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In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models. [ABSTRACT FROM AUTHOR]

The article "Quantum mechanical effects for a hydrogen atom confined in a dielectric spherical microcavity" (Wang et al., 2022) does not describe correctly the Shannon entropy for the states 3p and 4p for the hydrogen atom confined by a dielectric spherical microcavity. In this comment, we show the correct behavior for these states and conclude that there was an error in the data found by the authors of that article. Thus, looking for a physical meaning for the behavior observed in that paper makes no sense. [Display omitted] • Hydrogen atom confined by rigid walls. • Exact solution of the Schrödinger equation for the confined hydrogen atom. • Correct values of the Shannon entropy for the hydrogen atom confined by a dielectric spherical microcavity. [ABSTRACT FROM AUTHOR]

This research paper investigates the behavior of optical fiber pulses by studying the solitary wave solutions of the perturbed Chen–Lee–Liu (CLL) equation. The perturbed CLL equation is derived from the perturbed Schrödinger equation. Two analytical approaches for obtaining the innovative soliton-wave solutions are presented in this paper, and their reliability is examined by using a well-established numerical method. The accuracy of pulse wave analysis in an optical cable is demonstrated through a series of graphical representations. Our study introduces scientific novelty, as evidenced by the comparative analysis of our data with those of previous research papers. This research contributes to enhancing the understanding of the kinetics and physical behavior of pulses in optical fibers, which holds implications for the advancement of optical communication technologies. [ABSTRACT FROM AUTHOR]

In the present paper, we precisely conduct a quantum calculus method for the numerical solutions of PDEs. A nonlinear Schrödinger equation is considered. Instead of the known classical discretization methods based on the finite difference scheme, Adomian method, and third modified versions, we consider a discretization scheme leading to subdomains according to q-calculus and provide an approximate solution due to a specific value of the parameter q. Error estimates show that q-calculus may produce efficient numerical solutions for PDEs. The q-discretization leads effectively to higher orders of convergence provided with faster algorithms. The numerical tests are applied to both propagation and interaction of soliton-type solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland's variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in R N ∖ Ω ‾ for fractional order p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of sign-changing radial solutions with any prescribed numbers of zeros to the following Schrodinger equation with the critical exponential growth: \begin{equation*} \begin {cases} -\Delta u +u=\lambda ue^{u^2} \quad \quad \text {in } \quad \mathbb {R}^2,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0, \end{cases} \end{equation*} where 0<\lambda <1. Our proof relies on the shooting method, the Sturm's comparison theorem and a Liouville type theorem in exterior domain of \mathbb {R}^2. It seems to be the first existence result of sign-changing solution for Schrodinger equation with the critical exponential growth. [ABSTRACT FROM AUTHOR]

The Kaup-Newell system describes the pulse propagation in optical fibers. This paper applies the semi-inverse method to construct the system's variational formulations, two trial-variational formulations with an unknown function are established, and a detailed derivation is given to determine the unknown function. Finally, the Kaup-Newell system with two variables is converted to a partial differential equation with only one variable. [ABSTRACT FROM AUTHOR]

This paper examines whether a quantum computer can efficiently simulate the time evolution of the Schrödinger particle in a one-dimensional infinite potential well. In order to solve the Schrödinger equation in the quantum register, an algorithm based on the Quantum Discrete Sine Transform (QDST) is applied. The paper compares the results obtained in this way with the results given by the previous method (based on the QFT algorithm). [ABSTRACT FROM AUTHOR]

In the present paper, we give some remarks on the magnetic field operators ∇±iA. As its applications, we study the Schr¨odinger equation with a magnetic field -Δu+|A(x)|2u+iA(x) ·∇u = μu+|u|pu, x ∈ RN, where u is a complex-valued function and μ ∈ R. When N > 2, for 2 < p+2 < 2N N-2 or N = 2, for 2 < p+2 < +∞, the existence and nonexistence of minimizers of the corresponding minimization problem are given via constrained variational methods. As a by-product, the above equation admits a normalized solution. We point out that the condition divA(x) = 0 plays a crucial role in our study. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following semilinear Schrödinger equation: { -Δu + V(x)u = a(x)g(u) for x ∈ RN, u(x) → 0 as |x| → 8, where a(x) > ∞ for all RN. Under some different superlinear conditions on g(u), we obtain the existence of solutions for the above problem. In order to regain the compactness of the Sobolev embedding, a competing condition between a(x) and V(x) is introduced. [ABSTRACT FROM AUTHOR]

