Search Results (115)

Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of applied mathematics and mathematical physics. Here, we describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii–Vorob’ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely, a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$. Full article

The primary objective of this paper is to derive a non-relativistic system of equations for a spin-2 particle in the presence of an external Coulomb field, solve these equations, and determine the corresponding energy spectra. We begin with the known radial system of 39 equations formulated for a free spin-2 particle and modify it to incorporate the effects of the Coulomb field. By eliminating the 28 components associated with vector and rank-3 tensor fields, we reduce the system to a set of 11 second-order equations related to scalar and symmetric tensor components. In accordance with parity constraints, this system naturally groups into two subsystems consisting of three and eight equations, respectively. To perform the non-relativistic approximation, we employ the method of projective operators constructed from the matrix ${\Gamma }^0$ of the original matrix equation. This approach allows us to derive two non-relativistic subsystems corresponding to the parity restrictions, comprising two and three coupled differential equations. Through a linear similarity transformation, we further decouple these into five independent equations with a Schrödinger-type non-relativistic structure, leading to explicit energy spectra. Special attention is given to the case of the minimal quantum number of total angular momentum, $j=0$, which requires separate consideration. Full article

We investigate a generalized quantum Schrödinger equation in a comb-like structure that imposes geometric constraints on spatial variables. The model is extended by the introduction of nonlocal and fractional potentials to capture memory effects in both space and time. We consider four distinct scenarios: (i) a time-dependent nonlocal potential, (ii) a spatially nonlocal potential, (iii) a combined space–time nonlocal interaction with memory kernels, and (iv) a fractional spatial derivative, which is related to distributions asymptotically governed by power laws and to a position-dependent effective mass. For each scenario, we propose solutions based on the Green’s function for arbitrary initial conditions and analyze the resulting quantum dynamics. Our results reveal distinct spreading regimes, depending on the type of non-locality and the fractional operator applied to the spatial variable. These findings contribute to the broader generalization of comb models and open new questions for exploring quantum dynamics in backbone-like structures. Full article

In this paper we discuss some results on the existence of solutions for an elliptic system appearing in physical sciences. In particular the system appears when we look at standing wave solutions in the electrostatic situation for the Schrödinger equation coupled, with the minimal coupling rule, with the electromagnetic equations of Born–Infeld theory. Many difficulties appear, especially due to the fact we are in an unbounded domain (the whole space ${\mathbb{R}}^3$) and to the intrinsic nonlinear nature of the equations. We are able to prove the existence of a minimal energy solution by showing an approximating procedure that can be adapted depending on the value of the parameter p, which is in the nonlinearity. Full article

The dynamics of complex systems often exhibit multifractal properties, where interactions across different scales influence their evolution. In this study, we apply the Multifractal Theory of Motion within the framework of scale relativity theory to explore synchronization phenomena in complex systems. We demonstrate that the motion of such systems can be described by multifractal Schrödinger-type equations, offering a new perspective on the interplay between deterministic and stochastic behaviors. Our analysis reveals that synchronization in complex systems emerges from the balance of multifractal acceleration, convection, and dissipation, leading to structured yet highly adaptive behavior across scales. The results highlight the potential of multifractal analysis in predicting and controlling synchronized dynamics in real-world applications. Several applications are also discussed. Full article

The present review discusses the solution of the Heun, confluent, biconfluent, double confluent, and triconfluent equations in terms of polynomial expansions, and applies the results to generate exactly solvable Schrödinger potentials. Although there are more general approaches to solve these differential equations in terms of the expansions of certain special functions, the importance of polynomial solutions is unquestionable, as most of the known potentials are solvable in terms of the hypergeometric and confluent hypergeometric functions; i.e., Natanzon-class potentials possess bound-state solutions in terms of classical orthogonal polynomials, to which the (confluent) hypergeometric functions can be reduced. Since some of the Heun-type equations contain the hypergeometric and/or confluent hypergeometric differential equations as special limits, the potentials generated from them may also contain Natanzon-class potentials as special cases. A power series expansion is assumed around one of the singular points of each differential equation, and recurrence relations are obtained for the expansion coefficients. With the exception of the triconfluent Heun equations, these are three-term recurrence relations, the termination of which is achieved by prescribing certain conditions. In the case of the biconfluent and double confluent Heun equations, the expansion coefficients can be obtained in the standard way, i.e., after finding the roots of an (N + 1)th-order polynomial in one of the parameters, which, in turn, follows from requiring the vanishing of an (N + 1) × (N + 1) determinant. However, in the case of the Heun and confluent Heun equations, the recurrence relation can be solved directly, and the solutions are obtained in terms of rationally extended X₁-type Jacobi and Laguerre polynomials, respectively. Examples for solvable potentials are presented for the Heun, confluent, biconfluent, and double confluent Heun equations, and alternative methods for obtaining the same potentials are also discussed. These are the schemes based on the rational extension of Bochner-type differential equations (for the Heun and confluent Heun equation) and solutions based on quasi-exact solvability (QES) and on continued fractions (for the biconfluent and double confluent equation). Possible further lines of investigations are also outlined concerning physical problems that require the solution of second-order differential equations, i.e., the Schrödinger equation with position-dependent mass and relativistic wave equations. Full article

