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We consider the following higher-order Schrödinger equation involving supercritical growth and competing potentials: * (- Δ) m u + V (y) u = Q (y) u m ∗ - 1 + ε , u > 0 , in R N , u ∈ D m , 2 (R N) , where m ∗ = 2 N N - 2 m , N ≥ 4 m + 1 , m ≥ 2 is an integer, (y ′ , y ′ ′) ∈ R 2 × R N - 2 , V (y) = V (| y ′ | , y ′ ′) and Q (y) = Q (| y ′ | , y ′ ′) ≢ 0 are two bounded nonnegative functions. By using the finite-dimensional reduction argument and local Pohozaev-type identities, under some suitable assumptions on the potentials V and Q, we prove that for any small ε > 0 , the problem (∗) has a large number of bubble solutions whose functional energy is in the order ε - N - 4 m (N - 2 m) 2 . [ABSTRACT FROM AUTHOR]

The current study employs the Nikiforov-Uvarov method to solve the Schrödinger equation for quarkonium systems, utilizing the radial scalar power potential. The eigenvalues of energy and their corresponding wave functions are determined by including the spin–spin, spin–orbit, and tensor interactions in the radial scalar power potential. The mass spectra of charmonia, bottomonia, and bottom-charm in their S, P, D, and F states were determined. Our theoretical states for quarkonium systems align with experimental data across a range of spin levels, as evidenced by our comparison. The total percentage error of our work was computed, yielding a high level of accuracy. The cumulative percentage error for the meson masses of charmonia and bottomonia was determined to be 0.324% and 0.333%, respectively. The masses of the bottom-charm mesons had a total percentage error of 0.012%. Consequently, the present potential yields favorable outcomes for the quarkonium masses, surpassing previous theoretical studies and aligning well with experimental data. [ABSTRACT FROM AUTHOR]

We are concerned with the existence and mass collapse of standing waves with prescribed mass for the Schrödinger equation with general potential and nonlinearities. For local nonlinearities, this equation arises in the theory of Bose–Einstein condensates. For nonlocal nonlinearities, this equation is the Choquard euqation, which appears in the quantum theory of a polaron at rest. We used a unified approach (local minizing method) to study the existence of standing waves for the local, nonlocal and dipolar type cases. And then we established the mass collapse result. [ABSTRACT FROM AUTHOR]

We consider the existence of normalized solutions of the following nonlinear Schrödinger equation - Δ u + λ u = g (u) in Ω , ∫ Ω u 2 d x = ρ 2 , u ∈ H 0 1 (Ω) , where Ω ⊂ R N is a bounded domain with smooth boundary, N ≥ 3 , the nonlinearity g is Sobolev subcritical near infinity and at least mass critical growth near zero. We prove the existence of a solution, which is a local minimizer and obtain a second one of mountain pass type if Ω is a star-shaped domain in R N . Moreover, we uncover a relation between normalized solutions in R N and the corresponding normalized solutions on bounded domain B R (0) by analyzing the behavior of these solutions as R → ∞ . Our results are more general and we propose a different variational approach to deal with nonlinear Schrödinger equations on bounded domains with prescribed L 2 -norm. [ABSTRACT FROM AUTHOR]

We show that the dynamics of a quantum system can be represented by the dynamics of an underlying classical systems obeying the Hamilton equations of motion. This is achieved by transforming the phase space of dimension 2n into a Hilbert space of dimension n which is obtained by a peculiar canonical transformation that changes a pair of real canonical variables into a pair of complex canonical variables which are complex conjugate of each other. The probabilistic character of quantum mechanics is devised by treating the wave function as a stochastic variable. The dynamics of the underlying system is chosen so as to preserve the norm of the state vector. [ABSTRACT FROM AUTHOR]

Using a technique that accounts for disappearing phase-lag might lead to the elimination of phase-lag and all of its derivatives up to order four. The new technique known as the cost-efficient approach aims to improve algebraic order (AOR) and decrease function evaluations (FEvs). The one-of-a-kind approach is shown by Equation PF4DPHFITN142SPS. This method is endlessly periodic since it is P-Stable. The proposed method may be used to solve many different types of periodic and/or oscillatory problems. This innovative method was used to address the difficult issue of Schrödinger-type coupled differential equations in quantum chemistry. The new technique might be seen as a cost-efficient solution since it only requires 5FEvs to execute each step. We are able to greatly ameliorate our current situation with an AOR of 14. [ABSTRACT FROM AUTHOR]

