Search Results (82)

Exceptional points are parameter values at which two eigenvalues and their corresponding eigenfunctions coalesce, rendering the wave operator defective. They arise widely in non-Hermitian wave physics and disrupt the modal expansions on which semi-analytic scattering methods rely. For lined acoustic waveguides, an augmented mode-matching ansatz that restores completeness at such a degeneracy—by adjoining the generalised eigenfunction obtained from the derivative of the parametrised duct mode with respect to its transverse spectral parameter—has been established for junctions between duct sections with discrete modal sets. This article extends that ansatz to an open, radiating configuration: a rigid feed duct communicates through an impedance-lined throat, tuned to an order-two exceptional point, with a half-space bounded by a rigid flange. The radiating mouth replaces the discrete modal closure by a continuous spectrum, so the augmented basis must be matched against an improper spectral integral. The half-space field is generated by the aperture velocity, which builds the rigid-flange condition into the representation exactly, and the resulting improper integrals are rendered analytic by branch-aware substitutions whose cutoff is tied to the retained modal content. The formulation is validated on the matching and boundary conditions themselves: pointwise continuity of pressure and of normal velocity at the internal junction, pointwise pressure continuity at the radiating mouth, the vanishing of the normal velocity on the rigid flange, and the recovery of the classical flanged-duct radiation problem in the rigid limit, cross-checked against an independent implementation. The full lined problem, including the defective case, has been further verified against an independent finite-volume solution of the same boundary-value problem, whose grid-converged fractions agree with the mode-matching values to within $8\times {10}^{-5}$. A conserved-power identity is monitored as a necessary but not sufficient check. Numerical experiments confirm the known breakdown of the standard expansion at the exceptional point and the well-conditioned convergence of the augmented one in this radiating setting, and a scan of the complex admittance plane, refined by local optimisation and repeated across throat lengths and frequencies, shows that flange radiation detunes the absorption optimum away from the exceptional point, by an amount that grows with the radiated share of the power budget and vanishes as the throat lengthens. Full article

This comment provides a methodological clarification for the solution space of the quantum mechanical system with a hyperbolic potential, which has been previously investigated in our earlier study by using the Asymptotic Iteration Method (AIM). We clarify in this comment that the wavefunction structure (ansatz) and boundary condition used in the original study inherently restrict the solution space to odd-parity states. Therefore, this comment clarifies that the closed-form analytical energy eigenvalues, obtained as the “complete energy spectrum” in the original study, actually represent only the odd-parity states of the system. This structural feature does not affect the accuracy of the results obtained in our study; it mathematically defines the limits of algebraic validity for the reader. Full article

The time-resolved fluorescence spectrum of a driven two-level system is investigated using a time-dependent variational method. We first establish the transient build-up of the Mollow triplet via the Lindblad master equation under the rotating-wave approximation, providing a complete visualization of the spectral evolution from initial turn-on to steady state. To go beyond the perturbative regime, we employ the multiple Davydov D₂ ansatz (multi-D₂), which uses the ${\sigma }_z$ eigenstates as the basis and naturally accommodates arbitrary system–bath coupling types and spectral densities. The multi-D₂ method converges with M = 4 multiplicities in studied cases, outperforming the multi-D₁ ansatz (M = 8) for the ${\sigma }_x$ coupling benchmark. For pure ${\sigma }_z$ (dephasing) coupling under resonant driving, we find that the time-resolved spectrum reveals a distinct fluorescence peak at the Rabi frequency—a signature of dressed-state transitions induced by the dephasing channel that remains hidden in population dynamics. Under mixed ${\sigma }_x$–${\sigma }_z$ coupling, the spectrum exhibits combined features of both Mollow triplet and ${\sigma }_z$-mediated emission. The effects of sub-Ohmic, Ohmic, and super-Ohmic spectral densities are systematically compared. While the resonant spectral weight $J\left({\omega }_0\right)$ governs the overall dissipation rate, a controlled comparison at fixed $J\left({\omega }_0\right)$ reveals that the super-Ohmic regime exhibits an intrinsic shape-dependent suppression of sideband emission under resonant driving, highlighting an asymmetric role of the spectral density exponent in engineering transient fluorescence. Our work establishes the multi-D₂ ansatz as a versatile tool for simulating time-resolved fluorescence in complex bosonic environments. Full article

