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.

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.

Active matter exhibits collective motions at various scales. Geometric confinement has been identified as an effective way to control and manipulate active fluids, with much attention given to external factors. However, the impact of the inherent properties of active particles on collective motion under confined conditions remains elusive. Here, we use a highly tunable active nematics model to study active systems under confinement, focusing on the effect of the self-driven speed of active particles. We identify three distinct states characterized by unique particle and flow fields within confined active nematic systems, among which circular rotation emerges as a collective motion involving rotational movement in both particle and flow fields. The theoretical phase diagram shows that increasing the self-driven speed of active particles significantly enhances the region of the circular rotation state and improves its stability. Our results provide insights into the formation of high quality vortices in confined active nematic systems.

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.

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.

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.

The main focus of this paper is to address a generalized (2+1)-dimensional Hirota bilinear equation utilizing the bilinear neural network method. The paper presents the periodic solutions through a single-layer model of [3-4-1], followed by breather, lump and their interaction solutions by using double-layer models of [3-3-2-1] and [3-3-3-1], respectively. A significant innovation introduced in this work is the computation of periodic cross-rational solutions through the design of a novel [3-(2+2)-4-1] model, where a specific hidden layer is partitioned into two segments for subsequent operations. Three-dimensional and density figures of the solutions are given alongside an analysis of the dynamics of these solutions.

We connect magic (non-stabilizer) states, symmetric informationally complete positive operator valued measures (SIC-POVMs), and mutually unbiased bases (MUBs) in the context of group frames, and study their interplay. Magic states are quantum resources in the stabilizer formalism of quantum computation. SIC-POVMs and MUBs are fundamental structures in quantum information theory with many applications in quantum foundations, quantum state tomography, and quantum cryptography, etc. In this work, we study group frames constructed from some prominent magic states, and further investigate their applications. Our method exploits the orbit of discrete Heisenberg–Weyl group acting on an initial fiducial state. We quantify the distance of the group frames from SIC-POVMs and MUBs, respectively. As a simple corollary, we reproduce a complete family of MUBs of any prime dimensional system by introducing the concept of MUB fiducial states, analogous to the well-known SIC-POVM fiducial states. We present an intuitive and direct construction of MUB fiducial states via quantum T-gates, and demonstrate that for the qubit system, there are twelve MUB fiducial states, which coincide with the H-type magic states. We compare MUB fiducial states and SIC-POVM fiducial states from the perspective of magic resource for stabilizer quantum computation. We further pose the challenging issue of identifying all MUB fiducial states in general dimensions.

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.

Since the concept of quantum information masking was proposed by Modi et al (2018 Phys. Rev. Lett. 120, 230 501), many interesting and significant results have been reported, both theoretically and experimentally. However, designing a quantum information masker is not an easy task, especially for larger systems. In this paper, we propose a variational quantum algorithm to resolve this problem. Specifically, our algorithm is a hybrid quantum–classical model, where the quantum device with adjustable parameters tries to mask quantum information and the classical device evaluates the performance of the quantum device and optimizes its parameters. After optimization, the quantum device behaves as an optimal masker. The loss value during optimization can be used to characterize the performance of the masker. In particular, if the loss value converges to zero, we obtain a perfect masker that completely masks the quantum information generated by the quantum information source, otherwise, the perfect masker does not exist and the subsystems always contain the original information. Nevertheless, these resulting maskers are still optimal. Quantum parallelism is utilized to reduce quantum state preparations and measurements. Our study paves the way for wide application of quantum information masking, and some of the techniques used in this study may have potential applications in quantum information processing.

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.

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.

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.

In the realm of nonlinear integrable systems, the presence of decompositions facilitates the establishment of linear superposition solutions and the derivation of novel coupled systems exhibiting nonlinear integrability. By focusing on single-component decompositions within the potential BKP hierarchy, it has been observed that specific linear superpositions of decomposition solutions remain consistent with the underlying equations. Moreover, through the implementation of multi-component decompositions within the potential BKP hierarchy, successful endeavors have been undertaken to formulate linear superposition solutions and novel coupled KdV-type systems that resist decoupling via alterations in dependent variables.

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.

In this paper, the Drinfeld–Sokolov–Satsuma–Hirota (DSSH) system is studied by using residual symmetry and the consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the DSSH system is localized to Lie point symmetry in a properly prolonged system, based on which we get a new Bäcklund transformation for this system. New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system. Moreover, the DSSH system proves to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.

To solve the Poisson equation it is usually possible to discretize it into solving the corresponding linear system Ax = b. Variational quantum algorithms (VQAs) for the discretized Poisson equation have been studied before. We present a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix A. In detail, we decompose the matrices A and A² into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements. For the one-dimensional Poisson equation with different boundary conditions and the d-dimensional Poisson equation with Dirichlet boundary conditions, the number of decomposition terms is less than that reported in [Phys. Rev. A 2023 108, 032 418 ]. Based on the decomposition of the matrix, we design quantum circuits that efficiently evaluate the cost function. Additionally, numerical simulation verifies the feasibility of the proposed algorithm. Finally, the VQAs for linear systems of equations and matrix-vector multiplications with the K-banded Toeplitz matrix T_n^K are given, where ${T}_{n}^{K}\in {R}^{n\times n}$ and $K\in O(\mathrm{ploylog}n)$ .

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.

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.