Search Results (473)

In this article, we study a double-phase variable-exponent Kirchhoff problem and show the existence of at least three solutions. The proposed model, as a generalization of the Kirchhoff equation, is interesting since it is driven by a double-phase operator that governs anisotropic and heterogeneous diffusion associated with the energy functional, as well as encapsulating two different types of elliptic behavior within the same framework. To tackle the problem, we obtain regularity results for the corresponding energy functional, which makes the problem suitable for the application of a well-known critical point result by Bonanno and Marano. We introduce an n-dimensional vector inequality, not covered in the literature, which provides a key auxiliary tool for establishing essential regularity properties of the energy functional such as $C^1$-smoothness, the $\left(S_{+}\right)$-condition, and sequential weak lower semicontinuity. Full article

This study focuses on the analysis of soliton solutions within the framework of the time-fractional cubic–quintic nonlinear Schrödinger equation (TFCQ-NLSE), a powerful model with broad applications in complex physical phenomena such as fiber optic communications, nonlinear optics, optical signal processing, and laser–tissue interactions in medical science. The nonlinear effects exhibited by the model—such as self-focusing, self-phase modulation, and wave mixing—are influenced by the combined impact of the cubic and quintic nonlinear terms. To explore the dynamics of this model, we apply a robust analytical technique known as the sub-ODE method, which reveals a diverse range of soliton structures and offers deep insight into laser pulse interactions. The investigation yields a rich set of explicit soliton solutions, including hyperbolic, rational, singular, bright, Jacobian elliptic, Weierstrass elliptic, and periodic solutions. These waveforms have significant real-world relevance: bright solitons are employed in fiber optic communications for distortion-free long-distance data transmission, while both bright and dark solitons are used in nonlinear optics to study light behavior in media with intensity-dependent refractive indices. Solitons also contribute to advancements in quantum technologies, precision measurement, and fiber laser systems, where hyperbolic and periodic solitons facilitate stable, high-intensity pulse generation. Additionally, in nonlinear acoustics, solitons describe wave propagation in media where amplitude influences wave speed. Overall, this work highlights the theoretical depth and practical utility of soliton dynamics in fractional nonlinear systems. Full article

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function $\mathtt{p}(·)$. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter $\lambda =0$. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

An angled single-excitation elliptical vibration system for ultrasonic-assisted machining was developed in this paper, which was composed of a giant magnetostrictive transducer and an angled horn. Based on the continuous boundary conditions between the components, the frequency equation of the angled vibration system was derived, and the resonant frequencies of vibration systems with different angles were theoretically calculated. The finite element method was employed to investigate the impact of varying angles on the resonant frequency, elliptical trajectory, phase difference, and output amplitude of the vibration systems. The electrical impedance of the vibration system and the longitudinal and transverse vibration amplitudes at the end face of the horn were tested experimentally. The results show that the resonant frequency and phase difference in the vibration system decreased, the transverse amplitude of the output elliptical trajectory increased, and the longitudinal amplitude decreased with the increase in the included angle. The elliptical trajectories obtained from the test were generally consistent with the calculated results, and the calculated values of the resonant frequencies of the three angled vibration systems were in good agreement with the experimental test values. Full article

The objective of the present study is to detect chirped optical solitons of the perturbed Chen–Lee–Liu equation with full nonlinearity. By virtue of the traveling wave hypothesis, the discussed model is reduced to a simple form known as an elliptic equation. The latter equation, which is a second-order ordinary differential equation, is handled by the undetermined coefficient method of two forms expressed in terms of the hyperbolic secant and tangent functions. Additionally, the auxiliary equation method is applied to derive several miscellaneous solutions. Various types of chirped solitons are revealed such as W-shaped, bright, dark, gray, kink and anti-kink waves. Taking into consideration the existence conditions, the dynamical behaviors of optical solitons and their corresponding chirp are illustrated. The modulation instability of the perturbed CLL equation is examined by means of the linear stability analysis. It is found that all solutions are stable against small perturbations. These entirely new results, compared to previous works, can be employed to understand pulse propagation in optical fiber mediums and dynamic characteristics of waves in plasma. Full article