A formal development of the HypoEquilibrium (HE) state concept within the Steepest-Entropy-Ascent Quantum Thermodynamics (SEAQT) framework is presented, emphasizing its rigorous mathematical formulation. Using a general decomposition of the Hilbert space, HE states are defined in operator language and the reduced evolution of the associated intensive parameters for the regime where the dissipative dynamics commutes with the Hamiltonian is derived. It is proved that the M-th-order HE family (where M is the number of spectral sectors) constitutes an invariant manifold under the SEAQT equation of motion, ensuring that states initially representing a “mixture of canonicals” maintain this structure throughout their evolution. Furthermore, a formal connection is established between the HE ansatz and the rate-controlled constrained equilibrium (RCCE) method, identifying HE variables as constraint potentials. Finally, the model is extended to Non-Hamiltonian SEAQT (NH-SEAQT) interactions to describe thermodynamically consistent energy and entropy exchanges between subsystems and heat baths. This work provides the formal foundation for reduced-order modeling of far-from-equilibrium relaxation and transport processes, and supports a methodology previously applied across various physical and chemical systems. Full article

We study a special case of the extended Korteweg–de Vries (eKdV) equation, arising in the description of weakly nonlinear long waves with higher-order dispersive effects. The model incorporates both third- and fifth-order dispersion and quadratic nonlinearity and describes steeper and shorter waves than the classical KdV equation. First, we determine the Lie point symmetry algebra of the equation and show that it reduces to space–time translations, which in turn motivates a traveling-wave reduction. The reduced fifth-order ODE is then analyzed by means of a calibrated $(G^{\prime }/G)$-expansion ansatz. Although homogeneous balance suggests a degree $M=4$ for exact solutions, a degree-$M=2$ truncation already yields three coherent families of traveling waves—hyperbolic (solitary), trigonometric (periodic), and rational—distinguished by the discriminant of the auxiliary linear equation. Using the direct multiplier method, we construct four conservation laws, corresponding to mass, momentum, energy, and a higher-order dispersion invariant, with all ${\alpha }^2$ contributions retained. Direct substitution and numerical diagnostics demonstrate that, once the algebraic wave speed is imposed, the $M=2$ profiles satisfy the PDE with residuals of order ${10}^{-3}$ and preserve the conserved quantities to machine precision (below ${10}^{-13}\%$ relative variation) over extended integration domains. These results extend the known solution structure of the special eKdV equation and illustrate the effectiveness of the $(G^{\prime }/G)$ framework for higher-order dispersive models. Full article

The quantum approximate optimization algorithm (QAOA) is an efficient method for solving combinatorial optimization problems in quantum computing. These problems involve finding the best solution from a finite set of possibilities. At its core, the QAOA uses an Ansatz circuit composed of alternating unitary operators, the mixing and problem Hamiltonians, that are controlled by a set of parameters. Its goal is to find the optimal parameters so that the final quantum state of the circuit encodes the problem’s solution. While this parameter optimization is often handled by classical optimizers, including constrained optimization by linear approximations (COBYLA) and Nelder–Mead, these methods frequently present local extrema. Therefore, we developed a layerwise grid search (LGS) method as an alternative. Since a full grid search is too time-consuming, the LGS method significantly reduces the search time while still finding a good solution. To demonstrate its effectiveness, we present experimental results for the max-cut problem, comparing the performance of our LGS method against conventional classical optimizers. Full article

We analyze the spherically symmetric complex diffusion and special type of the complex reaction–diffusion equations. These equations are form invariant to the free Schrödinger equations and to the Schrödinger equations with power-law space-dependent potentials. Our new type of solutions are important because we found a new realm of solutions which lie between the solutions of the classical regular diffusion equation and the usual quantum mechanical solutions of the Schrödinger equation. As the solution method, we applied the the self-similar Ansatz, which reduces the original partial differential equation (PDE) to an ordinary differential equation (ODE) which can be solved analytically. The self-similar Ansatz couples the spatial and temporal variables together instead of the usual separation which has to be used in ordinary quantum mechanics for time-independent Hamiltonian. For the complex diffusion equation—without any additional source term—the solutions are the Kummer’s M and Kummer’s U functions. For some parameter values we found $L^2$ integrability, as in the Cartesian case. We interpret that this property can be a “quantum mechanical heritage” and can be a far relation to ordinary quantum mechanics. Therefore, in this sense, our solutions might have quantum mechanical interest in the future. For the complex reaction–diffusion-type equation we derived the Whittaker M and Whittaker W functions as solutions. These solutions have no $L^2$ integrability at all. All derived solutions have complex quadratic arguments. These kind of analytic solutions are new and cannot be found in the existing scientific literature. Finally, the role of the complex angular momentum was investigated as well. Full article

