This is a writeup of lectures delivered at the Asian Pacific Introductory School on Superstring and Related Topics in Beijing (2006) and an expanded version of these lectures given at the Third Summer School on Strings, Fields and Holography in Nanjing (2023). It aims to provide both a historical and pedagogical account of developments in finding 1/2 Bogomol'nyi-Prasad-Sommerfield (BPS) extended string solitons during the early stage of the so-called second string revolution, before which these objects were thought to be unrelated to strings. Non-supersymmetric solutions related to brane/anti brane systems or non-BPS systems are also discussed.

Renormalization group analysis has been proposed to eliminate secular terms in perturbation solutions of differential equations and thus expand the domain of their validity. Here we extend the method to treat periodic orbits or limit cycles. Interesting normal forms could be derived through a generalization of the concept ’resonance’, which offers nontrivial analytic approximations. Compared with traditional techniques such as multi-scale methods, the current scheme proceeds in a very straightforward and simple way, delivering not only the period and the amplitude but also the transient path to limit cycles. The method is demonstrated with several examples including the Duffing oscillator, van der Pol equation and Lorenz equation. The obtained solutions match well with numerical results and with those derived by traditional analytic methods.

Introducing ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potentials into the three-coupled nonlinear Gross–Pitaevskii equations offers a new way to seek stable soliton states in quasi-one-dimensional spin-1 Bose–Einstein condensates. In scenarios where the spin-independent parameter c₀ and the spin-dependent parameter c₂ vary, we use both analytical and numerical methods to investigate the three-coupled nonlinear Gross–Pitaevskii equations with ${ \mathcal P }{ \mathcal T }$-symmetric generalized Scarf-II potentials. We obtain analytical soliton states and find that simply modulating c₂ may change the analytical soliton states from unstable to stable. Additionally, we obtain numerically stable double-hump soliton states propagating in the form of periodic oscillations, exhibiting distinct behavior in energy exchange. For further investigation, we discuss the interaction of numerical double-hump solitons with Gaussian solitons and observe the transfer of energy among the three components. These findings may contribute to a deeper understanding of solitons in Bose–Einstein condensates with ${ \mathcal P }{ \mathcal T }$-symmetric potentials and may hold significance for both theoretical understanding and experimental design in related physics experiments.

We assume exponential corrections to the entropy of 5D charged AdS black hole solutions, which are derived within the framework of Einstein–Gauss–Bonnet gravity and nonlinear electrodynamics. Additionally, we consider two distinct versions of 5D charged AdS black holes by setting the parameters q → 0 and k → 0 (where q represents the charge, and k is the non-linear parameter). We investigate these black holes in the extended phase space, where the cosmological constant is interpreted as pressure, demonstrating the first law of black hole thermodynamics. The focus extends to understanding the thermal stability or instability, as well as identifying first and second-order phase transitions. This exploration is carried out through the analysis of various thermodynamic quantities, including heat capacity at constant pressure, Gibbs free energy (GFE), Helmholtz free energy (HFE), and the trace of the Hessian matrix. In order to visualize phase transitions, identify critical points, analyze stability and provide comprehensive analysis, we have made the contour plot of the mentioned thermodynamic quantities and observed that our results are very consistent. These investigations are conducted within the context of exponentially corrected entropies, providing valuable insights into the intricate thermodynamic behavior of these 5D charged AdS black holes under different parameter limits.

In this work, we consider the collapse of a ${\mathbb{D}}$-dimensional sphere in the framework of a higher-dimensional spherically symmetric space-time in which the gravitational action chosen is claimed to be somehow linked to the ${\mathbb{D}}$-dimensional modified term. This work investigates the criteria for the dynamical instability of anisotropic relativistic sphere systems with ${\mathbb{D}}$-dimensional modified gravity. The certain conditions are applied that lead to the collapse equation and their effects on adiabatic index Γ in both Newtonian (N) and Post-Newtonian (PN) regimes by using a perturbation scheme. The study explores that the Γ plays a crucial role in determining the degree of dynamical instability. This index characterizes the fluid’s stiffness and has a significant impact on defining the ranges of instability. This systematic investigation demonstrates the influence of various material properties such as anisotropic pressure, kinematic quantities, mass function, ${\mathbb{D}}$-dimensional modified gravity parameters, and the radial profile of energy density on the instability of considered structures during their evolution. This work also displays the dynamical behavior of spherically symmetric fluid configuration via graphical approaches.

