The paper starts from establishing an elliptic direct linearization (DL) scheme for the Kadomtsev-Petviashvili equation. The scheme consists of an integral equation (involving the Lam\'e function) and a formula for elliptic soliton solutions, which can be confirmed by checking Lax pair. Based on analysis of real-valuedness of the Weierstrass functions, we are able to construct a Marchenko equation for elliptic solitons. A mechanism to obtain nonsingular real solutions from this elliptic DL scheme is formulated. By utilizing elliptic $N$th roots of unity and reductions, the elliptic DL schemes, Marchenko equations and nonsingular real solutions are studied for the Korteweg-de Vries equation and Boussinesq equation. Illustrations of the obtained solutions show solitons and their interactions on a periodic background.

We present a method to construct high-order polynomial approximate invariants (AI) for non-integrable Hamiltonian dynamical systems, and apply it to modern ring-based particle accelerators. Taking advantage of a special property of one-turn transformation maps in the form of a square matrix, AIs can be constructed order-by-order iteratively. Evaluating AI with simulation data, we observe that AI's fluctuation is actually a measure of chaos. Through minimizing the fluctuations with control knobs in accelerators, the stable region of long-term motions could be enlarged.

In this paper we identify full list of Elementary Cellular Automata rules which can be simulated using a quantum circuit (there are 22 such rules). For every such rule we present quantum circuit implementing it with $O(N)$ gates.

A remarkable feature of dark matter consisting of ultralight bosonic particles is the emergence of superfluid Bose-Einstein condensate structures on galactic scales. We investigate the oscillations of the solitonic dark matter structure in the central galactic region by numerically solving the Bogoliubov-de Gennes problem, accounting for perturbations in the gravitational potential and local self-interactions. Our findings reveal that the central solitonic core, formed by the balance of gravitational attraction, quantum pressure, and repulsive interactions, exhibits significant oscillatory behaviour. These oscillations, characterized by distinct eigenmodes, provide insights into the dynamical properties of solitonic dark matter structures and their observational implications and contributions to galactic structure formation and evolution.

We obtain a two-dimensional nonlinear system of equations for the electrostatic potential envelope and the low-frequency magnetic field perturbation to describe the interaction of the upper hybrid wave propagating perpendicular to an external magnetic field with the dust-ion-magnetoacoustic (DIMA) wave in a magnetized dusty plasma. The equations contain both scalar and vector nonlinearities. A nonlinear dispersion relation is derived and the decay and modulation instability thresholds and growth rates are obtained. Numerical estimates show that instability thresholds can easily be exceeded in real dusty plasmas. In the static (subsonic) approximation, a two-dimensional (2D) soliton solution (ground state) is found numerically by the generalized Petviashvili relaxation method. The perturbations of the magnetic field and plasma density in the soliton are nonmonotonic in space and, along with the perturbation in the form of a well, there are also perturbation humps. Such peculiar radial soliton profiles differ significantly from previously known results on 2D solitons. The key point is that the presence of a gap in the DIMA wave dispersion due to the Rao cutoff frequency causes the nonlinearity to be nonlocal. We show that due to nonlocal nonlinearity the Hamiltonian is bounded below at fixed energy, proving the stability of the ground state.

This paper focuses on the analysis of a stochastic SAIRS-type epidemic model that explicitly incorporates the roles of asymptomatic and symptomatic infectious individuals in disease transmission dynamics. Asymptomatic carriers, often undetected due to the lack of symptoms, play a crucial role in the spread of many communicable diseases, including COVID-19. Our model also accounts for vaccination and considers the stochastic effects of environmental and population-level randomness using L\'evy processes. We begin by demonstrating the existence and uniqueness of a global positive solution to the proposed stochastic system, ensuring the model's mathematical validity. Subsequently, we derive sufficient conditions under which the disease either becomes extinct or persists over time, depending on the parameters and initial conditions. The analysis highlights the influence of random perturbations, asymptomatic transmission, and vaccination strategies on disease dynamics. Finally, we conduct comprehensive numerical simulations to validate the theoretical findings and illustrate the behavior of the model under various scenarios of randomness and parameter settings. These results provide valuable insights into the stochastic dynamics of epidemic outbreaks and inform strategies for disease management and control.