Search Results (95)

The nonlinear Schrödinger equation is a classical nonlinear evolution equation with wide applications. This paper explores the asymptotic behavior of solutions to the nonlinear Schrödinger equation with non-zero boundary conditions in the presence of a pair of second-order discrete spectra. We analyze the Riemann–Hilbert problem in the inverse scattering transform by the Deift–Zhou nonlinear steepest descent method. Then we propose a proper deformation to deal with the growing time term and give the conditions for the series in the process of deformation by the Laurent expansion. Finally, we provide the characterization of the interactions between the solitary waves corresponding to second-order discrete spectra and the coherent oscillations produced by the perturbation. Numerical verifications are also performed. Full article

The method of deriving fractional-order differential equations using Riesz fractional-order calculus is a new breakthrough, and it is possible to use the inverse scattering transform (IST) to solve the analytical solutions of the derived equations. However, there are still relatively few examples of using discrete Riesz fractional (DRF) order calculations. This article focuses on using discrete Riesz fractional-order calculations to derive a variable-coefficient fractional integrable semi-discrete nonlinear Schrödinger (vcfISDNLS) equation. On the one hand, we derive the vcfISDNLS equation through dispersion relation (DR) and DRF order calculations. On the other hand, we obtain the explicit expressions of the one-soliton solution, two-soliton solution, and three-soliton solution of this equation without reflection potentials (RPs). By deriving and solving the equation and displaying the obtained soliton solutions, possible evidential support can be provided for control engineering, automotive engineering, image processing, and optical communication systems. Full article

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation. Full article

Synthetic Aperture Radar (SAR) three-dimensional (3D) imaging has considerable potential in disaster monitoring and topographic mapping. Conventional 3D SAR imaging techniques for unmanned aerial vehicle (UAV) formations require rigorously regulated vertical or linear flight trajectories to maintain signal coherence. In practice, however, restricted collaboration precision among UAVs frequently prevents adherence to these trajectories, resulting in blurred scattering characteristics and degraded 3D localization accuracy. To address this, a 3D reconstruction technique based on horizontal-cross configurations is proposed, which establishes a new theoretical framework. This approach reduces stringent flight restrictions by transforming the requirement for vertical baselines into geometric flexibility in the horizontal plane. For dual-UAV subsystems, a geometric inversion algorithm is developed for initial scattering center localization. For multi-UAV systems, a multi-aspect fusion algorithm is proposed; it extends the dual-UAV inversion method and incorporates basis transformation theory to achieve coherent integration of multi-platform radar observations. Numerical simulations demonstrate an 80% reduction in implementation costs compared to tomographic SAR (TomoSAR), along with a 1.7-fold improvement in elevation resolution over conventional beamforming (CBF), confirming the framework’s effectiveness. This work presents a systematic horizontal-cross framework for SAR 3D reconstruction, offering a practical solution for UAV-based imaging in complex environments. Full article

Inverse Synthetic Aperture Radar (ISAR) imaging plays a crucial role in reconnaissance and target monitoring. However, the presence of uncertain factors often leads to indistinct component visualization and significant noise contamination in imaging results, where weak scattering components are frequently submerged by noise. To address these challenges, this paper proposes a Phase-Aware Multi-Scale Transformer network (PA-MSFormer) that simultaneously enhances weak component regions and suppresses noise. Unlike existing methods that struggled with this fundamental trade-off, our approach achieved 70.93 dB PSNR on electromagnetic simulation data, surpassing the previous best method by 0.6 dB, while maintaining only 1.59 million parameters. Specifically, we introduce a phase-aware attention mechanism that separates noise from weak scattering features through complex-domain modulation, a dual-branch fusion network that establishes frequency-domain separability criteria, and a progressive gate fuser that achieves pixel-level alignment between high- and low-frequency features. Extensive experiments on electromagnetic simulation and real-measured datasets demonstrate that PA-MSFormer effectively suppresses noise while significantly enhancing target visualization, establishing a solid foundation for subsequent interpretation tasks. Full article

