We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions \begin{align*} &q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &q(x,0)=q_{0}(x),\ \ \lim_{x\to \pm\infty} q_{0}(x)=q_{\pm}, \end{align*} where $|q_{\pm}|=1$ and $q_{+}=\delta q_{-}$, $\sigma\delta=-1$. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region $-6<\xi<6$ with $\xi=\frac{x}{t}$. In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for other solitonic regions $\xi<-6$ and $\xi>6$. Based on the Riemann-Hilbert problem of the the Cauchy problem, further using the $\bar{\partial}$ steepest descent method, we derive different long time asymptotic expansions of the solution $q(x,t)$ in above two different space-time solitonic regions. In the region $\xi<-6$, phase function $\theta(z)$ has four stationary phase points on the $\mathbb{R}$. Correspondingly, $q(x,t)$ can be characterized with an $\mathcal{N}(\Lambda)$-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function ${\rm Im}\nu(\zeta_i)$. In the region $\xi>6$, phase function $\theta(z)$ has four stationary phase points on $i\mathbb{R}$, the corresponding asymptotic approximations can be characterized with an $\mathcal{N}(\Lambda)$-soliton with diverse residual error order $\mathcal{O}(t^{-1})$.

We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (Commun Math Phys 276:773–798, 2007) that the quantum counterparts of gauge symmetries—the so-called Ward–Takahashi and Slavnov–Taylor identities—correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (Ann Phys 323:49–60, 2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang–Mills theory without reference to a particular renormalization scheme.

An extension of the Kadomtsev–Petviashvili (KP) hierarchy defined via scalar pseudo-differential operators was studied in Szablikowski and Blaszak (J. Math. Phys. 49(8), 082701, 20, 2008) and Wu and Zhou (J. Geom. Phys. 106, 327–341, 2016). In this paper, we represent the extended KP hierarchy into the form of bilinear equation of (adjoint) Baker–Akhiezer functions, and construct its additional symmetries. As a byproduct, we derive the Virasoro symmetries for the constrained KP hierarchies.

Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.

We prove essential self-adjointness of the spatial part of the linear Klein-Gordon operator with external potential for a large class of globally hyperbolic manifolds. The proof is conducted by a fusion of new results concerning globally hyperbolic manifolds, the theory of weighted Hilbert spaces and related functional analytic advances.

We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

In 1983, Klaus studied a class of potentials with bumps and computed the essential spectrum of the associated Schr{\"o}dinger operator with the help of some localisations at infinity. A key hypothesis is that the distance between two consecutive bumps tends to infinity at infinity. In this article, we introduce a new class of graphs (with patterns) that mimics this situation, in the sense that the distance between two patterns tends to infinity at infinity. These patterns tend, in some way, to asymptotic graphs. They are the localisations at infinity. Our result is that the essential spectrum of the Laplacian acting on our graph is given by the union of the spectra of the Laplacian acting on the asymptotic graphs. We also discuss the question of the stability of the essential spectrum in the appendix.

Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing equations for all correlation functions with disconnected boundary, thus completing the Schwinger-Dyson pyramid for quartic melonic (‘pillow’-vertices) models in arbitrary rank. We first study finite group actions that are parametrised by graphs and build the graph calculus on a suitable quotient of the monoid algebra mathcal {A}[G] corresponding to a certain function space mathcal {A} and to the free monoid G in finitely many graph variables; a derivative of an element of mathcal {A}[G] with respect to a graph yields its corresponding group action on mathcal {A}. The present result and the graph calculus have three potential applications: the non-perturbative large-N limit of tensor field theories, the solvability of the theory by using methods that generalise the topological recursion to the TFT setting and the study of ‘higher dimensional maps’ via Tutte-like equations. In fact, we also offer a term-by-term comparison between Tutte equations and the present Schwinger-Dyson equations.

We prove that Weyl quantization preserves constant of motion of the Harmonic Oscillator. We also prove that if $f$ is a classical constant of motion and $\mathfrak{Op}(f)$ is the corresponding operator, then $\mathfrak{Op}(f)$ maps the Schwartz class into itself and it defines an essentially selfadjoint operator on $L^2(\mathbb R^n)$. As a consequence, we provide detailed spectral information of $\mathfrak{Op}(f)$. A complete characterization of the classical constants of motion of the Harmonic Oscillator is given and we also show that they form an algebra with the Moyal product. We give some interesting examples and we analyze Weinstein average method within our framework.

We regard the Cauchy problem for a particular Whitham–Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify the traditional energy norm in use. Hamiltonian conservation provides with global well-posedness at least for small initial data in the one dimensional settings.