The Kuramoto–Sivashinsky (KS) equation and its fractional form (FKS) are widely used across scientific fields, including fluid dynamics, plasma physics, and chemical processes, to model nonlinear phenomena such as shock waves. It is worth emphasizing that this contribution is part (II) of a larger, systematic research program aimed at modeling, for the first time, completely nonintegrable, nonplanar, and fractional nonplanar evolutionary wave equations. This work focuses on the nonplanar KS framework and its applications to dust–acoustic shock waves in a complex plasma composed of inertial dust grains and inertialess nonextensive ions. This study analyzes both the nonplanar integer KS and nonplanar FKS equations, accounting for geometric effects. This is because the nonplanar model is most suitable for analyzing various nonlinear phenomena (e.g., shock waves) that arise and propagate in plasma physics, fluids, and other physical and engineering systems. Since the nonplanar KS equation is a fully non-integrable problem, its analysis poses a significant challenge for studying the properties of nonplanar shock waves in plasma physics. Therefore, the primary objective of this study is to analyze the nonplanar KS equation using the Ansatz method, thereby deriving semi-analytical solutions that simulate the propagation mechanism of nonplanar shock waves in various physical systems. Following this, we investigate the effect of the fractional factor on the profiles of nonplanar dust–acoustic shock waves to elucidate their propagation mechanism and assess the impact of the memory factor on their behavior. To achieve the second goal, we face a significant challenge because the model under study does not support exact solutions and is more complex than simpler physical models. Thus, the Tantawy technique is employed to overcome this challenge and to analyze this model for generating highly accurate analytical approximations suitable for modeling nonplanar fractional shock waves in various plasma models and in other physical and engineering systems. Full article

The (2+1)-dimensional Caudrey–Dodd–Gibbon (CDG) equation, which can frequently be used to be describe the propagations of shallow-water waves and plasma physics, is solved by various methods in this paper, thereby revealing its nonlinear dynamical behavior. First, through the linear superposition principle in conjunction with symmetric Hirota bilinear method, we obtain a bilinear form of the CDG equation, which possesses several symmetry properties, and construct the resonant solutions to exponential waves. Then, the one-rogue wave solutions to the CDG equation are constructed via the ansatz method. Finally, we show three-dimensional diagrams and density graphs of the yielded solutions to better show the dynamic characteristics. Full article

In recent years, several quantum algorithms have been proposed for addressing combinatorial optimization problems. Among them, the Quantum Approximate Optimization Algorithm (QAOA) has become a widely used approach. However, reported limitations of QAOA have motivated the development of multiple algorithmic variants, including recursive hybrid methods such as the Recursive Quantum Approximate Optimization Algorithm (RQAOA), as well as the Quantum-Informed Recursive Optimization (QIRO) framework. In this work, we integrate the Quantum Alternating Operator Ansatz within the QIRO framework in order to improve its quantum inference stage. Both the original and the enhanced versions of QIRO are applied to the Minimum Vertex Cover problem, an NP-complete problem of practical relevance. Performance is evaluated on a benchmark of Erdös-Rényi graph instances with varying sizes, densities, and random seeds. The results show that the proposed modification leads to a higher number of successfully solved instances across the considered benchmark, indicating that refinements of the variational layer can improve the effectiveness of the QIRO framework. Full article

The Kuramoto–Sivashinsky (KS) equation and its fractional generalizations (FKSs) arise as canonical models for a wide class of nonlinear dissipative–dispersive systems, including thin-film flows, combustion fronts, drift–wave turbulence in plasmas, and chemically reacting media, where shock-like and strongly localized structures play a central role in the dynamics. Despite their apparent simplicity, KS-type models become analytically intractable once higher-order dissipation, geometric effects, and memory (fractional) operators are incorporated, and standard perturbative or transform-based schemes often lead to cumbersome recursive structures, slow convergence, or severe restrictions on the initial data. In this work, a novel direct approximation procedure, referred to as the Tantawy Technique (TT), is developed and implemented to solve and analyze planar fractional KS-type equations and their Burgers-type reductions in a systematic manner. The central difficulty is to construct, for a given physically motivated initial profile, a rapidly convergent series in fractional time that remains stable for a broad range of the fractional order and transport coefficients, while still retaining a clear link to the underlying shock-wave physics. To overcome this, the TT combines (i) a Tanh-based exact shock solution of the planar integer-order KS equation, obtained first as a reference via the standard Tanh method, with (ii) a carefully designed fractional-time ansatz in powers of $t^{\rho }$, where the spatial coefficients are determined recursively from the governing equation in the Caputo sense. This construction yields closed-form expressions for the first few terms in the approximation hierarchy and allows one to monitor convergence through residual and absolute error measures. Full article