We present a short review of the methodology and applications of the Simple Equations Method (SEsM) for obtaining exact solutions to nonlinear differential equations. The applications part of the review is focused on the simple equations used, with examples of the use of the differential equations for exponential functions, for the function ${\left[{\displaystyle \frac{1}{p+exp\left(q\xi \right)}}\right]}^r$, for the function $1/{cosh}^n$, and for the function ${tanh}^n$. We list several propositions and theorems that are part of the SEsM methodology. We show how SEsM can lead to multisoliton solutions of integrable equations. Furthermore, we note that each exact solution to a nonlinear differential equation can, in principle, be obtained by the methodology of SEsM. The methodology of SEsM can be based on different simple equations. Numerous methods exist for obtaining exact solutions to nonlinear differential equations, which are based on the construction of a solution using certain known functions. Many of these methods are specific cases of SEsM, where the simple differential equation used in SEsM is the equation whose solution is the corresponding function used in these methodologies. We note that the exact solutions obtained by SEsM can be used as a basis for further research on exact solutions to corresponding differential equations by the application of methods that use the symmetries of the solved equation. Full article

This study employs the improved modified extended tanh method (IMETM) to derive exact analytical solutions of a higher-order nonlinear Schrödinger (HNLS) model, incorporating $\beta$-fractional derivatives in both time and space. Unlike classical methods such as the inverse scattering transform or Hirota’s bilinear technique, which are typically limited to integrable systems and integer-order operators, the IMETM offers enhanced flexibility for handling fractional models and higher-order nonlinearities. It enables the systematic construction of diverse solution types—including Weierstrass elliptic, exponential, Jacobi elliptic, and bright solitons—within a unified algebraic framework. The inclusion of fractional derivatives introduces richer dynamical behavior, capturing nonlocal dispersion and temporal memory effects. Visual simulations illustrate how fractional parameters $\alpha$ (space) and $\beta$ (time) affect wave structures, revealing their impact on solution shape and stability. The proposed framework provides new insights into fractional NLS dynamics with potential applications in optical fiber communications, nonlinear optics, and related physical systems. Full article

The design of emulsions at the nanoscale is a significant application of nanotechnology. For spherical droplets and a given volume of dispersed phase, the nanometre size of droplets inversely increases the total area, $A={\displaystyle \frac{3V}{r}}$, allowing greater contact with organic and inorganic materials during application. In topical applications, not only is cell contact increased, but also permeability in the cell membrane. Nanoemulsions typically achieve kinetic stability rather than thermodynamic stability, so their commercial application requires reasonable resistance to flocculation and coalescence, which can be affected by temperature changes. Therefore, their thermoresponsive characterisation becomes relevant. In this work, we analyse this response in an O/W nanoemulsion of Palmarosa for antibacterial purposes that has already shown stability for one year at controlled room temperature. We now study hysteresis processes and the behaviour of the statistical distribution in droplet size by Dynamic Light Scattering, obtaining remarkable stability under temperature changes up to 50 °C. This includes a maintained chemical composition observed using Fourier Transform Infrared Spectroscopy and the preservation of antibacterial properties analysed through optical density tests on cultures and the Spread-Plate technique for bacteria colony counting. We obtain practically closed hysteresis curves for some tracers of droplet size distributions through controlled thermal cycles between 10 °C and 50 °C, exhibiting a non-linear behaviour in their distribution. In general, the results show notable physical, chemical, and antibacterial stability, suitable for commercial applications. Full article

The increasing demand for sustainable disease management in aquaculture has intensified interest in plant-based therapeutics. This study evaluated the formulation and efficacy of andrographolide-loaded nanostructured lipid carriers (AND-NLCs) in Nile tilapia (Oreochromis niloticus) challenged with Streptococcus agalactiae ENC06. AND-NLCs were prepared by the phase-inversion technique and characterized by dynamic light scattering, transmission electron microscopy (TEM), Fourier-transform infrared spectroscopy (FTIR), and in vitro release profiling. Antibacterial activity was assessed by measuring inhibition zone diameters, minimum inhibitory concentration (MIC), and minimum bactericidal concentration (MBC). Growth performance, feed utilization, hepatosomatic index (HSI), and disease resistance were evaluated over a 60-day feeding trial. The AND-NLCs exhibited an optimal particle size (189.6 nm), high encapsulation efficiency (90.58%), sustained release, and structural stability. Compared to the free AND and control group, AND-NLC supplementation significantly improved growth, feed efficiency, HSI, and positive allometric growth. It also enhanced survival (73.3%) and relative percent survival (RPS = 65.6%) following S. agalactiae ENC06 infection. Antibacterial efficacy and physiological responses showed positive correlations with nanoparticle characteristics. These findings suggest that AND-NLCs enhance bioavailability and therapeutic efficacy, supporting their potential as a functional dietary additive to promote growth and improve disease resistance in tilapia aquaculture. Full article

