

We use on-shell amplitude methods to systematically analyze the possibility of extended supersymmetry for 4d Galileon models, expanding on previous $\mathcal{N}=1$ results. Assuming spins $\le 1$, we prove that there exists no $\mathcal{N}=4$ supersymmetric extension of any 4d Galileons. Thus the Galileons cannot be part of the effective action of a single flat maximally supersymmetric D3-brane, and that explains why such terms do not appear in the $α'$-expansion of the abelian open superstring amplitude. For $\mathcal{N}=2$ Galileons, we show that the complex scalar $Z = ϕ+ i χ$ of the vector supermultiplet cannot have $ϕ$ and $χ$ both enjoy enhanced shift symmetry; instead, $χ$ can at best be an $R$-axion with constant shift symmetry. Using the soft bootstrap, we demonstrate that the quartic DBI-Galileon is incompatible with $\mathcal{N}=2$ supersymmetry. A similar analysis performed at 7-point shows that a 2-parameter family of $\mathcal{N}=2$ supersymmetric quintic Galileons coupled with DBI passes the soft bootstrap. Finally, we show how supersymmetric coup\-lings between Galileons and gravitons arise in generalizations of our constructions, and we conclude with a discussion of Galileons and DBI-Galileons in the context of UV-completability vs.~the Swampland.

hep-th↗