This paper introduces a novel analytical technique called the “Variable Coefficient Second Degree Generalized Abel equation Method” (VCSDGAE), designed to tackle the complex Ginzburg–Landau equation. Unlike conventional methods that depend on constant coefficient ordinary differential equations (ODEs) and auxiliary ODEs, our approach employs variable coefficient ODEs within a sub-equation framework. We demonstrate the versatility of this method by successfully applying it to the complex Ginzburg–Landau equation. Through the presentation of analytical solutions, we showcase the method's effectiveness and efficiency, positioning it as a valuable resource for addressing complex nonlinear partial differential equations in fields such as fluid dynamics and wave propagation. The stability of the obtained soliton solutions is established through a linear stability analysis, confirming their robustness against small perturbations. This research not only broadens the spectrum of available analytical techniques but also contributes significantly to the advancement of solutions for various mathematical physics models.

The potential modified Korteweg–de Vries (pmKdV) equation possesses an infinite number of symmetries; however, the physical implications of these symmetries remain unexplored. Recent improvements have successfully elucidated that the physical significance of the infinite symmetries are related to wave number translation and wave center translation invariance. Following the ideas of these results, this paper aims to investigate the physical meanings associated with the infinite symmetries linked to the n-soliton solutions of the pmKdV equation. The findings indicate that the K-symmetries are related to the translational invariance of the wave center, while the τ-symmetry is associated with combined translational symmetries of both the wave center and wave number. Furthermore, these infinitely many symmetries are not entirely independent, only some of the special symmetries are autonomous. These results offer a novel method for deriving n-wave solutions of the pmKdV equation. Using this method, some special exact solutions of the pmKdV equation, such as the soliton, complexiton, breather and double-pole solutions, are presented.

In this paper, a generalized (3+1)-dimension variable-coefficient nonlinear evolution equation is investigated, which serves as a model for describing nonlinear wave behaviors in shallow water, ion-acoustic wave fluid mechanics and plasma physics. The Painlevé integrability is tested by the Weiss, Tabor and Carnevale (WTC) method with the simplified form of Krustal. The bilinear form of the equation is derived through the application of the Hirota bilinear method. Building on the bilinear equation, a broad range of analytical solutions are then obtained, including X-shaped and Y-shaped soliton solutions, lump solution, breather solution, and interaction solutions. In addition, another type of soliton solution, periodic solution, and ratio of trigonometric functions are derived.

Explosive synchronization (ES) describes an abrupt and hysteretic transition from incoherence to collective order and has been widely studied in static networks. Here, we show that temporal variability of network connectivity can fundamentally reshape this transition. We investigate inertial Kuramoto oscillators evolving on stochastically rewired random networks, where links are continuously replaced at controlled rates, allowing us to tune the interplay between inertia and topological dynamics. Our results reveal that temporal rewiring can both induce and suppress ES depending on the network density and the switching timescale. Sparse networks display ES only under very slow or very rapid switching, whereas denser networks exhibit robust explosive transitions across a broad parameter range. Increasing the rewiring probability generally promotes abrupt synchronization, but excessively frequent rewiring weakens hysteresis and reduces bistability. A systematic exploration across different degrees confirms that ES is most prominent when moderate-to-high rewiring probability is combined with rapid switching, whereas small rewiring probability favors continuous transitions. These findings demonstrate that temporal randomness is not merely a perturbation but a key control mechanism for abrupt collective behavior, representing how time-varying connectivity governs the onset, robustness, and disappearance of ES in dynamical networks.