The Profitable Tour Problem is a well-known NP-hard optimization challenge central to tourism planning, aiming to maximize collected profit while minimizing travel costs. While classical heuristics provide approximate solutions, they often struggle with finding globally optimal routes. This paper explores the application of near-term quantum computing to this problem. We propose a framework based on the Variational Quantum Eigensolver to find high-quality solutions for the Profitable Tour Problem. The core of our contribution is a novel methodology for constructing a constraint-aware variational ansatz that directly encodes the problem’s hard constraints. This approach circumvents the need for large penalty terms in the Hamiltonian problem, which are often a source of optimization challenges. We validate our method through numerical simulations on a representative tourism scenario of up to 25 qubits. The results demonstrate the viability of the approach, achieving high solution accuracy consistent with brute-force enumeration for smaller instances. This work serves as a proof-of-concept for applying Variational Quantum Eigensolver to complex tourism optimization problems and provides a basis for future exploration on real quantum hardware. Full article

We formalize an inverse, data-conditioned variant of the Variational Quantum Eigensolver (VQE) for clinical biomarker discovery. Given patient-encoded quantum states, we construct a task-specific Hamiltonian whose coefficients are inferred from clinical associations and interpret its expectation value as a calibrated energy score for prognosis and treatment monitoring. The method integrates coefficient estimation, ansatz specification with basis rotations, commuting-group measurements, and a practical shot budget analysis. Evaluated on public infectious disease datasets under severe class imbalance, the approach yields consistent gains in balanced accuracy and precision–recall over strong classical baselines, with stability across random seeds and feature ablations. This variational energy scoring framework bridges Hamiltonian learning and clinical risk modeling, offering a compact, interpretable, and reproducible route to biomarker prioritization and decision support. Full article

The present study models the deflection of nonlinear waves over a Kirchhoff plate underlying a Pasternak-like elastic foundation. A promising version of the tanh expansion analytical method has been deployed for the construction of regular exact solutions for the model, including the application of certain ansatz functions for validations and yet construction of more solutions. The resulting frequency equation and the modulation instability spectrum have been obtained for the linearized model, including the expressions for the related phase and group velocities. In addition, the study examines the equilibrium status of the resulting dynamical system with the help of the bifurcation analysis. Numerically, nonlinear deflection and dispersion of waves have been simulated through the acquired expressions and equations. Notably, the study notes that increasing both the Pasternak-like nonlinear parameter $\eta$ and time variation (for $x>0$) decreases the nonlinear deflection in the plate, while increasing the stiffness of the Winkler foundation increases deflection in the medium. In addition, the study establishes, concerning the determined frequency equation, that increasing the Winkler foundation stiffness increases the dispersion of nonlinear waves in the medium, while an opposite trend has been noted concerning the imposed Pasternak-like nonlinear foundation. In addition, both phase and group velocities, the gain function for modulation instability, and the resulting dynamical system have been noted to be greatly affected by the variation of the imposed foundational parameters. Lastly, this study has potential applications in various engineering fields while modeling and analysis of mechanical structures supported by additional structures. Full article

In this paper, we address the challenge of solving differential equations using physics-informed neural networks (PINNs), an innovative approach that integrates known physical laws into neural network training. The PINN approach involves three main steps: constructing a neural-network-based solution ansatz, defining a suitable loss function, and minimizing this loss via gradient-based optimization. We review two primary PINN formulations: the standard PINN I and an enhanced PINN II. The latter explicitly incorporates initial, final, or boundary conditions. Focusing on first-order differential equations, PINN II methods typically express the approximate solution as $\tilde{u}(x,\theta )=P\left(x\right)+Q\left(x\right)N(x,\theta )$, where $N(x,\theta )$ is the neural network output with parameters $\theta$, and $P\left(x\right)$ and $Q\left(x\right)$ are polynomial functions. We generalize this formulation by replacing the polynomial $Q\left(x\right)$ with a more flexible function $\varphi \left(x\right)$. We demonstrate that this generalized form yields a uniform approximation of the true solution, based on Cybenko’s universal approximation theorem. We further show that the approximation error diminishes as the loss function converges. Numerical experiments validate our theoretical findings and illustrate the advantages of the proposed choice of $\varphi \left(x\right)$. Finally, we outline how this framework can be extended to higher-order or other classes of differential equations. Full article