In this paper and the companion work [J. Funct. Anal. 281 (2021)], we prove that the Schrödinger map flows from Rd with d ≥ 2 to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. The main difficulty compared with the constant sectional curvature case is that the gauged equation now is not self-contained due to the curvature part. Our main idea is to use a novel bootstrap-iteration scheme to reduce the gauged equation to an approximate constant curvature system in finite times of iteration. This paper together with the companion work solves the open problem raised by Tataru. [ABSTRACT FROM AUTHOR]

In a recent paper Li et al. (Z Angew Math Phys 73:52, 2022. https://doi.org/10.1007/s00033-022-01681-4) have considered a generalized nonlinear Schrödinger equation which has extensive applications in various fields of physics and engineering. After proving Liouville integrability of this equation, they investigated the phenomenon of the modulational instability for the possible reason of the formation of the rogue waves. By means of the generalized ( 2 , N - 2 )-fold Darboux transformation, authors presented several mixed localized wave solutions, such as breathers, rogue waves and semi-rational solitons for their model equation, and accurately analyzed a number of important physical quantities. It is the aim of this Comment to point out that (i) the baseband modulation instability was developed in a wrong way and (ii) one of the two different types of Taylor series expansions for solution of Lax pair used in that article for building analytical solutions, especially the one obtained with ξ j = Z does not correspond to any solution of the spectral problem (2.1) when using u 0 x , t as the seed solution. Consequently, all mixed localized solutions that involve the mentioned Taylor series are invalid. [ABSTRACT FROM AUTHOR]

The KP‐I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution, which is a traveling wave, and the KP‐I equation in this case reduces to the Boussinesq equation. In this paper we classify all the 'lump‐type' solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman–Miller for the Schrödinger equation, we show that the lump‐type solutions are rational and their τ$\tau $ functions have to be polynomials of degree k(k+1)$k(k+1)$ for some integer k$k$. In particular, this implies that the lump solution is the unique ground state of the KP‐I equation (as conjectured by Klein–Saut). The problem studied in this paper was mentioned in Airault–McKean–Moser, our result can be regarded as a two‐dimensional analogy of their theorem on the classification of rational solutions for the KdV equation. [ABSTRACT FROM AUTHOR]

This paper presents triangular core fiber for single-mode operation. The fiber supports single mode at a wavelength of 1550-nm, and it has an extremely large mode area of 2000 μm². With peak power of 85.4-kW at 1550 nm wavelength, fiber design demonstrates undistorted delivery of 100 fs laser pulses. We have used Transfer matrix method to analyse the fiber. The splitstep Fourier method has been used to solve the non-linear Schrodinger equation. Raman's response and self-steepening have also been taken in to account. In this paper, we report the numerical demonstration of propagation of 100-fs pulse with peak power of 85.4-kW at 1550-nm wavelength through 2.5 m long fiber. The ratio of dispersion length to nonlinear length has been kept as 1 for the delivery of undistorted pulses. [ABSTRACT FROM AUTHOR]

The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 (| ∇ u | 2 + V (x) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x < + ∞ , where 0 < β < 2 , V (x) ≥ 0 and may vanish on an open set in R 2 . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in R 2 : (0.2) − Δ u + γ u = f (u) | x | β , where the nonlinearity f has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, "Concentration-compactness principle of singular Trudinger-Moser inequalities in R n and n -Laplace equations," Adv. Nonlinear Stud., vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in R 2 : (0.3) − Δ u + V (x) u = f (u) | x | β . [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of normalized solutions to the following fractional Schrödinger equation (- Δ) s u = λ u + g (u) + | u | 2 s ∗ - 2 u , x ∈ R N , ∫ R N u 2 = a 2 , where N ≥ 3 , s ∈ (0 , 1) , a > 0 , 2 s ∗ = 2 N / (N - 2 s) , λ ∈ R arises as a Lagrange multiplier, (- Δ) s is the fractional Laplace operator and g : R → R satisfies L 2 -supercritical conditions. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schrödinger equation on the product space R × T . We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the "auxiliary equation" and then on the application of the Crandall–Rabinowitz argument. [ABSTRACT FROM AUTHOR]