Nonlinear option pricing models are of significant research value as they better reflect the realities of financial markets, yet their numerical solution remains highly challenging. On the one hand, such models typically involve strong nonlinearity, multi-scale features of small parameters, and high sensitivity to initial data, which often make it difficult for traditional numerical methods to maintain stability and accuracy. On the other hand, many deep learning methods rely on boundary data, while in real financial markets boundary conditions are often unavailable, thereby limiting their applicability. Thus, deep learning methods that rely solely on initial data still face significant challenges in efficiently solving nonlinear option pricing models and achieving effective numerical predictions. To address these difficulties, this work employs the respecting causality physics-informed neural network (RCPINN), which depends solely on initial data and respects the inherent spatiotemporal causal structure of system evolution, enabling it to effectively handle the complex characteristics of nonlinear financial models. In the Ivancevic nonlinear option pricing model, the RCPINN successfully predicts the formation and evolution of both low-order and high-order financial rogue waves, revealing the underlying dynamical mechanisms of extreme localized waves. In the nonlinear Black–Scholes transaction cost model, the RCPINN effectively captures the dynamical evolution of European call and put option prices. This work not only validates the effectiveness and applicability of RCPINN in predicting financial rogue waves and European option prices within the framework of nonlinear option pricing models, but also demonstrates its advantages in handling initial data sensitivity, multi-scale features, and strong nonlinearity in financial problems. More importantly, this approach provides a novel technical pathway for both early warning of extreme financial risks and the simulation of European option price evolution, while the findings offer valuable insights for applications in derivative pricing, exchange rate forecasting and financial market risk monitoring and early warning.

The Navier–Stokes (NS) equation with Coriolis force and density-dependent viscosity is an important physical model, which has been widely used to understand and analyze a wide array of phenomena, including behaviors of the Gulf stream, dynamics of hurricanes, operation of chemical reactors and functionality of rotating machines. In this paper, based on the matrix and curve integration techniques, we build a sufficient condition for the existence of Cartesian vector solutions u = b(t) + A(t)x for the N-dimensional NS equation, in which A satisfies appropriate matrix equations. Then, we discuss two special cases of A and thereby explicit analytical solutions are obtained. To shed light on these solutions, we give some illustrative examples. Among them, some examples form the generalization previously obtained by other authors and some examples are quite new. Finally, we analyze the properties of Cartesian vector solutions in a special case.

This paper explores the application of Whitham modulation theory to the third-order focusing Kaup–Newell model, offering a complete classification of solutions for step-like initial value problems. Using the finite-gap integration method, we derive periodic solutions and the corresponding Whitham modulation equations. By analyzing the distribution of Riemann invariants, we identify the fundamental wave structures emerging from the step-like initial value problem. Furthermore, we provide a complete classification of solutions for this problem. Our results demonstrate that Whitham modulation theory serves as an effective analytical framework for studying initial value discontinuities in the third-order focusing KN model, offering new insights into its nonlinear dynamical behavior. Moreover, the direct numerical simulations show remarkable agreement with the results from Whitham modulation theory.

In this work, we propose a new deep learning approach based on physics-informed neural networks (PINNs), termed parallel parity-time-symmetric PINNs (PPTS-PINNs), to address the inverse problem of determining the PT-symmetric potential function in the nonlinear Schrödinger equation (NLSE). By incorporating PT-symmetry constraints and a gradient enhancement strategy, the method effectively improves both the accuracy and stability of solving inverse problems, while preserving the consistency of the physical structure. We conduct systematic numerical experiments on two representative NLSEs under both noise-free and noisy conditions (with noise levels of 1% and 5%) to evaluate the reconstruction performance of the model given fixed observational data. The results demonstrate that PPTS-PINNs can robustly reconstruct the complex-valued potential function across different noise levels, achieving an overall error on the order of 10^-3. Notably, under high-noise conditions, the combination of PT-symmetry constraints and the gradient enhancement strategy significantly enhances the model's robustness by mitigating error propagation. Overall, the proposed method exhibits strong adaptability and generalization capabilities in PT-symmetric modeling and noise-resilient learning, offering a novel perspective for solving more complex physical inverse problems.

In this article, the large-time asymptotic wave dynamics of rogue curves are analytically investigated and numerically confirmed in the Davey-Stewartson (DS) I equation. We show that, when time in bilinear expressions of the rogue curves is large, a certain number of localized lump-shaped waves would arise on the uniform background, exhibiting various wave patterns. We further show that, as time increases, the individual lump-shaped wave asymptotically evolves into a line soliton on the constant background that persist at large time. By performing large-time asymptotic analysis, we reveal that such wave patterns as well as the numbers of lump-shaped waves can be analytically determined by the structure of nonzero roots of the Wronskian-Hermite polynomials. Our asymptotic predictions are compared to true solutions quantitatively and excellent agreement is obtained.