The normalization of the polar functions Θ ℓ , m (θ) for the solution of Schrödinger's equation for the Coulomb potential usually proceeds by appealing to the properties of Associated Legendre polynomials. We show how to achieve the normalization directly from the overall form of the solution and the recursion relation for its series part. When combined with a previous such normalization for the radial part of the solution, the entire hydrogen atom solution can be normalized without having to invoke any properties of special functions. [ABSTRACT FROM AUTHOR]

We prove a general existence theorem for non-self-adjoint Sturm-Liouville problems and we derive quite general asymptotic formulae for their eigenvalues and the corresponding eigenfunctions. The derived formulae are general, very accurate and remain valid for a large class of complex potentials with singularities. These results are obtained by using He's homotopy perturbation method (HPM) with an auxiliary parameter. It will be shown that this method is easy to use and makes the study of these problems more simple and more efficient. In order to illustrate the theory, interesting asymptotic and numerical results are discussed and presented from a wide range of examples. [ABSTRACT FROM AUTHOR]

This study offers a comprehensive analysis of the Perturbed Schrödinger -Hirota Equation (PSHE), crucial for understanding soliton dynamics in modern optical communication systems. We extended the traditional Nonlinear Schrödinger Equation (NLSE) to include higher-order nonlinearities and spatiotemporal dispersion, capturing the complexities of light pulse propagation. Employing the modified auxiliary equation method and Adomian Decomposition Method (ADM), we derived a spectrum of exact traveling wave solutions, encompassing exponential, rational, trigonometric, and hyperbolic functions. These solutions provide insights into soliton behaviors across diverse parameters, essential for optimizing fiber optic systems. The precision of our analytical solutions was validated through numerical solutions, and we explored modulation instability, revealing conditions for soliton formation and evolution. The findings have significant implications for the design and optimization of next-generation optical communication technologies. [ABSTRACT FROM AUTHOR]

The paper presents an accelerated algorithm for solving the direct scattering problem for the continuous spectrum of the Manakov system associated with the vector nonlinear Schrödinger equation of the Manakov model. The numerical formulation requires fast calculation of the products of polynomials depending on the spectral parameter of the problem. For localized solutions, a so-called super-fast algorithm for solving the direct scattering problem of the second order of accuracy is presented, based on the convolution theorem and the fast Fourier transform; for a discrete grid of size N, it requires asymptotically arithmetic operations. To speed up the calculation of the spectra of reflection coefficients, a matrix version of the fast Fourier transform is proposed and tested, in which the coefficients of the discrete Fourier transform are non-commuting matrices. Numerical simulation using the example of the exact solution of the Manakov system (hyperbolic secant) confirmed the high speed of calculations and the second order of accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasilinear Schrödinger equation: - ε 2 Δ u + V (x) u - ε 2 Δ (u 2) u = g (u) , x ∈ R 2 , where ε > 0 is a small parameter, V ∈ C (R 2 , R) is uniformly positive and allowed to be unbounded from above, and g ∈ C (R , R) has a critical exponential growth at infinity. In the autonomous case, when ε > 0 is fixed and V (x) ≡ V 0 ∈ R + , we first present a remarkable relationship between the existence of least energy solutions and the range of V 0 without any monotonicity conditions on g. Based on some new strategies, we establish the existence and concentration of positive solutions for the above singularly perturbed problem. In particular, our approach not only permits to extend the previous results to a wider class of potentials V and source terms g, but also allows a uniform treatment of two kinds of representative nonlinearities that g has extra restrictions at infinity or near the origin, namely lim inf | t | → + ∞ t g (t) e α 0 t 4 or g (u) ≥ C q , V u q - 1 with q > 4 and C q , V > 0 is an implicit value depending on q, V and the best constant of the embedding H 1 (R 2) ⊂ L q (R 2) , considered in the existing literature. To the best of our knowledge, there have not been established any similar results, even for simpler semilinear Schrödinger equations. We believe that our approach could be adopted and modified to treat more general elliptic partial differential equations involving critical exponential growth. [ABSTRACT FROM AUTHOR]

We are concerned with the normalized ℓ -peak solutions to the nonlinear Schrödinger equation - ε 2 Δ v + V (x) v = f (v) + λ v , ∫ R N v 2 = α ε N. Here λ ∈ R will arise as a Lagrange multiplier, V has a local maximum point, and f is a general L 2 -subcritical nonlinearity that satisfies a nonlipschitzian property such that lim s → 0 f (s) / s = - ∞ . The peaks of solutions that we construct cluster around a local maximum of V as ε → 0 . Since there is no information about the uniqueness or nondegeneracy of the limiting system, a sensitive lower gradient estimate should be made when the local centroids of the functions are away from the local maximum of V. We introduce a new method to obtain this estimate, which differs significantly from the ideas of del Pino and Felmer [22] (Math. Ann. 2002), where a special gradient flow with high regularity is used, and in Byeon and Tanaka [7-8] (J. Eur. Math. Soc. 2013 & Mem. Amer. Math. Soc. 2014), where an additional translation flow is introduced. We also give the existence of ground state solutions for the autonomous problem, i.e., the case V ≡ 0 . The ground state energy is not always negative and the strict subadditivity of the ground state energy is achieved here by strict concavity. [ABSTRACT FROM AUTHOR]