Understanding the effects of point liquid loading on transversely isotropic poroelastic media is crucial for advancing geomechanics and biomechanics, where precise modeling of fluid-structure interactions is essential. This paper presents a comprehensive analysis of infinite transversely isotropic poroelasticity under a fluid source, based on Biot’s theory, aiming to uncover new and previously unexplored insights in the literature. We begin our study by deriving a general solution for fluid-saturated, transversely isotropic poroelastic materials in terms of harmonic functions that satisfy sixth-order homogeneous partial differential equations, using potential theory and Almansi’s theorem. Based on these general solutions and potential functions, we construct a Green’s function for a point fluid source, introducing three new harmonic functions with undetermined constants. These constants are determined by enforcing continuity and equilibrium conditions. Substituting these into the general solution yields fundamental solutions for poroelasticity that provide crucial support for a wide range of project problems. Numerical results and comparisons with existing literature are provided to illustrate physical mechanisms through contour plots. Our observations reveal that all components tend to zero in the far field and become singular at the concentrated source. Additionally, the contours exhibit rapid changes near the point fluid source but display gradual variations at a distance from it. These findings highlight the intricate behavior of the system under point liquid loading, offering valuable insights for further research and practical applications.

Deep learning combining the physics information is employed to solve the Boussinesq equation with second-order time derivative. High prediction accuracies are achieved by adding a new initial loss term in the physics-informed neural networks along with the adaptive activation function and loss-balanced coefficients. The numerical simulations are carried out with different initial and boundary conditions, in which the relative L₂-norm errors are all around 10⁻⁴. The prediction accuracies have been improved by two orders of magnitude compared to the former results in certain simulations. The dynamic behavior of solitons and their interaction are studied in the colliding and chasing processes for the Boussinesq equation. More training time is needed for the solver of the Boussinesq equation when the width of the two-soliton solutions becomes narrower with other parameters fixed.

Recently, large-scale trapped ion systems have been realized in experiments for quantum simulation and quantum computation. They are the simplest systems for dynamical stability and parametric resonance. In this model, the Mathieu equation plays the most fundamental role for us to understand the stability and instability of a single ion. In this work, we investigate the dynamics of trapped ions with the Coulomb interaction based on the Hamiltonian equation. We show that the many-body interaction will not influence the phase diagram for instability. Then, the dynamics of this model in the large damping limit will also be analytically calculated using few trapped ions. Furthermore, we find that in the presence of modulation, synchronization dynamics can be observed, showing an exchange of velocities between distant ions on the left side and on the right side of the trap. These dynamics resemble that of the exchange of velocities in Newton’s cradle for the collision of balls at the same time. These dynamics are independent of their initial conditions and the number of ions. As a unique feature of the interacting Mathieu equation, we hope this behavior, which leads to a quasi-periodic solution, can be measured in current experimental systems. Finally, we have also discussed the effect of anharmonic trapping potential, showing the desynchronization during the collision process. It is hoped that the dynamics in this many-body Mathieu equation with damping may find applications in quantum simulations. This model may also find interesting applications in dynamics systems as a pure mathematical problem, which may be beyond the results in the Floquet theorem.

In this paper, the nonlinearization of the Lax pair and the Darboux transformation method are used to construct the rogue wave on the elliptic function background in the reduced Maxwell–Bloch system, which is described by four component nonlinear evolution equations (NLEEs). On the background of the Jacobian elliptic function, we obtain the admissible eigenvalues and the corresponding non-periodic eigenfunctions of the model spectrum problem. Then, with the help of the one-fold Darboux transformation and two-fold Darboux transformation, rogue waves on a dn-periodic background and cn-periodic background are derived, respectively. Finally, the corresponding complex dynamical properties and evolutions of the four components are illustrated graphically by choosing suitable parameters.