Superoscillations are a phenomenon in physics, where linear combinations of low-frequency plane waves interfere almost destructively in such a way that the resulting wave has a higher frequency than any of the individual waves. The evolution of superoscillatory initial datum under the time dependent Schrödinger equation is stable in free space, but in general it is unclear whether it can be preserved in the presence of an external potential. In this paper, we consider the two-dimensional problem of superoscillations interacting with a half-plane barrier, where homogeneous Dirichlet or Neumann boundary conditions are imposed on the negative x 2 -semiaxis. We use the Fresnel integral technique to write the wave function as an absolute convergent Green’s function integral. Moreover, we introduce the propagator of the Schrödinger equation in form of an infinite order differential operator, acting continuously on the function space of exponentially bounded entire functions. In particular, this operator allows to prove that the property of superoscillations is preserved in the form of a similar phenomenon called supershift, which is stable over time. [ABSTRACT FROM AUTHOR]

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales. [ABSTRACT FROM AUTHOR]

This paper introduces a novel model proposed by Wazwaz et al. in 2023, in the nonlinear optics literature. This contributes to advancing optical devices and technologies, particularly in telecommunications and laser systems. The characteristics of bright, dark, and periodic soliton solutions for the (2+1)-dimensional nonlinear Schrödinger equation with fourth-order nonlinearity and dispersion are explored in this paper. The relevance of these solutions lies in the study of nonlinear waves propagating in an inhomogeneous optical fiber. The soliton solutions are obtained through the implementation of three analytical methods: the Kudryashov method, the Bernoulli Sub-ODE method, and the Extended Direct Algebraic method. The bright, dark, and periodic soliton solutions are constructed by utilizing bilinear forms. Furthermore, the impact of variable coefficients on the structures of these solitons is analyzed. Graphical illustrations depict the propagation of bright, dark, and periodic solitons. [ABSTRACT FROM AUTHOR]

In this paper, we prove the multiplicity and concentration behavior of complex-valued solutions for the following Kirchhoff–Schrödinger equation with magnetic field - (a ε 2 + b ε [ u ] A / ε 2) Δ A / ε u + V (x) u = f (| u | 2) u , x ∈ R 2 , where ε > 0 is a small parameter, the nonlinearity f is involved in critical exponential growth in the sense of Trudinger–Moser inequality and both V : R 2 → R and A : R 2 → R 2 are continuous potential and magnetic potential, respectively. Imposing a local constraint of potential V(x) first introduced from del Pino and Felmer, we get the multiplicity of solutions by way of the relationship between the number of the solutions and the topology of the set with V attaining the minimum. Our strategy of main proof is based on the variational methods combined with the penalization technique, the Trudinger–Moser inequality and Ljusternik–Schnirelmann theory, and our result is still new even without magnetic effect. [ABSTRACT FROM AUTHOR]

In this paper we will prove the existence of a positive solution for a class of Schrödinger logarithmic equation of the form. - Δ u + u = Q (x) u log u 2 , in Ω , B u = 0 on ∂ Ω , where Ω ⊂ R N , N ≥ 3 , is an exterior domain, i.e., Ω c = R N \ Ω is a bounded smooth domain where B u = u or B u = ∂ u ∂ ν . We have used new approach that allows us to apply the usual C 1 -variational methods to get a nontrivial solutions for these classes of problems. [ABSTRACT FROM AUTHOR]

The design of a solid-state quantum processor is nowadays a hot research topic in microelectronics. Like the logic gates in a classical processor, quantum gates serve as the fundamental building blocks for quantum processors. The main goal of the present paper is to deduce the matrix of the main one- and two-qubit quantum gates from the Schrödinger equation. The mathematical formalism is kept as comfortable as possible for electronics engineers. This paper does not cover topics such as dissipations, state density, coherence, and state purity. In a similar manner, this paper also deals with the quantum nature of a quantum processor by leveraging the concept of a finite-state machine, which is a background notion for any electronics engineer. [ABSTRACT FROM AUTHOR]