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, H ħ , V : = - ħ - 2 L ħ + V on the lattice ħ Z n , where V is a positive multiplication operator and L ħ is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit ħ → 0 . [ABSTRACT FROM AUTHOR]

It is well-known that diatomic molecules are molecules that consist of two atoms bonded together chemically. The use of diatomic molecules is broad and has various applications in different fields of study, such as physical sciences and life sciences. Therefore, in this study, the effects of magnetic and AB-flux fields on thermodynamic properties of hydrogen (H2), lithium hydride (LiH), hydrogen chloride (HCl) and carbon monoxide (CO) diatomic molecules are investigated. The analytical expressions for the partition function are derived using the energy equation by employing the Euler–Maclaurin summation formula. These properties obtained are thoroughly analyzed utilizing graphical representations as a function of temperature. It was observed that the entropy of the CO diatomic molecule exhibits a paramagnetic behavior which agrees with the Linde cycle when the system is subjected to the magnetic and AB-flux fields. Our findings will be valuable in various technological and scientific fields such as magnetic refrigeration, magnetic levitation, magnetic separation, magnetic storage, magnetic resonance imaging and magnetic force microscopy. [ABSTRACT FROM AUTHOR]

The oscillator bases expansion stands as an efficient approximation method for the time-independent Schrödinger equation. The method, originally formulated with one non-linear variational parameter, can be extended to incorporate two such parameters. It handles both non- and semi-relativistic kinematics with generic two-body interactions. In the current work, focusing on systems of three identical bodies, the method is generalised to include the management of a given class of three-body forces. The computational cost of this generalisation proves to not exceed the one for two-body interactions. The accuracy of the generalisation is assessed by comparing with results from Lagrange mesh method and hyperspherical harmonic expansions. Extensions for systems of N identical bodies and for systems of two identical particles and one distinct are also discussed. [ABSTRACT FROM AUTHOR]

We reformulate the time-independent Schrödinger equation as a Maurer–Cartan equation on the superspace of eigensystems of the former equation. We then twist the differential such that its cohomology becomes the space of solutions with a fixed energy. A perturbation of the Hamiltonian corresponds to a deformation of the twisted differential, leading to a simple recursive relation for the eigenvalue and eigenfunction corrections. [ABSTRACT FROM AUTHOR]

Context: This work presents analytical expressions for ro-vibrational energy models of diatomic molecules by introducing fractional parameters to improve molecular interaction analysis. Thermodynamic models, including Helmholtz free energy, mean thermal energy, entropy, and isochoric heat capacity, are formulated for diatomic molecules such as CO (X 1∑+), Cs2 (3 3∑g+), K2 (X 1∑g+), 7Li2 (6 1Πu), 7Li2 (1 3Δg), Na2 (5 1Δg), Na2 (C(2) 1Πu), and NaK (c 3∑+). The incorporation of fractional parameters improves predictive accuracy for vibrational energies, as shown by reductions in percentage average absolute deviations from 0.5511 to 0.2185% for CO. Findings indicate a linear decrease in Helmholtz free energy and an initial increase in heat capacity with rising temperature, providing valuable insights for characterizing materials and optimizing molecular processes in chemistry, material science, and chemical engineering. The results obtained show strong agreement with established theoretical predictions and experimental data, validating the robustness and applicability of the proposed models. Methods: The energy equations are derived by solving the radial Schrödinger equation for a variant of the Tietz potential using the generalized fractional Nikiforov-Uvarov (GFNU) method in addition to a Pekeris-type approximation for the centrifugal term. The canonical partition function is derived using the modified Poisson series formula, which serves as a basis for calculating other thermodynamic functions. All computations are carried out using MATLAB programming software. [ABSTRACT FROM AUTHOR]

In this paper, we study the scattering theory for the cubic inhomogeneous Schrödinger equations with inverse square potential i u t + Δ u - a | x | 2 u = λ | x | - b | u | 2 u with a > - 1 4 and 0 < b < 1 in dimension three. In the defocusing case (i.e. λ = 1 ), we establish the global well-posedness and scattering for any initial data in the energy space H a 1 (R 3) . While for the focusing case(i.e. λ = - 1 ), we obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson and Murphy (Proc Am Math Soc 145:4859–4867, 2017) and Campos and Cardoso (Proc Am Math Soc 150:2007–2021, 2022) that avoids the use of interaction Morawetz estimate. We also address the existence and the non-existence of normalized solutions of the above Schrödinger equation in dimension N for the focusing and defocusing cases. [ABSTRACT FROM AUTHOR]