In this study, we propose that many different thermodynamic modeling approaches, including the general equation for the non-equilibrium reversible-irreversible coupling (GENERIC), Onsager's variational principle, the energetic variational approach, and classical irreversible thermodynamics, can all be cast into the gradient-conservative structure (GCS). GCS enjoys many nice mathematical properties, has close connection with the large deviations principle and gradient flows in Wasserstein space, and fulfills laws of thermodynamics. Our results demonstrate that the GCS may serve as a unified theoretical framework to model various non-equilibrium thermodynamic processes.

Under study in this paper is a nonlinear Schrödinger equation with local and nonlocal nonlinearities, which originates from the parity-symmetric reduction of the Manakov system and has applications in some physical systems with the parity symmetry constraint between two fields/components. Via the Riemann-Hilbert method, the theory of inverse scattering transform with the presence of double poles is extended for this equation under nonzero boundary conditions (NZBCs). Also, the double-pole soliton solutions with NZBCs are derived in the reflectionless case. It is shown that the quasi-periodic beating solitons can be obtained when the double pole lies off the circle $\Gamma$ centered at the origin with radius $\sqrt{2}{q}_{0}$ (where q₀ is the modulus of NZBCs) on the spectrum plane. Moreover, using the improved asymptotic analysis method, the asymptotic solitons are found to be located in some logarithmic curves of the xt plane.

We establish a quantum dynamics framework for curved submanifolds embedded in higher-dimensional spaces. Through rigorous dimensional reduction, we derive the first complete Schrödinger and Klein-Gordon equations incorporating non-perturbative geometric interactions-resolving ambiguities in constrained quantization. Crucially, extrinsic curvature of the ambient manifold governs emergent low-dimensional quantum phenomena. Remarkably, this mechanism generates scalar field masses matching Kaluza-Klein spectra while eliminating periodic compactification Requirements. We hypothesize that geometric induction can produce Higgs-mechanism-type potentials. Under this working hypothesis, particle masses arise solely from submanifold embedding geometry and matter-field couplings are encoded in curvature invariants. If this hypothesis holds, it would enable experimental access to higher-dimensional physics at all energy scales through geometric induction. We also discuss the Higgs vacuum near small-mass black holes.

This paper presents a geometric perspective that connects reciprocal transformations with multidimensional integrable deformations. By interpreting conservation laws as closed 1-forms, we formalize reciprocal transformations as induced local diffeomorphisms on the jet bundle. This allows us to characterize higher-dimensional deformations as systematic fiber bundle extensions, where fiber coordinates are generated by potential functions of the conservation laws. This perspective provides an interpretation for the covariant lifting of Lax pairs to higher dimensions and reveals that auto-Bäcklund transformations are composite diffeomorphisms. These results are applied to several classical integrable models.

This study investigates the dimensionless quasi-geostrophic potential vorticity (QG-PV) equation with external sources. Employing the Gardner–Morikawa transformation and weakly nonlinear perturbation expansion, we derive the nonlinear Boussinesq equation with external sources. We demonstrate the existence of explicit zero-order and first-order Wronskian solutions for the model equation when α₄ = 0. Furthermore, using a modified Jacobi elliptic function method, we obtain soliton-like solutions for both α₄ = 0 and α₄ ≠ 0. Analysis of these solutions reveals that the generalized β-plane approximation and shear flow are significant factors in inducing nonlinear Rossby waves, and that external sources play a crucial role in influencing Rossby wave behavior.

The paper considers applications of Rota–Baxter algebras to renormalization in quantum field theory and quantum integrability to obtain new solutions for the Yang–Baxter equations which can be studied by the method of Renormalization Group.

In this paper, we investigate data-driven bright soliton solutions of the nonlocal reverse-time nonlinear Schrödinger (NLS) equation and the parameter identification using the physically informed neural networks (PINNs) algorithm. Accurate simulations and comparative analyses of relative and absolute errors are performed for two-soliton and four-soliton solutions including linear solitary waves and periodic waves. In the training process, the standard PINNs scheme is employed for linear solitary wave solutions, while the prior information is added at local sharp regions for periodic wave solutions due to the complicated collision behaviors. For the parameter identification, we accurately recognize the nonlinear coefficients of the nonlocal NLS equation from known solutions with different noises. These results reinforce the application of deep learning with the PINNs framework to successfully study nonlocal integrable systems.