Gas voids inevitably form during the solidification of phase change materials (PCMs) due to volumetric contraction and thus deteriorate the thermal conductivity of solidified PCMs. In this work, the gas void morphology and distribution in solidified pure paraffin within a cubic thermal energy storage unit are experimentally studied. The three-dimensional structure of the solidified pure paraffin is reconstructed via computed tomography (CT) scanning with a resolution of up to 25 µm. Four distinct morphological types of gas voids are found, including irregular elliptical gas voids, elongated “needle-like” gas voids, micro gas voids, and large circular gas voids. The formation mechanisms of each type are analyzed. The morphology and distribution of gas voids indicate that the solidified pure paraffin structure is anisotropic. The effective thermal conductivity (ETC) of this solid–gas structure is numerically evaluated using lattice Boltzmann simulations, and a two-term power equation is fitted. The results show that the ETC in the vertical direction is significantly lower than in the horizontal direction and the ETC could be reduced by as much as 31.5% due to the presence of gas voids. Full article

In a large deep lake, the generation of internal Kelvin waves and internal Poincaré waves due to wind stress on the lake surface is a significant phenomenon. These internal waves play a crucial role in material transport within the lake and have profound effects on its ecosystem and environment. Our study, which investigated the modes of internal waves in Lake Biwa using the vertical temperature distribution from field observations, has yielded important findings. We have demonstrated the applicability of the frequency equation solutions, considering the Coriolis force. The period of the internal Poincaré waves, as observed in the field, was found to match the solutions of the frequency equation. For example, observational data collected in late October revealed excellent agreement with the theoretical solutions derived from the frequency equation, showing periods of 14.7 h, 11.8 h, 8.2 h, and 6.3 h compared to the theoretical values of 14.4 h, 11.7 h, 8.5 h, and 6.1 h, respectively. However, the periods of the internal Kelvin waves in the field observation results were longer than those of the theoretical solutions. The Modified Mathew function uses a series expansion around $q_i=0$, making it difficult to estimate the periods of internal Kelvin waves under conditions where $\left|q_i\right|>1.0$. Furthermore, in lakes with an elliptical shape, such as Lake Biwa, the elliptical cylinder showed better reproducibility than the circular cylinder. These findings have significant implications for the rapid estimation of internal wave periods using the frequency equation. Full article

This study investigates a stochastic production planning problem with regime-switching parameters, inspired by economic cycles impacting production and inventory costs. The model considers types of goods and employs a Markov chain to capture probabilistic regime transitions, coupled with a multidimensional Brownian motion representing stochastic demand dynamics. The production and inventory cost optimization problem is formulated as a quadratic cost functional, with the solution characterized by a regime-dependent system of elliptic partial differential equations (PDEs). Numerical solutions to the PDE system are computed using a monotone iteration algorithm, enabling quantitative analysis. Sensitivity analysis and model risk evaluation illustrate the effects of regime-dependent volatility, holding costs, and discount factors, revealing the conservative bias of regime-switching models when compared to static alternatives. Practical implications include optimizing production strategies under fluctuating economic conditions and exploring future extensions such as correlated Brownian dynamics, non-quadratic cost functions, and geometric inventory frameworks. In contrast to earlier studies that imposed static or overly simplified regime-switching assumptions, our work presents a fully integrated framework—combining optimal control theory, a regime-dependent system of elliptic PDEs, and comprehensive numerical and sensitivity analyses—to more accurately capture the complex stochastic dynamics of production planning and thereby deliver enhanced, actionable insights for modern manufacturing environments. Full article

In this article, we present the extended simple equations method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The expansions made to the original SEsM algorithm are implemented in several directions: (1) In constructing analytical solutions: exact solutions to FNPDE systems are presented by simple or complex composite functions, including combinations of solutions to two or more different simple equations with distinct independent variables (corresponding to different wave velocities); (2) in selecting appropriate fractional derivatives and appropriate wave transformations: the choice of the type of fractional derivatives for each system of FNPDEs depends on the physical nature of the modeled real process. Based on this choice, the range of applicable wave transformations that are used to reduce FNPDEs to nonlinear ODEs has been expanded. It includes not only various forms of fractional traveling wave transformations but also standard traveling wave transformations. Based on these methodological enhancements, a generalized SEsM algorithm has been developed to derive exact solutions of systems of FNPDEs. This algorithm provides multiple options at each step, enabling the user to select the most appropriate variant depending on the expected wave dynamics in the modeled physical context. Two specific variants of the generalized SEsM algorithm have been applied to obtain exact solutions to two time-fractional shallow-water-like systems. For generating these exact solutions, it is assumed that each system variable in the studied models exhibits multi-wave behavior, which is expressed as a superposition of two waves propagating at different velocities. As a result, numerous novel multi-wave solutions are derived, involving combinations of hyperbolic-like, elliptic-like, and trigonometric-like functions. The obtained analytical solutions can provide valuable qualitative insights into complex wave dynamics in generalized spatio-temporal dynamical systems, with relevance to areas such as ocean current modeling, multiphase fluid dynamics and geophysical fluid modeling. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. 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