This paper investigates a higher-order modified nonlinear Schrödinger equation with higher-order dispersion and self-steepening effects, which can be used to study the dynamics of asymmetric and steepened optical pulse transmission in optical fibres. The modulation instability of the plane-wave solution has been analysed, and the state transition of the rogue waves under the higher-order dispersion effect is presented. The findings demonstrate that the self-steepening effect may also produce an asymmetric unstable frequency band. Combined with the modulation instability, the rogue wave, rational soliton and mixed interaction of localized waves have a quantitative relationship with plane-wave parameters. Various exact solutions can be accurately located and obtained according to the generalized Darboux transformation. The asymptotic analysis of rational solutions demonstrates the state transition of higher-order rogue waves. This paper illustrates the significance of modulation instability for studying the integrable systems by the Darboux transformation. It is a new guidance to solve the difficulty of exact excitation of asymmetric localized waves in complex integrable systems. The results have prospective applications and references for the emergence, amplification and compression of asymmetric pulses in optical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study the solution of a singularly perturbed inhomogeneous mixed problem on the half-axis for the Schrödinger equation in the presence of a "strong" turning point for the limit operator on time interval that do not contain focal points. Based on the ideas of the regularization method for asymptotic integration of problems with an unstable spectrum, it is shown how regularizing functions should be constructed for this type of singularity. The paper describes in detail the formalism of the regularization method, justifies the algorithm, constructs an asymptotic solution of any order in a small parameter, and proves a theorem on the asymptotic convergence of the resulting series. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new scheme for the integration of the periodic nonlinear Schrödinger equation and rigorously prove convergence rates at low regularity. The new integrator has decisive advantages over standard schemes at low regularity. In particular, it is able to handle initial data in Hs for 0 < s ≤ 1. The key feature of the integrator is its ability to distinguish between low and medium frequencies in the solution and to treat them differently in the discretization. This new approach requires a well-balanced filtering procedure which is carried out in Fourier space. The convergence analysis of the proposed scheme is based on discrete (in time) Bourgain space estimates which we introduce in this paper. A numerical experiment illustrates the superiority of the new integrator over standard schemes for rough initial data. [ABSTRACT FROM AUTHOR]

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. To construct the hierarchical equations, we use the monodromy matrix method, as first proposed by B.A. Dubrovin. The method we use consists of solving the following sequence of problems. First, using the Lax operator, we find the monodromy matrix, which is a polynomial in the spectral parameter. More precisely, we find a sequence of monodromy matrices dependent on the degree of this polynomial. Each Lax operator has its own sequence of monodromy matrices. Then, using the terms from the decomposition of the monodromy matrix, we construct a sequence of second operators from a Lax pair. A hierarchy of evolutionary integrable nonlinear equations follows from the conditions of compatibility of the sequence of Lax pairs. Also, knowledge of the monodromy matrix allows us to find stationary equations that are analogs of the Novikov equations for the Korteweg–de Vries equation. In addition, the characteristic equation of the monodromy matrix corresponds to the spectral curve equation of the relevant multiphase solution for the integrable nonlinear equation. Since the coefficients of the spectral curve equation are integrals of the hierarchical equations, they can be utilized to find the simplest solutions of the constructed integrable nonlinear equations. In this paper, we demonstrate the operation of this method, starting with the assignment of the Lax operator and ending with the construction of the simplest solutions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show the existence of positive solutions of the following Schrodinger equations with potential and indefinite nonlinearity: ... h changes sign in R³ Our approach combines variational techniques based on critical point theory and some analysis techniques. [ABSTRACT FROM AUTHOR]

