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1. On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

   https://doi.org/10.1090/cams/32  

   Lührmann, Jonas; Schlag, Wilhelm (February 2024, Communications of the American Mathematical Society)

   We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]). The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172]. 

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2. Integrable nonlocal derivative nonlinear Schrödinger equations

   https://doi.org/10.1088/1361-6420/ac5f75  

   Ablowitz, Mark J; Luo, Xu-Dan; Musslimani, Ziad H; Zhu, Yi (April 2022, Inverse Problems)

   Abstract Integrable standard and nonlocal derivative nonlinear Schrödinger equations are investigated. The direct and inverse scattering are constructed for these equations; included are both the Riemann–Hilbert and Gel’fand–Levitan–Marchenko approaches and soliton solutions. As a typical application, it is shown how these derivative NLS equations can be obtained as asymptotic limits from a nonlinear Klein–Gordon equation. 

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3. Global solutions for massive Maxwell-Klein-Gordon equations

   https://doi.org/10.1002/cpa.21864  

   Klainerman, S; Wang, Q; Yang, S (August 2019, Communications on pure and applied mathematics)

   Vardhan, S (Ed.)

   We derive the asymptotic properties of the mMKG system (Maxwell coupled with a massive Klein-Gordon scalar field) in the exterior of the domain of influence of a compact set. This complements the previous well-known results, restricted to compactly supported initial conditions, based on the so-called hyperboloidal method. That method takes advantage of the commutation properties of the Maxwell and Klein-Gordon equations with the generators of the Poincaré group to resolve the difficulties caused by the fact that they have, separately, different asymptotic properties. Though the hyperboloidal method is very robust and applies well to other related systems, it has the well-known drawback of requiring compactly supported data. In this paper we remove this limitation based on a further extension of the vector field method adapted to the exterior region. Our method applies, in particular, to nontrivial charges. The full problem can then be treated by patching together the new estimates in the exterior with the hyperboloidal ones in the interior. This purely physical space approach introduced here maintains the robust properties of the old method and can thus be applied to other situations such as the coupled Einstein Klein-Gordon equation. 

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4. Tropical cyclone rapid intensification and the excitation of inertia-gravity waves on the edges of an evolving potential vorticity structure

   https://doi.org/10.3389/feart.2022.1038351  

   Schubert, Wayne H.; Taft, Richard K. (November 2022, Frontiers in Earth Science)

   The problem of tropical cyclone rapid intensification is reduced to a potential vorticity (PV) equation and a second order, inhomogeneous, partial differential equation for the azimuthal wind. The latter equation has the form of a Klein-Gordon equation, the right-hand side of which involves the radial derivative of the evolving PV field. When the PV field evolves rapidly, inertia-gravity waves are excited at the edges of the evolving PV structure. In contrast, when the PV field evolves slowly, the second order time derivative term in the Klein-Gordon equation is negligible, inertia-gravity waves are not excited, and the equation reduces to an invertibility principle for the PV. The above concepts are presented in the context of an axisymmetric shallow water model, in both its linear and nonlinear forms. The nonlinear results show a remarkable sensitivity of vortex intensification to the percentage of mass that is diabatically removed from the region inside a given absolute angular momentum surface. 

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5. An integer basis for celestial amplitudes

   https://doi.org/10.1007/JHEP08(2023)192  

   Cotler, Jordan; Miller, Noah; Strominger, Andrew (August 2023, Journal of High Energy Physics)

   We present a discrete basis of solutions of the massless Klein-Gordon equation in 3 + 1 Minkowski space which transform as 𝔰𝔩(2, ℂ) Lorentz/conformal primaries and descendants, and whose elements all have integer conformal dimension. We show that the basis is complete in the sense that the Wightman function can be expressed as a quadratic sum over the basis elements. 

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