This study investigates the impact of biogas slurry ratio, hole diameter and depth under hole irrigation on the soil wetting front migration distance and cumulative infiltration. In this study, a model describing the water transport characteristics of biogas slurry hole irrigation was developed based on the HYDRUS model. Results demonstrated that the HYDRUS model can be used for biogas slurry hole irrigation (NSE > 0.952, PBIAS ≤ ±0.34). Furthermore, the study revealed that the soil cumulative infiltration and soil wetting front migration distance decreased gradually with an increase in the biogas slurry ratio, while they increased gradually with an increase in the hole diameter and depth. The lateral and vertical wetting front migration distances exhibited a well-defined power function relationship with the soil’s stable infiltration rate and infiltration time (R² ≥ 0.977). The soil wetting front migration distance curve can be represented by an elliptic curve equation (R² ≥ 0.957). Additionally, there was a linear relationship between the cumulative infiltration and soil wetted body area (R² ≥ 0.972). Soil wetting front migration distance model ($X=4.442f_0^{0.375}{t}^{0.24}, Z=11.988f_0^{0.287}{t}^{0.124}, f_0=96.947K_s^{1.151}{D}^{0.236}{H}^{1.042}$, NSE > 0.976, PBIAS ≤ ±0.13) and cumulative infiltration model ($I=0.3365S$, NSE > 0.982, PBIAS ≤ ±0.10) established under biogas slurry hole irrigation exhibited good reliability. This study aims to determine optimal hole diameter, depth, and irrigation volume for biogas slurry hole irrigation by establishing a model for soil wetting front migration distance and cumulative infiltration based on crop root growth patterns, thereby providing a scientific basis for its practical application. Full article

This paper aims to present the application of variational methods in studying the existence and related properties of solutions for Klein–Gordon–Maxwell systems through a literature review. First, we give an introduction and variational framework for the Klein–Gordon–Maxwell system. Second, we review the existence and nonexistence of solutions for autonomous Klein–Gordon–Maxwell systems under subcritical growth, critical growth and zero-mass conditions. Third, we introduce studies on the existence and properties of solutions for Klein–Gordon–Maxwell systems classified according to potential functions. Finally, we review the existence of solutions for Klein–Gordon–Maxwell systems in two-dimensional space. Full article

This paper proposes the generalized modified unstable nonlinear Schrödinger’s equation with applications in modulated wavetrain instabilities. The extended direct algebra and generalized Ricatti equation methods are applied to find innovative soliton solutions to the equation. The solutions are obtained in the form of elliptic, hyperbolic, and trigonometric functions. Moreover, a Galilean transformation is used to convert the problem into a dynamical system. We use the theory of planar dynamical systems to derive the equilibrium points of the dynamical system and analyze the Hamiltonian polynomial. We further investigate the bifurcation phase portrait of the system and study its chaotic behaviors when an external force is applied to the system. Graphical 2D and 3D plots are explored to support our mathematical analysis. A sensitivity analysis confirms that the variation in initial conditions has no substantial effect on the stability of the solutions. Furthermore, we give the modulation instability gain spectrum of the considered model and graphically indicate its dynamics using 2D plots. The reported results demonstrate not only the dynamics of the analyzed equation but are also conceptually relevant in establishing the temporal development of modest disturbances in stable or unstable media. These disturbances will be critical for anticipating, planning treatments, and creating novel mechanisms for modulated wavetrain instabilities. Full article

High-order harmonic generation in atomic systems driven by laser fields with tailored symmetries provides a powerful approach for producing structured ultrafast light sources. In this work, we theoretically investigate the ellipticity control of high-order harmonics emitted from helium atoms exposed to orthogonally polarized two-color laser pulses with a 1:3 frequency ratio. The polarization properties of the harmonics are governed by the interplay between the spatial symmetry of the driving field and the atomic potential. By numerically solving the time-dependent Schrödinger equation, we show that fine-tuning the relative phase and amplitude ratio between the fundamental and third-harmonic components enables selective symmetry breaking, resulting in the emission of elliptically and circularly polarized harmonics. Remarkably, we achieve near-perfect circular polarization (ellipticity ≈ 0.995) for the 5th harmonic, as well as highly circularly polarized 17th (0.945), 21st (0.96), and 23rd (0.935) harmonics, demonstrating a level of polarization control and efficiency that exceeds previous schemes. Our results highlight the advantage of using a 1:3 frequency ratio orthogonally polarized two-color laser field over the conventional 1:2 configuration, offering a promising route toward tunable attosecond light sources with tailored polarization characteristics. Full article