A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schrödinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy presented in the paper is worded out. Finally, we further enlarge the three-dimensional loop algebra into a six-dimensional loop algebra so that a new isospectral integrable hierarchy which is a type of extended integrable model is produced whose bi-Hamiltonian structure is also derived from the vector-product trace identity. This new approach presented in the paper possesses extensive applications in the aspect of generating integrable hierarchies of evolution equations. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field. With K we associate the ring of finite adeles K. In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group K n. Specifically, we consider matrix-valued Hamiltonian operators H K n = Δ K n ⊗ Id + V , where Δ K n is the Vladimirov operator and V is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by − H K n . This formula is given in terms of the ordered time exponentials. [ABSTRACT FROM AUTHOR]

We are revisiting the problem of solving a discrete nonlinear Schrödinger equation by the inverse scattering transform method, by use of the recently developed ExactMPF package within MAPLE Software. ExactMPF allows for an exact Wiener-- Hopf factorization of matrix polynomials regardless of the partial indices of the matrix. The package can be widely used in various problems, where Wiener--Hopf factorization as one of the effective mathematical tools is required, as its code has already been disclosed. The analysis presented in this paper contains not only numerical examples of its use, but is also supported by appropriate and accurate a priori estimations. The procedure itself guarantees that the ExactMPF package produces all computations arithmetically exactly, and a detailed numerical analysis of various aspects of the computational algorithm and approximation strategies is provided in the case of a finite initial impulse. [ABSTRACT FROM AUTHOR]

In this paper, we consider matrix Schrödinger equation, dynamical Schrödinger equation and matrix KdV. We construct their explicit solutions using our GBDT version of Bäcklund–Darboux transformation and square roots of the generalised matrix eigenvalues. A separate section is dedicated to several examples including the case of strongly singular potentials. [ABSTRACT FROM AUTHOR]

The excitation energy transfer (EET) process for photosynthetic antenna complexes consisting of subunits, each comprised of multiple chromophores, remains challenging to describe. The multichromophoric Förster resonance energy transfer theory is a popular method to describe the EET process in such systems. This paper presents a new time-domain method for calculating energy transfer based on the combination of multichromophoric Förster resonance energy transfer theory and the Numerical Integration of the Schrödinger Equation method. After validating the method on simple model systems, we apply it to the Light-Harvesting antenna 2 (LH2) complex, a light harvesting antenna found in purple bacteria. We use a simple model combining the overdamped Brownian oscillators to describe the dynamic disorder originating from the environmental fluctuations and the transition charge from the electrostatic potential coupling model to determine the interactions between chromophores. We demonstrate that with this model, both the calculated spectra and the EET rates between the two rings within the LH2 complex agree well with experimental results. We further find that the transfer between the strongly coupled rings of neighboring LH2 complexes can also be well described with our method. We conclude that our new method accurately describes the EET rate for biologically relevant multichromophoric systems, which are similar to the LH2 complex. Computationally, the new method is very tractable, especially for slow processes. We foresee that the method can be applied to efficiently calculate transfer in artificial systems as well and may pave the way for calculating multidimensional spectra of extensive multichromophoric systems in the future. [ABSTRACT FROM AUTHOR]

Quantum mechanics has a deterministic Schrödinger equation for the wave function. The Göttingen–Copenhagen statistical interpretation is based on the Born Rule that interprets the wave function as a "probability amplitude." A precept of this interpretation is the lack of determinism in quantum mechanics. The Bohm interpretation is that the wave function is a source of a field experienced by the electrons, thereby attributing determinism to quantum theory. In this paper, we present a new perspective on such determinism. The ideas are based on the equations of motion or "Quantal Newtonian" Laws obeyed by each electron. These Laws, derived from the temporal and stationary-state Schrödinger equation, are interpreted in terms of "classical" fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. According to the Second Law, each electron experiences an external field—the quantal Coulomb-Lorentz law. It also experiences an internal field representative of properties of the system: correlations due to Coulomb repulsion and Pauli principle; the density; kinetic effects; and an internal magnetic field component. There is a response field. The First Law states that the sum of the external and internal fields experienced by each electron vanishes. These fields are akin to those of classical physics: They pervade all space; their structure is descriptive of the quantum system; the energy of the system is stored in these fields. It is in the classical behavior of these fields, which arise from quantal sources that one may then speak of determinism in quantum mechanics. [ABSTRACT FROM AUTHOR]

