This paper investigates the characteristics of a class of n + d-dimensional volume-preserving maps and flows derived from perturbations of integrable action-angle maps and flows. Although action-angle volume-preserving maps have a counterpart to the Kolmogorov–Arnold–Moser theorem, we demonstrate here the nonexistence of invariant manifolds for these maps and flows. Additionally, we explore the persistence and stability of periodic orbits in action-angle maps, including the persistence and stability of fixed points.

In this note, we extend our exploration of the essential S-spectrum and introduce the class of Jeribi essential pseudo S-spectrum for bounded right linear operators over the right quaternionic Hilbert space. We explore several properties and the stability of the essential S-spectrum and essential pseudo S-spectrum of this class. Furthermore, we study relationships between the Jeribi essential S-spectrum and other essential S-spectra while investigating the correlation of the essential S-spectrum of this class for 2 × 2 block operator matrices with their components and formulating a characterization of the Jeribi essential S-spectrum for the summation of two bounded linear operators. The study concludes by introducing the notion of essential pseudo S-spectrum of this class and exploration of some of the algebraic properties.

The main objective of this paper is to study the phase transition of the liquid 4He system with non-homogenous density, which is an extension of the work in [Yang and Zhang, Discrete Cont. Dyn.-B, 24, 5107–5120 (2019)]. Based on the new Gibbs free energy and the unified mathematical model for the statistical physics proposed by Ma and Wang, we derive the general dynamic model of the liquid 4He system with non-homogenous density. By using the dynamical phase transition theory, the superfluidity transition occurs for the liquid 4He system, when parameters go through the critical condition depending on the scale of occupying domain.

This paper deals with rare events in a general interacting gas at high temperature, by means of Large Deviations Principles (LDP). The main result is an LDP for the tagged empirical field, which features the competition of an energy term and an entropy term. The approach to proving this LDP is to first deduce one for the tagged empirical field of non-interacting particles at high temperature, and upgrade that result to interacting particle systems.

We consider quantum lattice Hamiltonians and derive recursive spectral relations bridging successive particle number sectors. One relation gives conditions under which the charge gap dominates the neutral gap. We verify these conditions under a triad of symmetries (translation-invariance, charge and dipole conservation) that are present, e.g., in periodic fractional quantum Hall systems. Thus, this gap domination, previously observed numerically, is a universal feature imposed by symmetry. A second relation yields a new induction-on-particle-number method for deriving spectral gaps. The results cover both bosons and fermions.

We classify Toda-type tt*-structures in terms of the anti-symmetry condition. A Toda-type tt*-structure is a flat bundle whose flatness condition is the tt*-Toda equation (Guest–Its–Lin). We show that the Toda-type tt*-structure can be described as a fixed point of e−12πn+1-multiplication and this “intrinsic” description reduces the possibilities of the anti-symmetry condition to only two cases. We give an application to the relation between tt*-Toda equations and representation theory.

In this work, we develop the generalized Fourier transform and construct action-angle variables for the Lakshmanan-Porsezian-Daniel (LPD) equation. This framework mathematically characterizes ultrashort femtosecond soliton propagation in dispersive nonlinear optical fibers. By establishing completeness relations for squared eigenfunctions in the spectral problem, we prove that solutions of the Lax system span the Schwartz function space, thereby rigorously formulating the generalized Fourier transform. Furthermore, explicit Poisson brackets construction for scattering data generates canonical action-angle variables governed by spectral parameters. Crucially, the Hamiltonian formulation reveals a duality: conservation laws admit equivalent representations through both spectral parameters and potential functions, establishing a direct correspondence between scattering data and nonlinear potentials. This framework not only deepens the analytical understanding of the LPD equation’s integrability but also provides theoretical foundations for controlling optical solitons in defocusing media with higher-order nonlinear effects.

A brief overview of fluctuation-induced forces in statistical systems with film geometry at the critical point and the calculation of Casimir amplitudes, which characterize these forces quantitatively, is presented. Particular attention is paid to the special features of strongly anisotropic m-axis systems at the Lifshitz point, specifically, in the case of a “perpendicular” orientation of surfaces with free boundary conditions. Beyond the simplest one-loop approximation, calculations of Casimir amplitudes are impossible without knowledge of the Gaussian propagator, which corresponds to the lines of Feynman diagrams in the perturbation theory. We present an explicit expression for such a propagator in the case of an anisotropic system confined by parallel surfaces perpendicular to one of the anisotropy axes. Using this propagator, we reproduce the one-loop result derived earlier in an essentially different way. The knowledge of the propagator provides the possibility of higher-order calculations in perturbation theory, especially in the context of the epsilon expansion around the upper critical dimension.

We investigate the time-evolution problem associated with the Klein–Gordon equation, using superoscillations as initial data. Additionally, the Segal–Bargmann transform is used to derive integral representations of the resulting solutions.

In this paper, we study a class of retarded semilinear parabolic equations driven by colored noise in terms of the phase spaces. We first show the existence of C1 inertial manifolds of these equations. And then, we prove the convergence and C1 smoothness convergence of inertial manifolds of these retarded semilinear parabolic equations in the varying phase spaces.