We investigate the entanglement entropy in quantum states featuring repeated sequential excitations of unit patterns in momentum space. In the scaling limit, each unit pattern contributes independently and universally to the entanglement entropy, leading to a characteristic volume-law scaling. Crucially, this universal contribution remains identical for both free and interacting models, enabling decomposition of the total entanglement into pattern-specific components. Numerical verification in fermionic and bosonic chains confirms this volume-law fragmentation phenomenon. For fermionic systems, we derive analytical expressions where many-body entanglement becomes expressible through few-body entanglement components. Notably, this analytical framework extends to spin-1/2 XXZ chains through appropriate identifications.

This paper proposes an innovative form of group reduction or similarity transformation involving off-diagonal block matrices. The proposed method is applied to the Ablowitz–Kaup–Newell–Segur (AKNS) matrix spectral problem, leading to the generation of reduced matrix AKNS integrable hierarchies. As a result, a variety of reduced multiple-component integrable nonlinear Schrödinger and modified Korteweg–de Vries models are derived from the analysis of the reduced AKNS matrix spectral problem.

In this paper, we investigate the integrable fractional coupled Gerdjikov–Ivanov equation and derive its explicit form by employing the completeness relation of squared eigenfunctions. Based on the Riemann–Hilbert method, we construct the fractional N-soliton solutions. We find that as the power ε of the Riesz fractional derivative increases, the amplitudes of the fractional soliton solutions remain invariant, while their widths decrease and the absolute values of the wave velocity, group velocity, and phase velocity increase. Additionally, we examine the long-time asymptotic behavior of the fractional N-soliton solution. The results show that as t →±∞, the solution can be approximated by the sum of N fractional one-soliton solutions, with each soliton's amplitude and velocity remaining constant, whereas both position and phase shifts are observed.

Urban rail transit is an efficient and environmentally friendly mode of transport, which is an important means of transportation for passengers. From a holistic point of view, this paper constructs an urban rail transit interchange topology (URTIT) network based on the interchange relationships among lines. We investigate a unique influence propagation mechanism to explore the impact of applying new technologies on the passenger travel behavior of urban rail transit. We analyze the influence from three aspects: the influence range, the influence propagation path, and the influence intensity. Based on the Dijkstra algorithm, the influence propagation paths are found according to the shortest transfer time. The improved path−based gravity model is applied to measure the influence intensity. The case study on urban rail transit in Beijing, China is carried out. The influence propagation mechanism of a single line in the Beijing URTIT network is analyzed, considering that Beijing Subway Line S1 is equipped with magnetic levitation technology. We not only quantify the impact of technologies on passenger travel behavior of urban rail transit, but also perform the sensitivity analysis. To avoid randomness, the influence propagation mechanisms of all lines are explored in this paper. The research results correspond to the situation in reality, which can provide certain references for urban rail transit operation and planning.

In order to investigate physically meaning localized nonlinear waves on the periodic background defined by Weierstrass elliptic ℘-function for the (n + 1)-dimensional generalized Kadomtsev–Petviashvili equation by Darboux transformation, the associated linear spectral problem with the Weierstrass function as the external potential is studied by utilizing the Lamé function. The degenerate solutions of the nonlinear waves have also been obtained by approaching the limits of the half-periods ω₁ and ω₂ of ℘(x). At the same time, the evolution and nonlinear dynamics of various nonlinear waves under different parameter regimes are systematically discussed. The findings may open avenues for related experimental investigations and potential applications in various nonlinear science domains, such as nonlinear optics and oceanography.

This paper investigates the physical significance of the infinitely many K- and τ-symmetries associated with the soliton and complex solutions of the sine-Gordon (sG) equation. It is shown that the K-symmetries are linear combinations of wave center translation symmetries, while the τ-symmetries combine both wave center translation and wave number translation symmetries. Only a subset of the K- and τ-symmetries are independent, indicating that these symmetries are not incomplete. A special one-soliton solution of the sG equation is derived by using the generalized symmetries.