We simulate the gravitational redshift of quantum matter waves with a long de Broglie wavelength by tracing particle beams along geodesics, when they propagate within the rotation plane of binary black holes. The angular momentum of the binary black hole causes an asymmetric gravitational redshift distribution around the two black holes. The gravitational redshift changes the frequency of quantum matter waves and their wavelength, resulting in the different interference patterns of quantum matter waves with respect to different wavelengths. The interference pattern demonstrates strong contrast intensity and spatial order with respect to different wavelengths and the rotational angle of the binary black hole. A bright semicircular arc emerges from the interference pattern to bridge the two black holes, when they rotate to certain angles, which provides a theoretical understanding on the gravitational lensing effect of quantum matter waves.

This study delves into the role of the neuromuscular junction in communication between nerves and muscles, as well as the importance of sarcomeres in muscle contraction. A mechanical device and circuit model is developed to simulate the movement of sarcomeres and the biophysical properties of skeletal muscles, including membrane potential and channel currents. The model integrates electromagnetic, kinetic, and elastic potential energy, which is verified by Helmholtz’s theorem. By using memristors to simulate the neuromuscular junction, the coupling of neuronal circuits with muscle cell circuits is achieved, and dynamic analysis is conducted. Adjusting Hamiltonian energy parameters can modulate oscillation patterns and beam displacement, optimizing the coupling strength between neurons and muscle cells. The study demonstrates that by manipulating energy ratios, it is possible to control the interactions between muscle cells.

We study fundamental dark-bright solitons and the interaction of vector nonlinear Schrödinger equations in both focusing and defocusing regimes. Classification of possible types of soliton solutions is given. There are two types of solitons in the defocusing case and four types of solitons in the focusing case. The number of possible variations of two-soliton solutions depends on this classification. We demonstrate that only special types of two-soliton solutions in the focusing regime can generate breathers of the scalar nonlinear Schrödinger equation. The cases of solitons with equal and unequal velocities in the superposition are considered. Numerical simulations confirm the validity of our exact solutions.

The aim of this paper is to study an extended modified Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff (mKdV-CBS) equation and present its Lax pair with a spectral parameter. Meanwhile, a Miura transformation is explored, which reveals the relationship between solutions of the extended mKdV-CBS equation and the extended (2+1)-dimensional Korteweg–de Vries (KdV) equation. On the basis of the obtained Lax pair and the existing research results, the Darboux transformation is derived, which plays a crucial role in presenting soliton solutions. In addition, soliton molecules are given by the velocity resonance mechanism.

This study explores the dynamics of charged Hayward black holes, focusing on the effects of electric charge and the length factor on accretion disk characteristics. Our results show that increasing both parameters reduces the size of the event horizon and innermost stable circular orbits (ISCO) radius, with the electric charge exerting a more pronounced influence. Additionally, the length factor and electric charge can effectively replicate the spin of a Kerr black hole. Both parameters also affect the electromagnetic radiation emitted from the accretion disk, increasing the flux, temperature, and radiative efficiency. The peak radiation occurs in the soft x-ray band, with higher values of electric charge and length factor enhancing disk luminosity and shifting the peak to higher frequencies. These findings can offer valuable insights into the accretion processes around black holes and their observable signatures, particularly in x-ray astronomy.

In this work, we investigate the thermodynamic variables of a harmonic oscillator in a conical geometry metric. Moreover, we introduce an external field in the form of a Wu–Yang magnetic monopole (WYMM) and an inverse square potential into the system and analyze the results. Using an analytical approach, we obtain the energy level and study the thermodynamics at finite temperature. Our findings demonstrate that thermodynamic variables, except for the specific heat and entropy, are influenced by the topological parameters, the strength of the WYMM, and the inverse square potential.

In this paper, the physics informed neural network (PINN) deep learning method is applied to solve two-dimensional nonlocal equations, including the partial reverse space y-nonlocal Mel'nikov equation, the partial reverse space-time nonlocal Mel'nikov equation and the nonlocal two-dimensional nonlinear Schrödinger (NLS) equation. By the PINN method, we successfully derive a data-driven two soliton solution, lump solution and rogue wave solution. Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small, which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations. Moreover, the parameter discovery of the partial reverse space-time nonlocal Mel'nikov equation is analysed in terms of its soliton solution for the first time.