We are concerned with Liouville-type theorems for the nonlinear Schrödinger equation - Δ u + λ | x | α u = | x | β | u | p - 1 u in R + N , with ∂ u ∂ ν = | x | γ | u | q - 1 u on Σ 1 , where Σ 1 : = { x = (x 1 , ... x N) ∈ R N ; x N = 0 , x 1 > 0 } . Here, N ≥ 2 , p , q > 1 , α , β , γ > - 2 and λ is a positive real parameter. We prove the nonexistence of weak sign-changing solutions which are stable or with finite Morse index, possibly unbounded. The main ideas we use here are integral estimates, a Pohozaev type identity, and a monotonicity formula. [ABSTRACT FROM AUTHOR]

In this note, we show that the Schrödinger equation in quantum mechanics is mathematically equivalent to the Euler-Bernoulli equation for vibrating beams and plates in elasticity theory, with dependent initial data. Remarks are made on potential applications of this equivalence for symplectic and quantum computing, the two-slit experiment using vibrating beams and plates, and the -adic Euler-Bernoulli equation. [ABSTRACT FROM AUTHOR]

Incorporating the inherent heterogeneity of living systems into models of active nematics is essential to provide a more realistic description of biological processes such as bacterial growth, cell dynamics and tissue development. Spontaneous flow of a confined active nematic is a fundamental feature of these systems, in which the role of heterogeneity has not yet been considered. We therefore determine the form of spontaneous flow transition for an active nematic film with heterogeneous activity, identifying a correspondence between the unstable director modes and solutions to Schrödinger's equation. We consider both activity gradients and steps between regions of distinct activity, finding that such variations can change the signature properties of the flow. The threshold activity required for the transition can be raised or lowered, the fluid flux can be reduced or reversed and interfaces in activity induce shear flows. In a biological context fluid flux influences the spread of nutrients while shear flows affect the behaviour of rheotactic microswimmers and can cause the deformation of biofilms. All the effects we identify are found to be strongly dependent on not simply the types of activity present in the film but also on how they are distributed. Incorporating the inherent heterogeneity of living systems into models of active nematics is essential to provide a realistic description of important biological processes. The authors determine the form of spontaneous flow transition for a heterogeneous active nematic film, finding a correspondence with Schrodinger's equation and changes to signature properties of the flow and transition threshold. [ABSTRACT FROM AUTHOR]

We formulate and analyze a fully discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate the existence and uniqueness of the approximation, and prove that the scheme is of optimal accuracy in the L 2 , H 1 and L ∞ norms in space and second-order accurate in time. Numerical results are presented which support the theory. [ABSTRACT FROM AUTHOR]

Properties of molecular electron density functionals are studied. The Kohn potential can be considered as the potential density of the system′s particles whose integration gives the energy of the molecule′s quantum state. Unambiguous expressions for molecular electron energies are obtained in the form of the Kohn potential integrals using a simplified wave density model based on the nuclear wave density isolation. A one-electron Schrödinger equation with an effective potential for calculating molecular orbitals is presented. The equation can be used to calculate 3D electron density of molecules in ground and excited states. [ABSTRACT FROM AUTHOR]

In this paper, we discuss how to reduce the interference that noise introduces into the scalar input signal of a quantum gate. Non-separable quantum gates can be made by making a small potential change to the Hamiltonian and then using perturbation theory to figure out the evolution operator. It is assumed that a scalar, temporally varying signal modulates the potential. To lessen the impact of noise on the design of the gate, we here take into account an extra noise component in the input signal and process it with a linear time-invariant filter. In order to meet these requirements, the Frobenius norm of the difference between the realized gate and the theoretical gate is minimized while taking into account the energy of the signal and the energy of the filter. Results from a computer simulation have been obtained by discretizing the resulting equations. The simulation results show that the proposed method effectively reduces the impact of noise on the gate design and improves its performance. This approach can be useful in designing gates for various applications, including signal processing and communication systems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger equation: - Δ u = σ f (u) + λ u , in R N , ∫ R N | u | 2 d x = a , u ∈ H 1 (R N) , where N ≥ 3 , a > 0 , σ > 0 , and λ ∈ R appears as a Lagrange multiplier. Assume that the nonlinear term f satisfies conditions only in a neighborhood of zero. For f has a subcritical growth, we prove the existence of the positive normalized solution for the equation with sufficiently small σ > 0 . For f has a supercritical growth, we derive the existence of the positive normalized solution for the equation with σ > 0 large enough. In addition, we also obtain infinitely many normalized solutions with sufficiently small σ > 0 for the subcritical case. [ABSTRACT FROM AUTHOR]