The time fractional Schrödinger equation contributes to our understanding of complex quantum systems, anomalous diffusion processes, and the application of fractional calculus in physics and cubic B-spline is a versatile tool in numerical analysis and computer graphics. This paper introduces a numerical method for solving the time fractional Schrödinger equation using B-spline functions and the Atangana-Baleanu fractional derivative. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while a θ-weighted scheme is used to discretize the space directions. The efficiency of the method is demonstrated through numerical results, and error norms are examined at various values of the non-integer parameter, temporal directions, and spatial directions. [ABSTRACT FROM AUTHOR]

In this paper, we solve the bound state problem for the Varshni-Hellmann potential via a useful technique. In our technique, we obtain the bound state solution of the Schrödinger equation for the Varshni-Hellmann potential via ansatz method. We obtain the energy eigenvalues and the corresponding eigenfunctions. Also, the behavior of the energy spectra for both the ground and the excited state of the two body systems is illustrated graphically. The similarity of our results to the accurate numerical values is indicative of the efficiency of our technique. [ABSTRACT FROM AUTHOR]

The paper focuses on investigating the time-fractional (3 + 1)-dimensional cubic and quantic nonlinear Schrödinger equation. We adopt the novel Kudryashov method to generate a distinct class of optical solutions for the current conformable fractional derivative problem. Our method explores various solution forms, including dark, wave, mixed dark-bright, and singular solutions. The soliton solutions we construct are visually represented to illustrate the influence of the fractional order derivative. Further, we elucidate the influence of solution parameters on the wave envelope, providing clear interpretations through 2D graphics presentations. The results underscore the efficacy of our approach in discovering exact solutions for nonlinear partial differential equations, especially in cases where alternative methods prove ineffective. The significance of the present paper lies in its contribution to advancing the understanding of the behavior of optical solutions in nonlinear systems, providing valuable insights for both theoretical and practical applications. In the field of nonlinear optics, this equation can describe the propagation of optical pulses in nonlinear media. It helps in understanding the behavior of intense laser beams as they propagate through materials exhibiting nonlinear optical effects such as self-focusing, self-phase modulation, and optical solitons. [ABSTRACT FROM AUTHOR]

This paper investigates dynamical behaviors and controllability of some nonautonomous localized waves based on the Gross–Pitaevskii equation with attractive interatomic interactions. Our approach is a relation constructed between the Gross–Pitaevskii equation and the standard nonlinear Schrödinger equation through a new self-similarity transformation which is to convert the exact solutions of the latter to the former's. Subsequently, one can obtain the nonautonomous breather solutions and higher-order rogue wave solutions of the Gross–Pitaevskii equation. It has been shown that the nonautonomous localized waves can be controlled by the parameters within the self-similarity transformation, rather than relying solely on the nonlinear intensity, spectral parameters, and external potential. The control mechanism can induce an unusual number of loosely bound higher-order rogue waves. The asymptotic analysis of unusual loosely bound rogue waves shows that their essence is energy transfer among rogue waves. Numerical simulations test the dynamical stability of obtained localized wave solutions, which indicate that modifying the parameters in the self-similarity transformation can improve the stability of unstable localized waves and prolong their lifespan. We numerically confirm that the rogue wave controlled by the self-similarity transformation can be reproduced from a chaotic initial background field, hence anticipating the feasibility of its experimental observation, and propose an experimental method for observing these phenomena in Bose–Einstein condensates. The method presented in this paper can help to induce and observe new stable localized waves in some physical systems. [ABSTRACT FROM AUTHOR]

In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing nonlinear Schrödinger equation with various (deterministic and random) initial data. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see [C. Fan and Z. Zhao, Discrete Contin. Dyn. Syst., 41 (2021), pp. 3973--3984] and the references therein), and, moreover, we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the L¹ -data assumption (see [C. Fan and Z. Zhao, Proc. Amer. Math. Soc., 151 (2023), pp. 2527--2542] for the necessity of the L¹-data assumption). [ABSTRACT FROM AUTHOR]