The investigation centers on a generalized Toda lattice equation featuring four potentials. Initially, the continuous counterpart is first established using the continuous limit approach. Subsequently, leveraging its 3 × 3 matrix spectrum problem, the discrete generalized (m, 3N − m)-fold Darboux transformation (DT) is formulated for this discrete system. Finally, the derived generalized DT is employed to generate precise rational solutions and hybrid rational-exponential solutions, which are thoroughly analyzed and graphically illustrated. These novel findings may offer insights into the lattice dynamical phenomena described by Toda lattice equation.

Official and civil information, as distinct information sources, significantly influence public behavior and the dynamics of epidemic transmission. In this paper, we propose a three-layer ${U}_{1}{A}_{1}{U}_{1}-{U}_{2}{A}_{2}{U}_{2}-SIS$ coupled model to analyze the co-evolution process of official information dissemination, civil information dissemination and epidemic transmission, considering the interdependencies between the information dissemination channels. The first layer describes the official information dissemination process. The second layer models the civil information dissemination process, considering the effects of perceived risk costs and the role of the correlation between official and civil information. The third layer represents the epidemic transmission process, highlighting the impact of the correlation between official and civil information on epidemic transmission. Then, using the microscopic Markov chain approach, we describe the information-epidemic coupled dynamics and derive the epidemic outbreak threshold. Our research demonstrates that a stronger positive correlation between official and civil information raises the epidemic threshold and suppresses the scale of epidemic transmission. Furthermore, individuals’ adoption of civil information should involve a more thorough assessment of the infection risk based on their personal circumstances, which can contribute to more effective epidemic control. Moreover, enhancing infected individuals’ accurate comprehension of official information can effectively curb the transmission of the epidemic. Our study highlights the importance of both official and civil information dissemination in epidemic management and provides insights for policymakers in developing effective public health and communication strategies.

In this paper, we show a general procedure to nonlinearize bilinear equations by using the Bell polynomials. As applications, we obtain nonlinear forms of some integrable bilinear equations (in the sense of having three-soliton solutions) of the KdV type and mKdV type that were found by Jarmo Hietarinta in the 1980s. Examples of non-integrable bilinear equations of the KdV type are also given.

A polynomial scheme is proposed here to compute exact solutions of nonlinear partial differential equations (NPDEs) based on a series expansions of solutions and a renormalization group (RG) related resummation. The most salient feature of the current approach is that only linear algebraic equations need to be solved to implement the resummation for closed-form exact solution and parameter dependence, which does not require any sophisticated analysis like Cole–Hopf transformation or Painlevé test. New exact solutions of typical NPDEs are computed with this novel method, including one- and two-soliton (solitary wave) solutions, periodic solutions of exponential or elliptic function type. Moreover, exact reduced equations may also be conveniently computed for further analysis.

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.

This paper focuses on the analytical technique based on nonlocal symmetry and consistent tanh expansion method for constructing abundant analytical solutions of a new extended Kadomtsev–Petviashvili–Benjamin–Ono (eKP–BO) equation in (2+1) dimensions. First, commencing with the Painlevé analysis, the integrability of the (2+1)-dimensional eKP–BO equation and its nonlocal symmetry are discussed. Second, the localization of the nonlocal symmetry of the extended system is determined by means of the prolongation method. Furthermore, through this localization process, the initial value problem of the extended system is solved, thereby providing a finite symmetry transformation of the (2+1)-dimensional eKP–BO equation. Finally, we follow the consistent tanh expansion method to unveil the interaction solutions of the soliton-cnoidal type and resonant soliton type to the eKP–BO equation, and we study their dynamic properties in a visual manner.

We propose schemes for realizing various forms of bright solitons, bright vortices, and breathing solitons in a non-resonant, incoherently pumped exciton-polariton condensate system by introducing a two-dimensional Moiré lattice external potential. The symmetric shape of the soliton, at the center of the potential field is determined by the rotation angle of the two-dimensional Moiré lattice external potential. Within a specific range of rotation angles, the stability of the soliton is governed by the depth of the second sub-lattice. These two parameters mutually influence and constrain the soliton’s characteristics, and under certain rotation angles and sub-lattice depths, a bright vortex can be formed. At low pumping levels and with carefully chosen peak-to-valley positions in the external potential, the rotation angle becomes the primary factor controlling the distinct forms of breathing bright solitons. Our proposal provides effective schemes for the formation and control of various types of bright solitons and bright vortices in systems employing Moiré lattice external potentials. This scheme for realizing polariton Bose–Einstein condensates (BECs) within a Moiré lattice external potential also holds promise for advancing research in fields such as superfluidity and superconductivity.