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in ${\mathbb{R}}^2,$ but it does not in ${\mathbb{R}}^n,$ for $n>2.$ Lord Kelvin overcame this problem by defining a new (at the time) inversion, the so-called Kelvin’s inversion (or transformation). This inversion has many good properties, making it extremely useful in each case where the geometry of the original problem raises issues. But by using Kelvin’s inversion, these issues are transformed into easier ones, due to a simpler geometry. In this review paper, we study Kelvin’s inversion, deploying its basic properties. Moreover, we present some applications, where its use enables scientists to solve difficult problems in scattering, electrostaticity, thermoelasticity, potential theory and bioengineering. 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

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

A new generalized (3+1)-dimensional Kadomtsev–Petviashvil (3dKP) equation is derived from the inverse scattering transform method. This equation can be reduced to the standard KP equation and the well-know (3+1)-dimensional equation. In making use of the Lax pair transformation, a Bäcklund transformation of the generalized (3+1)-dimensional KP equation is constructed and some soliton solutions are produced. Finally, a superposition formula is singled out as well by making use of the Bäcklund transformation. As far as we know, the work presented in this paper has not been studied up to now. Full article

The quality of ISAR (Inverse Synthetic Aperture Radar) images has a significant impact on the detection and recognition of targets. Therefore, ISAR image quality assessment is a fundamental prerequisite and primary link in the utilization of ISAR images. Previous ISAR image quality assessment methods typically extract hand-crafted features or use simple multi-layer networks to extract local features. Hand-crafted features and local features from networks usually lack the global information of ISAR images. Furthermore, most deep neural networks obtain feature representations by abridging the prediction quality score and the ground truth, neglecting to explore the strong correlations between features and quality scores in the stage of feature extraction. This study proposes a Gramin Transformer to explore the similarity and diversity of features extracted from different images, thus obtaining features containing quality-related information. The Gramin matrix of features is computed to obtain the score token through the self-attention layer. It prompts the network to learn more discriminative features, which are closely associated with quality scores. Despite the Transformer architecture’s ability to extract global information, the Channel Attention Block (CAB) can capture complementary information from different channels in an image, aggregating and mining information from these channels to provide a more comprehensive evaluation of ISAR images. ISAR images are formed from target scattering points with a background containing substantial silent noise, and the Inter-Region Attention Block (IRAB) is utilized to extract local scattering point features, which decide the clarity of target. In addition, extensive experiments are conducted on the ISAR image dataset (including space stations, ships, aircraft, etc.). The evaluation results of our method on the dataset are significantly superior to those of traditional feature extraction methods and existing image quality assessment methods. Full article

Balancing efficiency and accuracy is often challenging in the numerical solution of three-dimensional (3D) point source acoustic wave equations for layered media. To overcome this, an efficient solution method in the spatial-wavenumber domain is proposed, utilizing the Non-Uniform Fast Fourier Transform (NUFFT) to achieve arbitrary non-uniform sampling. By performing a two-dimensional (2D) Fourier transform on the 3D acoustic wave equation in the horizontal direction, the 3D equation is transformed into a one-dimensional (1D) space-wavenumber-domain ordinary differential equation, effectively simplifying significant 3D problems into one-dimensional problems and significantly reducing the demand for memory. The one-dimensional finite-element method is applied to solve the boundary value problem, resulting in a pentadiagonal system of equations. The Thomas algorithm then efficiently solves the system, yielding the layered wavefield distribution in the space-wavenumber domain. Finally, the wavefield distribution in the spatial domain is reconstructed through a 2D inverse Fourier transform. The correctness of the algorithm was verified by comparing it with the finite-element method. The analysis of the half-space model shows that this method can accurately calculate the wavefield distribution in the air layer considering the air layer while exhibiting high efficiency and computational stability in ultra-large-scale models. The three-layer medium model test further verified the adaptability and accuracy of the algorithm in calculating the distribution of acoustic waves in layered media. Through a sensitivity analysis, it is shown that the denser the mesh node partitioning, the higher the medium velocity, and the lower the point source frequency, the higher the accuracy of the algorithm. An algorithm efficiency analysis shows that this method has extremely low memory usage and high computational efficiency and can quickly solve large-scale models even on personal computers. Compared with traditional FEM, the algorithm has much higher advantages in terms of memory usage and efficiency. This method provides a new approach to the numerical solution of partial differential equations. It lays an essential foundation for background field calculation in the scattering seismic numerical simulation and full-waveform inversion of acoustic waves, with strong theoretical significance and practical application value. Full article