Let \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential \Upsilon belonging to the reverse Hölder class B_{Q/2}, where Q is the homogeneous dimension of the stratified Lie group \mathbb {G}. Inspired by Shen's pioneer work and Li's work, we study fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for \mathcal {L}-harmonic functions on \mathbb {G}. [ABSTRACT FROM AUTHOR]

With this follow-up paper, we continue developing a mathematical framework based on information geometry for representing physical objects. The long-term goal is to lay down informational foundations for physics, especially quantum physics. We assume that we can now model information sources as univariate normal probability distributions N (μ , σ 0) , as before, but with a constant σ 0 not necessarily equal to 1. Then, we also relaxed the independence condition when modeling m sources of information. Now, we model m sources with a multivariate normal probability distribution N m (μ , Σ 0) with a constant variance–covariance matrix Σ 0 not necessarily diagonal, i.e., with covariance values different to 0, which leads to the concept of modes rather than sources. Invoking Schrödinger's equation, we can still break the information into m quantum harmonic oscillators, one for each mode, and with energy levels independent of the values of σ 0 , altogether leading to the concept of "intrinsic". Similarly, as in our previous work with the estimator's variance, we found that the expectation of the quadratic Mahalanobis distance to the sample mean equals the energy levels of the quantum harmonic oscillator, being the minimum quadratic Mahalanobis distance at the minimum energy level of the oscillator and reaching the "intrinsic" Cramér–Rao lower bound at the lowest energy level. Also, we demonstrate that the global probability density function of the collective mode of a set of m quantum harmonic oscillators at the lowest energy level still equals the posterior probability distribution calculated using Bayes' theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. While these new assumptions certainly add complexity to the mathematical framework, the results proven are invariant under transformations, leading to the concept of "intrinsic" information-theoretic models, which are essential for developing physics. [ABSTRACT FROM AUTHOR]

In this paper, we study the discrete nonlinear Schrödinger equation - Δ u + V (x) - ρ (| x | 2 + 1) u = f (x , u) , u ∈ ℓ 2 (Z N) , where N ≥ 3 , V is a bounded periodic potential and 0 lies in a spectral gap of the Schrödinger operator - Δ + V . The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is the lack of compactness of the Cerami sequence. We overcome these two major difficulties by the generalized linking theorem and Lions lemma. This enables us to establish the existence and asymptotic behavior of ground state solutions for small ρ ≥ 0 . [ABSTRACT FROM AUTHOR]

This study investigates some novel solitary wave solutions of the perturbed Chen–Lee–Liu ( ℒ ℒ) equation to explain the physical and dynamic behavior of the pulse in optical fiber. The perturbed ℒ ℒ equation is one of the icon models that have been derived from the well-known Schrödinger equation. Two analytical techniques are used to obtain novel solitary wave solutions. Then, using a well-known numerical scheme, these solutions are looked at objectively to figure out why they work. Several graphs in different styles are used to show how the pulse waves in an optical fiber are analyzed and how accurate the analysis is. The scientific novelty of this research is given by explaining the comparison figure out of our results that have been published in some recently published research papers. [ABSTRACT FROM AUTHOR]

In general, Schrödinger's equations have lots of applications in quantum mechanics. Accordingly, in this paper, some new forms of the exact traveling wave solutions have been obtained for two different types of Schrödinger's equations, namely the unstable generalized nonlinear Schrödinger's equation and the hyperbolic nonlinear Schrödinger's equation. In the literature one can see different forms of the solution by various existing methods. However, the applied methodology or obtained results are sometimes complicated in nature. So here to obtain the new and simple forms of the solutions for both of the equations we have used a very simple and important method known as sine-cosine method. In addition, the probability density functions (PDFs) for both of the considered problems have also been computed. To visualize the impact of the solutions, graphical representations have been made with respect to various parameters. From the results one can see, they are in terms of complex-valued functions. In special cases comparison has also been given. Furthermore, the results are computed and validated using the Maple software. [ABSTRACT FROM AUTHOR]