This review summarizes the application of physics-informed neural networks (PINNs) for solving higher-order nonlinear partial differential equations belonging to the nonlinear Schrödinger equation (NLSE) hierarchy, including models with external potentials. We analyze recent studies in which PINNs have been employed to solve NLSE-type evolution equations up to the fifth order, demonstrating their ability to obtain one- and two-soliton solutions, as well as other solitary waves with high accuracy. To provide benchmark solutions for training PINNs, we employ analytical methods such as the nonisospectral generalization of the AKNS scheme of the inverse scattering transform and the auto-Bäcklund transformation. Finally, we discuss recent advancements in PINN methodology, including improvements in network architecture and optimization techniques. Full article

The objective of this manuscript is to investigate the (2+1)-dimensional Chiral nonlinear Schrödinger equation (CNLSE). We employ the traveling wave transformation to convert the nonlinear partial differential equation (NLPDE) into the nonlinear ordinary differential equation (NLODE). Utilizing the two new vital techniques to derive the solitary wave solutions, the generalized Arnous method and the Riccati equation method, we obtained various types of waves like bright solitons, dark solitons, and periodic wave solutions. Sensitivity analysis is also discussed using different initial conditions. Sensitivity analysis refers to the study of how the solutions of the equations respond to changes in the parameters or initial conditions. It involves assessing the impact of variations in these factors on the behavior and properties of the solutions. To better comprehend the physical consequences of these solutions, we showcase them through different visual depictions like 3D, 2D, and contour plots. The findings of this study are original and hold significant value for the future exploration of the equation, offering valuable directions for researchers to deepen knowledge on the subject. Full article

The method of particular solutions using polynomial basis functions (MPS-PBF) has been extensively used to solve various types of partial differential equations. Traditional methods employing radial basis functions (RBFs)—such as Gaussian, multiquadric, and Matérn functions—often suffer from accuracy issues due to their dependence on a shape parameter, which is very difficult to select optimally. In this study, we adopt the MPS-PBF to solve the time-dependent Schrödinger equation in two dimensions. By utilizing polynomial basis functions, our approach eliminates the need to determine a shape parameter, thereby simplifying the solution process. Spatial discretization is performed using the MPS-PBF, while temporal discretization is handled via the backward Euler and Crank–Nicolson methods. To address the ill conditioning of the resulting system matrix, we incorporate a multi-scale technique. To validate the efficacy of the proposed scheme, we present four numerical examples and compare the results with known analytical solutions, demonstrating the accuracy and robustness of the scheme. Full article

We address the scientific “time” concept in the context of more general relaxation processes toward the Wärmetod of thermodynamic equilibrium. More specifically, we sketch a construction of a conceptual ladder of chemical reaction steps that can rigorously bridge a description from the microscopic domain of molecular quantum chemistry to the macroscopic materials domain of Gibbsian thermodynamics. This conceptual reformulation follows the pioneering work of Kenichi Fukui (Nobel 1981) in rigorously formulating the intrinsic reaction coordinate (IRC) pathway for controlled description of non-equilibrium passages between reactant and product equilibrium states of an overall material transformation. Elementary chemical reaction steps are thereby identified as the logical building-blocks of an integrated mathematical framework that seamlessly spans the gulf between classical (pre-1925) and quantal (post-1925) scientific conceptions and encompasses both static and dynamic aspects of material change. All modern chemical reaction rate studies build on the apparent infallibility of quantum-chemical solutions of Schrödinger’s wave equation and its Dirac-type relativistic corrections. This infallibility may now be properly accepted as an added“inductive law” of Gibbsian chemical thermodynamics, the only component of 19th-century physics that passed intact through the revolutionary quantum upheavals of 1925. Full article