We investigate phase-controlled bound states in a one-dimensional photonic waveguide coupled to an artificial giant atom at two distant sites. Specifically, we identify the bound state out of the continuum (BOC) and the bound state in the continuum (BIC) and derive the exact existence condition for the BOC. Furthermore, we analytically determine the BIC’s frequency and photonic distribution profile. Remarkably, our analysis reveals quantum beats in both atomic and photonic dynamics, arising from coherent oscillations between the BIC and BOC. These results establish a novel approach for manipulating waveguide quantum electrodynamics via engineered bound states, with potential applications in quantum information processing.

This work demonstrates that once a large number of pion is condensed in a high-energy hadron collision, the gamma-ray spectrum from π⁰ decay takes on a typical broken power-law shape, which has been documented in many astronomical observations, but we have not yet recognized it. We show that this pion condensation is caused by a large number of soft gluons condensed in protons.

The QCD axion bubbles can form due to an explicit breaking of the Peccei–Quinn symmetry in the early Universe. In this paper, we investigate the modified formation of a QCD axion bubble in the presence of an axionlike particle (ALP), considering its resonant conversion to a QCD axion. We consider a general scenario where the QCD axion mixes with ALP before the QCD phase transition. In this scenario, the energy density of the ALP can be adiabatically transferred to the QCD axion at a temperature T_R, resulting in the suppression of the cosmic background temperature T_B at which the energy density of the QCD axion equals that of the radiation. The QCD axion bubbles form when the QCD axions arise during the QCD phase transition. Finally, we briefly discuss the impact of the formation of QCD axion bubbles on the formation of primordial black holes.

We analyze the steady-state characteristics of a damped harmonic oscillator (system) in the presence of a non-Markovian bath characterized by Lorentzian spectral density. Although Markovian baths presume memoryless dynamics, the introduction of complex temporal connections by a non-Markovian environment radically modifies the dynamics of the system and its steady-state behaviour. We obtain the steady-state Green’s function and correlation functions of the system using the Schwinger–Keldysh formalism. In both rotating and non-rotating wave approximation, we analyzed various emergent properties like effective temperature and distribution function. We also explore the impact of dissipation and non-Markovian bath on the quantum Zeno and anti-Zeno effects. We show that a transition between Zeno to anti-Zeno effect can be tuned by bath spectral width and the strength of dissipation.

We investigate the application of the on-shell unitarity method to compute the anomalous dimensions of effective field theory operators. We compute one-loop anomalous dimensions for the dimension-7 operator mixing in low-energy effective field theory (LEFT). The on-shell method significantly simplifies the construction of scattering amplitudes. By leveraging the correspondence between the anomalous dimensions of operator form factors and the double-cut phase-space integrals, we bypass the need for direct loop integral calculations. The resulting renormalization group equations derived in this work provide crucial insights into the scale dependence of the LEFT dimension-7 Wilson coefficients, which will aid in precision experimental fitting of these coefficients.

The theory of statistical physics relies on ergodicity, whereby in large or interacting systems lacking integrability, trajectories eventually explore nearly all points in the phase space. It has been believed that chaotic dynamics provide a possible pathway to ergodicity. Here, we examine the phase space density distributions and their recurrence in the harmonic oscillator, the linear and nonlinear Mathieu equations, the Lorenz attractor, and the Nosé–Hoover model. We show that in models with periodic or quasiperiodic dynamics, sharp peaks can be found in the phase space density distributions. However, for the chaotic dynamics, their distributions display totally different behaviors. We understand these differences using recurrence plots. Our results show that while chaotic dynamics provide an efficient way for the trajectory to explore a large portion of the phase space, which is necessary for ergodicity, the chaotic dynamics are not sufficient for this goal. For instance, despite the Nosé–Hoover model being chaotic, it is not sufficiently large for ergodicity. Therefore, our results may lead to an important conclusion, which is that ergodicity may be realized from large chaotic systems. These findings in these simple models can be explored in experiments in the future, which may provide some key insights into ergodic dynamics.