In this paper, in the classical framework, the lower bounds for the sensitivities of the generalized Lemaitre Tolman Bondi metric are evaluated. The calculated lower bounds via the linear dynamical systems L∂∂θ$L_{\frac{\partial }{\partial \theta }}$, L∂∂r$L_{\frac{\partial }{\partial r}}$, and L∂∂ϕ$L_{\frac{\partial }{\partial \phi }}$ are −ln2+ln|(ṘB)2−(R′)2|−2ln|B|$-\ln 2+\ln|{(\dot{R}B)}^{2}-{(R^{\prime })}^{2}|-2\ln|B|$, 2ln|Ḃ|−ln2$2\ln|\dot{B}|-\ln 2$ and −ln2−2ln|B|+ln|(Ṙ2B2−R′2)sin2θ−B2cos2θ|$-\ln 2-2\ln|B|+ \ln |(\dot{R}^{2}B^{2}-R^{\prime 2})\sin ^{2}\theta -B^{2}\cos ^{2}\theta|$ respectively. The sensitivities and the lower sensitivities via L∂∂t$L_{\frac{\partial }{\partial t}}$ are zero are also shown. In the quantum framework, the properties of the Einstein‐Vaz shells which are the final result of the quantum gravitational collapse arising from the Lemaitre Tolman Bondi discussed by Vaz in 2014 are analyzed. In fact, Vaz showed that continued collapse to a singularity can only be obtained if one combines two independent and entire solutions of the Wheeler‐DeWitt equation. Forbidding such a combination leads naturally to matter condensing on the Schwarzschild surface during quantum collapse. In that way, an entirely new framework for black holes (BHs) has emerged. The approach of Vaz was also consistent with Einstein's idea in 1939 of the localization of the collapsing particles within a thin spherical shell. Here, following an approach of oned of us (CC), we derive the BH mass and energy spectra via a Schrodinger‐like approach, by further supporting Vaz's conclusions that instead of a spacetime singularity covered by an event horizon, the final result of the gravitational collapse is an essentially quantum object, an extremely compact "dark star". This "gravitational atom" is held up not by any degeneracy pressure but by quantum gravity in the same way that ordinary atoms are sustained by quantum mechanics. Finally, the time evolution of the Einstein‐Vaz shells is discussed. In this paper, in the classical framework, the lower bounds for the sensitivities of the generalized Lemaitre Tolman Bondi metric are evaluated. In the quantum framework, the properties of the Einstein‐Vaz shells which are the final result of the quantum gravitational collapse arising from the Lemaitre Tolman Bondi discussed by Vaz in 2014 are analyzed. In fact, Vaz showed that continued collapse to a singularity can only be obtained if one combines two independent and entire solutions of the Wheeler‐DeWitt equation. Forbidding such a combination leads naturally to matter condensing on the Schwarzschild surface during quantum collapse. In that way, an entirely new framework for black holes (BHs) has emerged. The approach of Vaz was also consistent with Einstein's idea in 1939 of the localization of the collapsing particles within a thin spherical shell. Here, following an approach of oned of us (CC), we derive the BH mass and energy spectra via a Schrodinger‐like approach, by further supporting Vaz's conclusions that instead of a spacetime singularity covered by an event horizon, the final result of the gravitational collapse is an essentially quantum object, an extremely compact "dark star". This "gravitational atom" is held up not by any degeneracy pressure but by quantum gravity in the same way that ordinary atoms are sustained by quantum mechanics. Finally, the time evolution of the Einstein‐Vaz shells is discussed. [ABSTRACT FROM AUTHOR]

This paper was concerned with a new class of Schrödinger equations involving double phase operators with variable exponent in RN. We gave the corresponding Musielak-Orlicz Sobolev spaces and proved certain properties of the double phase operator. Moreover, our main tools were the topological degree theory and Galerkin method, since the equation contained a convection term. By using these methods, we derived the existence of weak solution for the above problems. Our result extended some recent work in the literature. [ABSTRACT FROM AUTHOR]

The existence of L2–normalized solutions is studied for the equation − Δ u + μ u = f (x , u) in R N , ∫ R N u 2 d x = m. Here m > 0 and f(x, s) are given, f(x, s) has the L2-subcritical growth and (μ, u) ∈ R × H1(RN) are unknown. In this paper, we employ the argument in Hirata and Tanaka ("Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches," Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka ("Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches," Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry. [ABSTRACT FROM AUTHOR]