In this short note, we give an easy proof of the following result: for n ≥ 2 , lim t → 0 e i t Δ f x + γ (t) = f (x) almost everywhere whenever γ is an α -Hölder curve with 1 2 ≤ α ≤ 1 and f ∈ H s (R n) , with s > n 2 (n + 1) . This is the optimal range of regularity up to the endpoint. [ABSTRACT FROM AUTHOR]

We introduce the spaces A α , β p (Ω) of L p -solutions to the Vekua equation (generalized monogenic functions) D w = α w ¯ + β w in a bounded domain in R n , where D = ∑ i = 1 n e i ∂ i is the Moisil–Teodorescu operator, α and β are bounded functions on Ω . The main result of this work consists of a Hodge decomposition of the L 2 solutions of the Vekua equation. From this orthogonal decomposition arises an operator associated with the Vekua operator, which in turn factorizes certain Schrödinger operators. Moreover, we provide an explicit expression of the ortho-projection over A α , β p (Ω) in terms of the well-known ortho-projection of L 2 monogenic functions and an isomorphism operator. Finally, we prove the existence of component-wise reproductive Vekua kernels and the interrelationship with the Vekua projection in Bergman’s sense. [ABSTRACT FROM AUTHOR]

This paper studies the inhomogeneous bi-harmonic nonlinear Schrödinger equation i u ˙ + Δ 2 u ± | x | − τ | u | p − 1 u = 0 , where u : = u (t , x) : R × R N → C , τ > 0 and p > 1 . One proves the existence of global solutions with datum in Lebesgue space L ϱ (R N) , for a certain 1 < ϱ < 2 . This work complements the known global well-posedness result in the mass-sub-critical regime, namely 1 < p < 1 + 8 − 2 τ N and with finite mass. This work aims to develop a local theory in non-Hilbert Lebesgue spaces. Moreover, one complements the local solution to a global one under some suitable assumptions on the parameters. To this end, one uses a Strichartz-type estimate based on L ϱ . Moreover, one breaks down the data into two parts: the first is in L 2 , and the second has a small Strichrtz norm under the free bi-harmonic Schrödinger propagator. Then, one uses a standard fix point argument coupled with Strichartz estimates. First, one resolves the above problem in L 2 and then takes the considered equation with a perturbed source term. The local solution is complemented by a global one with induction. Since the free associated kernel e i ⋅ Δ 2 is unitary in L 2 , the Schrödinger equations seem more adapted to be mathematically studied in functional spaces based on L 2 , such as the Sobolev spaces W s , 2 (R N) . The main novelty here is investigating the above inhomogeneous bi-harmonic nonlinear Schrödinger problem in some Lebesgue spaces different from L 2 . One essential challenge is overcoming the lack of conservation laws, namely the mass and the energy, representing standard tools in the Schrödinger context. The second technical difficulty is the presence of an inhomogeneous source term, namely the singular decaying term | x | − τ , which causes some serious complications. In order to deal with the inhomogeneous term, one uses Lorentz spaces with the property | x | − τ ∈ L N τ , ∞ . [ABSTRACT FROM AUTHOR]

This work presents a new method called Dimensionless Fluctuation Balance (DFB), which makes it possible to obtain distributions as solutions of Partial Differential Equations (PDEs). In the first case study, DFB was applied to obtain the Boltzmann PDE, whose solution is a distribution for Boltzmann gas. Following, the Planck photon gas in the Radiation Law, Fermi–Dirac, and Bose–Einstein distributions were also verified as solutions to the Boltzmann PDE. The first case study demonstrates the importance of the Boltzmann PDE and the DFB method, both introduced in this paper. In the second case study, DFB is applied to thermal and entropy energies, naturally resulting in a PDE of Boltzmann's entropy law. Finally, in the third case study, quantum effects were considered. So, when applying DFB with Heisenberg uncertainty relations, a Schrödinger case PDE for free particles and its solution were obtained. This allows for the determination of operators linked to Hamiltonian formalism, which is one way to obtain the Schrödinger equation. These results suggest a wide range of applications for this methodology, including Statistical Physics, Schrödinger's Quantum Mechanics, Thin Films, New Materials Modeling, and Theoretical Physics. [ABSTRACT FROM AUTHOR]