Significant new progress has been made in nonlinear integrable systems with Riesz fractional-order derivative, and it is impressive that such nonlocal fractional-order integrable systems exhibit inverse scattering integrability. The focus of this article is on extending this progress to nonlocal fractional-order Schrödinger-type equations with variable coefficients. Specifically, based on the analysis of anomalous dispersion relation (ADR), a novel variable-coefficient Riesz fractional-order generalized NLS (vcRfgNLS) equation is derived. By utilizing the relevant matrix spectral problems (MSPs), the vcRfgNLS equation is solved through the inverse scattering transform (IST), and analytical solutions including n-soliton solution as a special case are obtained. In addition, an explicit form of the vcRfgNLS equation depending on the completeness of squared eigenfunctions (SEFs) is presented. In particular, the 1-soliton solution and 2-soliton solution are taken as examples to simulate their spatial structures and analyze their structural properties by selecting different variable coefficients and fractional orders. It turns out that both the variable coefficients and fractional order can influence the velocity of soliton propagation, but there is no energy dissipation throughout the entire motion process. Such soliton solutions may not only have important value for studying the super-dispersion transport of nonlinear waves in non-uniform media, but also for realizing a new generation of ultra-high-speed optical communication engineering. Full article

Recently, we introduced random complex and phase screen methods as powerful tools for numerically investigating the evolution of partially coherent pulses (PCPs) in nonlinear dispersive media. However, these methods are restricted to the Schell model type. Non-Schell model light has attracted growing attention in recent years for its distinctive characteristics, such as self-focusing, self-shifting, and non-diffraction properties as well as its critical applications in areas such as particle trapping and information encryption. In this study, we incorporate the Monte Carlo method into the pseudo-mode superposition method to derive the random electric field of any PCPs, including non-Schell model pulses (nSMPs). By solving the nonlinear Schrödinger equations through numerical simulations, we systematically explore the propagation dynamics of nSMPs in nonlinear dispersive media. By leveraging the nonlinearity and optical coherence, this approach allows for effective control over the focal length, peak power, and full width at half the maximum of the pulses. We believe this method offers valuable insights into the behavior of coherence-related phenomena in nonlinear dispersive media, applicable to both temporal and spatial domains. Full article

Assimilating complex systems to multifractal-type objects reveals continuous and non-differentiable curve dynamics, aligning with the Multifractal Theory of Motion. Two scenarios, a Schrödinger-type and a Madelung-type multifractal scenario, are possible in this setting. If the Madelung scenario employs maximized information entropy for a distribution density, then Newtonian and oscillator-type forces can be determined. In the presence of these forces and a matter background, we analyze the two-body problem. The obtained results are as follows: a generalized Hubble-type law, a dependence of Newton’s constant on the epoch and background density, a generalization of Lorentz transform (involving the Hubble constant, Newton’s constant, the speed of light, and cosmic matter density), etc. Moreover, in the same scenario, the functionality of a diffusion-type equation implies instabilities, such as period doubling, through an SL(2R) invariance. Thus, multiple infragalactic and extragalactic instabilities are exemplified. Full article

We give some of our results over the past few years about rogue waves concerning some partial differential equations, such as the focusing nonlinear Schrödinger equation (NLS), the Kadomtsev–Petviashvili equation (KPI), the Lakshmanan–Porsezian–Daniel equation (LPD) and the Hirota equation (H). For the NLS and KP equations, we give different types of representations of the solutions, in terms of Fredholm determinants, Wronskians and degenerate determinants of order $2N$. These solutions are called solutions of order N. In the case of the NLS equation, the solutions, explicitly constructed, appear as deformations of the Peregrine breathers $P_N$ as the last one can be obtained when all parameters are equal to zero. At order N, these solutions are the product of a ratio of two polynomials of degree $N(N+1)$ in x and t by an exponential depending on time t and depending on $2N-2$ real parameters: they are called quasi-rational solutions. For the KPI equation, we explicitly obtain solutions at order N depending on $2N-2$ real parameters. We present different examples of rogue waves for the LPD and Hirota equations. Full article

This paper is devoted to establishing multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials. The main features of the paper are the appearance of the Hardy potential and nonlocal Kirchhoff coefficients, and the absence of the compactness condition of the Palais–Smale type. To demonstrate the multiplicity results, we exploit the fountain theorem and the dual fountain theorem as the main tools, respectively. Full article