In this article, a Generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation is studied, serving as an extended shallow water wave model in higher dimensions. Firstly, utilizing the Bell polynomial method, the bilinear form of the equation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws are derived, confirming the equation’s complete integrability in the context of the Lax pair. Subsequently, the nonlinear superposition formula of the equation is constructed based on the derived bilinear Bäcklund transformation and an array of infinite superposition soliton solutions of the equation are formulated using this nonlinear superposition formula. Lastly, leveraging the obtained bilinear equation, infinite superposition solutions of various functional types are constructed. Their dynamic characteristics are analyzed through illustrated solution images. It is noteworthy that this paper not only uncovers a multitude of properties through the Bell polynomial method but also derives both infinite linear and nonlinear superposition solutions, enriching the diversity of solutions, these aspects have not been previously explored in existing literature.

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.

A direct renormalization method without spectrum theory is proposed to compute the perturbation of solitons in nearly integrable systems with multiple small parameters. The evolution equations of these parameters in unperturbed solitons are obtained as the renormalization equations. Compared with routine methods, the advantages of the renormalization method are that the formulation is only based on a clear and simple mathematical theory, namely the Taylor expansion at a general point, the secular terms in perturbation series are eliminated automatically, any priori physical assumption on the form of the solution is avoided, multiple time scales arise naturally from the final naive perturbation expansion, and the Green’s function and corresponding spectrum of linear differential operators are not needed. As applications, the perturbation of solitons for KDV, MKdV and nonlinear Schrodinger equations, are obtained.

Neuronal membranes exhibit capacitive properties. By adjusting parameters and applying external stimulus currents, nonlinear circuits can achieve various oscillation modes. In this study, we developed a double-membrane neuron model that is sensitive to both light and magnetic fields. This was achieved by incorporating a phototube and a Josephson junction into a Chua’s circuit. Analysis of the neuron model revealed the following findings: (1). Fluctuations in light and magnetic fields induce changes in the neuronal membrane potential. (2). The proportion of stored energy in the model devices varies under different discharge modes. (3). By integrating the principles of ‘lower energy equals higher stability’and ‘exponential gain,’we proposed a method to adjust the structural parameters of neurons, achieving targeted regulation of two key parameters. This research provides valuable insights into the energy regulation mechanisms of neuronal current channels and membrane structures.

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.

In the paper we derive new solutions for the discrete and continuous Schwarzian Korteweg-de Vries (SKdV) equations. These solutions are characterized by trigonometric functions as backgrounds. For the discrete SKdV equation, its solutions are derived by using trigonometric function seeds and Bäcklund transformation. Solutions for the continuous SKdV equation are obtained by taking continuum limits.

The nonisospectral effect λ_t=α(t)λ satisfied by spectral parameter λ opens up a new scheme for constructing localized waves to some nonlinear partial differential equations. In this paper, we perform this effect on a complex nonisospectral nonpotential sine-Gordon equation by the bilinearization reduction method. From an integrable nonisospectral Ablowitz-Kaup-Newell-Segur equation, we construct some exact solutions in double Wronskian form to the reduced complex nonisospectral nonpotential sine-Gordon equation. These solutions, including soliton solutions, Jordan-block solutions and interaction solutions, exhibit localized structure, whose dynamics are analyzed with graphical illustration. The research ideas and methods in this paper can be generalized to other negative order nonisospectral integrable systems.

In this paper, we investigate the (2+1)-dimensional three-component long-wave-short-wave resonance interaction system, which describes complex systems and nonlinear wave phenomena in physics. By employing the Hirota bilinear method, we derive the general nondegenerate N-soliton solution of the system, where each short-wave component contains N arbitrary functions of the independent variable y. The presence of these arbitrary functions in the analytical solution enables the construction of a wide range of nondegenerate soliton types. Finally, we illustrate the structural features of several novel nondegenerate solitons, including M-shaped, multiple double-hump, and sawtooth double-striped solitons, as well as interactions between nondegenerate solitons, such as dromion-like solitons and solitoffs, with the aid of figures.

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.

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.

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.