A unique behaviour of the Schrödinger map equation, a geometric partial differential equation, is presented by considering its evolution for regular polygonal curves in both Euclidean and hyperbolic spaces. The results are consistent with those for the vortex filament equation, an equivalent form of the Schrödinger map equation in the Euclidean space. Thus, with all possible choices of regular polygons in a given setting, our analysis not only provides a novel extension to its usefulness as a pseudorandom number generator but also complements the existing results. [ABSTRACT FROM AUTHOR]

Applying a method with vanished phase–lag might potentially eliminate the phase–lag and its first, second, and third derivatives. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost–efficient approach. Equation PF3DPHFITN142SPS demonstrates the unique method. The suggested approach is P–Stable, meaning it is indefinitely periodic. The suggested approach is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost–efficient approach. With an AOR of 14, we can significantly improve our present predicament. [ABSTRACT FROM AUTHOR]

We study the nonlinear dynamics of perturbed, spectrally stable T-periodic stationary solutions of the Lugiato–Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. It is known that for each N ∈ N , such a T-periodic wave train is asymptotically stable against NT-periodic, i.e. subharmonic, perturbations, in the sense that initially nearby data will converge at an exponential rate to a (small) spatial translation of the underlying wave. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential rates of decay depend on N and, in fact, tend to zero as N → ∞ , leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in N. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in N. This in turn yields an improved nonuniform subharmonic stability result providing an N-independent ball of initial perturbations which eventually exhibit exponential decay at an N-dependent rate. Finally, we argue that our results connect in the limit N → ∞ to previously established stability results against localized perturbations, thereby unifying existing theories. [ABSTRACT FROM AUTHOR]

We study a one-dimensional nonstationary Schrödinger equation with a potential slowly depending on time. The corresponding stationary operator depends on time as on a parameter. It has finitely many negative eigenvalues and absolutely continuous spectrum filling . The eigenvalues move with time to the edge of the continuous spectrum and, having reached it, disappear one after another. We describe the asymptotic behavior of a solution close at some moment to an eigenfunction of the stationary operator, and, in particular, the phenomena occurring when the corresponding eigenvalue approaches the absolutely continuous spectrum and disappears. [ABSTRACT FROM AUTHOR]

We begin by deriving the nonlinear Schrödinger equation (NLSE) in the presence of nonlinear microstructures, where the refractive index undergoes periodic modulation in the transverse direction for photonic crystal containing circular shape. Our investigation delves deeply into the intricate mechanisms behind transverse and longitudinal modulation of the refractive indexes, which aligns with the design of numerous manufactured slab microstructure waveguides. Theory of solitary waves in nonlinear microstructure, here unconventional fiber has been studied and examined in detail. In this work, composite methods leading both transverse and longitudinal modulation of refractive indexes have been presented in detail. The work also presents the diverse characteristics of this NLSE for both homogeneous and nonhomogeneous medium, encompassing various orders of nonlinearity specific to the nonlinear microstructure under consideration. Additionally, we analyze the wave propagation profiles for wide signals, which are wider than the periodicity of micro-structured photonic crystal. We also conduct a perturbation analysis for narrow signals that are even narrower than the periodicity of photonic crystal in transverse direction. [ABSTRACT FROM AUTHOR]

Last December 27th, 2023, Ebru Cavlak Aslan et al. have published the article in the Journal "Optical and Quantum Electronics" the article titled "Analysing of different wave structures to the dissipative NLS equation and modulation instability" (Opt Quantum Electron 56:254, 2024. https://doi.org/10.1007/s11082-023-06035-6) where they tried to present analytical solutions of a dissipative nonlinear Schrödinger equation and study the modulational instability of the continuous wave solution of that equation. After analyzing their results, we found a number of shortcomings about both the modulational instability study and the analytical solutions. For analytical solutions, we found that all their found exact solutions were about the standard nonlinear Schrödinger equation and may contain errors/mistakes since at least one of their found dark soliton solutions was done in the region of modulational instability of the standard nonlinear Schrödinger equation. We also found that all their results about the modulational instability were obtained on an erroneous plane wave solution of their model equation, leading thus to obsolete results. It is the aim of the present comment to correct all shortcomings found in that study on the modulational instability. [ABSTRACT FROM AUTHOR]

In this article, we propose an alternate approach to study confined two-electron systems using the modified form of the Le Sech wavefunction. In the present approach, rather than using the cut-off factor in the variational wavefunction, we determine it directly by solving Schrödinger like equation. The results for kinetic energies, electron-nucleus interaction, electron–electron interaction, total energies, densities, ionization energies, and moments of confined H - and He atom are compared with the most accurate values found in the literature to show the effectiveness of our method. The present approach applies to a wide range of confinement potentials. We demonstrate it by showing the results for Coulomb, harmonic oscillator, and soft-confinement potentials. [ABSTRACT FROM AUTHOR]

Context: The Analyzing of energetic bond spectra of diatomic compounds is crucial to understanding their qualities because it allows one to evaluate their attributes. Diatoms compounds' spectral properties and bound energies are presented in this study. These energies are found by solving the Schrodinger equation while making consideration of the employing of the Kratzer Feus potential. Method: This study focuses on the calculation of bound states for diatomic molecules using the WKB approximation. The final energy spectrum equation is utilized to compute the bound states of specific diatomic molecules for varying quantum numbers n and l through the utilization of the Mathematica software. The method produced the desired and anticipated results, as shown by a comparison of the eigenvalue results with earlier studies. [ABSTRACT FROM AUTHOR]

Context: Accurate functions to analytically represent the potential energy interactions of CO + diatomic system in X 2 Σ + , A 2 Π , and B 2 Σ + electronic states are proposed. The new functions depend upon only four parameters directly obtained from experimental data, without any fitting procedure. These functions have been developed from the modified generalized potential proposed by Araújo and Ballester. The function for the X 2 Σ + electronic state represents a significant improvement to the previously proposed model. To quantify the accuracy of the potential energy functions, the Lippincont test is used. The novel potential was also compared with the classical Morse potential and with the recently proposed Improved Generalized Pöschl-Teller potential. Furthermore, the main spectroscopic constants and vibrational energy levels are calculated and compared for all potentials. The present results agree excellently with the experiment Rydberg-Klein-Rees (RKR) potentials. Methods: The rovibrational energy levels of the proposed diatomic potentials were asserted by solving radial the Schrödinger equation of the nuclear motion with the aid of the LEVEL program. [ABSTRACT FROM AUTHOR]

This article examines a compact scheme employing fuzzy transform via exponential basis to solve nonlinear stationary convection–diffusion equations. The scheme executes approximated fuzzy components which estimate the solution values with fourth-order accuracy in an optimal computing time. Such an arrangement associates the approximated fuzzy components with solution values by a linear system. The Jacobian matrices in the scheme are monotone and irreducible. The proof of convergence is briefly discussed. Numerical simulations with nonlinear and linear convection–diffusion equations occurring in quantum mechanics and rheological Carreau fluid will be examined to corroborate the new scheme's utility and efficiency of computational convergence order. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the local discontinuous Galerkin (LDG) methods coupled with multistep implicit–explicit (IMEX) time discretization to solve one-dimensional and two-dimensional nonlinear Schrödinger equations. In this approach, the nonlinear terms are treated explicitly, while the linear terms are handled implicitly. By the skew symmetry property of LDG operators and the properties of Gauss–Radau projections, we obtain error estimates for the prime and auxiliary variables, as well as the estimate for the time difference of the prime variables. These results, together with a carefully chosen numerical initial condition, allow us to obtain the optimal error estimate in both space and time for the fully discrete scheme. Numerical experiments are performed to verify the accuracy and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying a fractional Schrödinger equation of the form { (− Δ) s u + V (x) u = f (x , u) in Ω , u > 0 in Ω , u = 0 in R n ∖ Ω , where 0 < s < 1 , n > 2 s , Ω is a smooth bounded domain in R n , (− Δ) s denotes the fractional Laplacian of order s, f (x , t) is a function in C (Ω ‾ × R) , and f (x , t) / t is nondecreasing in t and converges uniformly to an L ∞ function q (x) as t approaches infinity. The potential energy V satisfies appropriate assumptions. In the first part of our study, we analyze the asymptotic linearity of the nonlinearity and investigate the occurrence of the bifurcation phenomenon. We employ variational techniques and a "mountain pass" approach in our proof, notable for not assuming the Ambrosetti–Rabinowitz condition or any replacement condition on the nonlinearity. Additionally, we extend our methods to handle cases where the function f (x , t) exhibits superlinearity in t at infinity, represented by q (x) ≡ + ∞ . [ABSTRACT FROM AUTHOR]

This paper investigates a generalized form of the nonlinear Schrödinger equation characterized by a logarithmic nonlinearity. The nonlinear Schrödinger equation, a fundamental equation in nonlinear wave theory, is applied across various physical systems including nonlinear optics, Bose–Einstein condensates, and fluid dynamics. We specifically explore a logarithmic variant of the nonlinear Schrödinger equation to model complex wave phenomena that conventional polynomial nonlinearities fail to capture. We derive four distinct forms of the nonlinear Schrödinger equation with logarithmic nonlinearity and provide exact solutions for each, encompassing bright, dark, and kink-type solitons, as well as a range of periodic solitary waves. Analytical techniques are employed to construct bounded and unbounded traveling wave solutions, and the dynamics of these solutions are analyzed through phase portraits of the associated dynamical systems. These findings extend the scope of the nonlinear Schrödinger equation to more accurately describe wave behaviors in complex media and open avenues for future research into non-standard nonlinear wave equations. [ABSTRACT FROM AUTHOR]

Applying a phase-fitting method might potentially vanish the phase-lag and its first derivative. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost-efficient approach. Equation PF2DPHFITN142SPS demonstrates the unique method. The suggested approach is P-Stable, meaning it is indefinitely periodic. The proposed method is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost-efficient approach. With an AOR of 14, we can significantly improve our present predicament. [ABSTRACT FROM AUTHOR]

Applying a phase-fitting method might potentially vanish the phase-lag and its first derivative. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost-efficient approach. Equation PF1DPHFITN142SPS demonstrates the unique method. The suggested approach is P-Stable, meaning it is indefinitely periodic. The proposed method is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost-efficient approach. With an AOR of 14, we can significantly improve our present predicament. [ABSTRACT FROM AUTHOR]

The quaternionic Schrödinger equation ∂ ∂ t | f ⟩ = - A | f ⟩ is the crucial part of the study of quaternionic quantum mechanics and plays indispensable roles in related fields. One of the practical and special cases that has received more attention from mathematicians and physicists is that A is a Hermitian quaternion matrix. The problem can be equivalent to a Hermitian quaternion right eigenvalue problem A α = α λ by discretization. This paper, by means of a complex representation method, studies the Hermitian quaternion Schrödinger equation problem, and proposes a novel algebraic method (complex structure-preserving method) for right eigenvalue problems of Hermitian quaternion matrices. Moreover, the complex structure-preserving method is superior and formally simple compared to previous methods, and numerical experiments also demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of ultrashort pulses in a real-world system featuring periodically distributed dispersion and nonlinearity. We present a precise solution resembling a chirped soliton for the higher-order nonlinear Schrödinger equation (HNLS). This solution incorporates group velocity dispersion (GVD), stimulated Raman scattering (SRS), a third-order dispersion (TOD) term, cubic nonlinear effects and self-steepening (SS) effects with spatially varying coefficients. The derivation is based on specific parametric conditions, accounting for both linear and nonlinear absorption and amplification. The investigation reveals the stability of solitary-like solutions as they propagate over long distances in the medium. Additionally, we furnish parametric conditions that dictate the existence of chirped solitons. The numerical results closely corroborate the outcomes obtained through analytical approaches. [ABSTRACT FROM AUTHOR]

The study of the static properties of isotopes, such as ground-state energy of the system and the energy spectra, electric quadrupole, as well as charge radii and so many others, is one of the significant areas of interest in nuclear physics. Therefore, in this field, the ground-state energy of a nucleus is an essential part of the study. In addition, understanding the nuclear force depends critically on understanding the nuclear charge radii. In this work, we considered Li isotopes in the study of two-body system configuration, and then we calculated the ground-state energy and charge radii for some of them. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension. [ABSTRACT FROM AUTHOR]

Achieving a perfectly spherical nanostructures might not be feasible in experiments due to factors like growth kinetics, solvent's nature and surface effects. This numerical investigation examines how the shape of a surrounding shell (prolate, spherical or oblate) in core/shell quantum dots (CSQDs) buried into two commonly used dielectric oxides (SiO2 or HfO2), microscopically influences their electro-optical characteristics. In the context of the effective mass approach (EMA) and the density matrix formalism (DMF), we have reached the stationary eigenstates and their matching wave functions by solving the Schrödinger equation. Our computations revealed that the discrete electronic states can fluctuate with ellipticity parameter as a consequence of different quantum confinement origins along the major and minor axes. The shell anisotropy provided an effective opportunity to finely adjust resonant frequencies and calibrate the magnitude order of the Quadratic electro-optic effects (QEOEs), electro-absorption (EA) process, optical absorption characteristics (OACs) and refractive index changes (RICs) within QD/oxide interfacs. Computed coefficients have experienced red/blue shift contingent upon variations in the inner core radius, ellipticity parameter and the types of capping oxides. [ABSTRACT FROM AUTHOR]

In this paper, the fractional Chiral nonlinear Schrödinger equation with time-dependent coefficients (FCNSE-TDCs) is considered. The mapping method is applied in order to get hyperbolic, elliptic, trigonometric and rational fractional solution. These solutions are vital for understanding some fundamentally complicated phenomena. The obtained solutions will be very helpful for applications such as optics, plasma physics and nonlinear quantum mechanics. Finally, the influence of the time-dependent coefficients and the conformable fractional derivative order on the exact solutions of the FCNSE-TDCs is presented. [ABSTRACT FROM AUTHOR]