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Journal articles on the topic 'Equation de Schrödinger non-linéaire'

Author: Grafiati

Published: 4 June 2021

Last updated: 19 October 2024

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1

Oh, Tadahiro, Philippe Sosoe, and Leonardo Tolomeo. "Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus." Inventiones mathematicae 227, no. 3 (November 8, 2021): 1323–429. http://dx.doi.org/10.1007/s00222-021-01080-y.

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AbstractWe study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrödinger equation on the one-dimensional torus. In an influential paper, Lebowitz et al. (J Stat Phys 50(3–4):657–687, 1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (Ann Inst H Poincaré Anal Non Linéaire 31(6):1267–1288, 2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz et al. (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schrödinger equation on the one-dimensional torus.

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2

Feng, Wei, and Song-Lin Zhao. "Soliton solutions to the nonlocal non-isospectral nonlinear Schrödinger equation." International Journal of Modern Physics B 34, no. 25 (September 9, 2020): 2050219. http://dx.doi.org/10.1142/s0217979220502197.

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In this paper we study the nonlocal reductions for the non-isospectral Ablowitz-Kaup-Newell-Segur equation. By imposing the real and complex nonlocal reductions on the non-isospectral Ablowitz-Kaup-Newell-Segur equation, we derive two types of nonlocal non-isospectral nonlinear Schrödinger equations, in which one is real nonlocal non-isospectral nonlinear Schrödinger equation and the other is complex nonlocal non-isospectral nonlinear Schrödinger equation. Of both of these two equations, there are the reverse time nonlocal type and the reverse space nonlocal type. Soliton solutions in terms of double Wronskian to the reduced equations are obtained by imposing constraint conditions on the double Wronskian solutions of the non-isospectral Ablowitz-Kaup-Newell-Segur equation. Dynamics of the one-soliton solutions are analyzed and illustrated by asymptotic analysis.

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3

Fa, Kwok Sau. "Integro-differential Schrödinger equation and description of unstable particle." Modern Physics Letters B 28, no. 30 (December 10, 2014): 1450234. http://dx.doi.org/10.1142/s0217984914502340.

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The description of a particle in the quantum system is probabilistic. In the ordinary quantum mechanics the total probability of finding the particle is conserved, i.e. the probability is normalized for all the times. To find a non-constant total probability an imaginary term should be added to the potential energy which is not physical. Recently, generalizations of the ordinary Schrödinger equation have been proposed by using the Feynman path integral and analogy between the Schrödinger equation and diffusion equation. In this work, an integro-differential Schrödinger equation is proposed by using analogy between the Schrödinger equation and diffusion equation. The equation is obtained from the continuous time random walk model with diverging jump length variance and generic waiting time probability density. The equation generalizes the ordinary and fractional Schrödinger equations. One can show that the integro-differential Schrödinger equation can describe a non-constant total probability for a free particle, and it includes the exponential decay which is fundamental for the description of radioactive decay.

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4

Malham, Simon J. A. "Integrability of local and non-local non-commutative fourth-order quintic non-linear Schrödinger equations." IMA Journal of Applied Mathematics 87, no. 2 (March 17, 2022): 231–59. http://dx.doi.org/10.1093/imamat/hxac002.

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Abstract We prove integrability of a generalized non-commutative fourth-order quintic non-linear Schrödinger equation. The proof is relatively succinct and rooted in the linearization method pioneered by Ch. Pöppe. It is based on solving the corresponding linearized partial differential system to generate an evolutionary Hankel operator for the ‘scattering data’. The time-evolutionary solution to the non-commutative non-linear partial differential system is then generated by solving a linear Fredholm equation which corresponds to the Marchenko equation. The integrability of reverse space-time and reverse time non-local versions, in the sense of Ablowitz and Musslimani (2017, Integrable nonlocal nonlinear equations, Stud. Appl. Math. 139, 7–59), of the fourth-order quintic non-linear Schrödinger equation are proved contiguously by the approach adopted. Further, we implement a numerical integration scheme based on the analytical approach above, which involves solving the linearized partial differential system followed by numerically solving the linear Fredholm equation to generate the solution at any given time.

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5

Gaspard, P., and M. Nagaoka. "Non-Markovian stochastic Schrödinger equation." Journal of Chemical Physics 111, no. 13 (October 1999): 5676–90. http://dx.doi.org/10.1063/1.479868.

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6

Arnbak, H., P. L. Christiansen, and Yu B. Gaididei. "Non-relativistic and relativistic scattering by short-range potentials." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, no. 1939 (March 28, 2011): 1228–44. http://dx.doi.org/10.1098/rsta.2010.0330.

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Relativistic and non-relativistic scattering by short-range potentials is investigated for selected problems. Scattering by the δ ′ potential in the Schrödinger equation and δ potentials in the Dirac equation must be solved by regularization, efficiently carried out by a perturbation technique involving a stretched variable. Asymmetric regularizations yield non-unique scattering coefficients. Resonant penetration through the potentials is found. Approximative Schrödinger equations in the non-relativistic limit are discussed in detail.

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7

Nathiya, N., and C. Amulya Smyrna. "Infinite Schrödinger networks." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 4 (December 2021): 640–50. http://dx.doi.org/10.35634/vm210408.

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Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

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8

Modanese, Giovanni. "Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current." Mathematics 6, no. 9 (September 4, 2018): 155. http://dx.doi.org/10.3390/math6090155.

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In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.

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9

SHI, HAIPING, and YUANBIAO ZHANG. "Existence results of solitons in discrete non-linear Schrödinger equations." European Journal of Applied Mathematics 27, no. 5 (February 15, 2016): 726–37. http://dx.doi.org/10.1017/s0956792516000036.

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The discrete non-linear Schrödinger equation is one of the most important inherently discrete models, having a crucial role in the modelling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology. In this paper, a class of discrete non-linear Schrödinger equations are considered. Using critical point theory in combination with periodic approximations, we establish some new sufficient conditions on the existence results for solitons of the equation. The classical Ambrosetti–Rabinowitz superlinear condition is improved.

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10

Vinokurov, N. A. "Derivation of the non-stationary Schrödinger equation from the stationary one." SIBERIAN JOURNAL OF PHYSICS 18, no. 3 (February 22, 2024): 104–12. http://dx.doi.org/10.25205/2541-9447-2023-18-3-104-112.

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A derivation of the time-dependent Schrödinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent Schrödinger equation. It is shown that the standard time-dependent Schrödinger equation can be obtained for the semiclassical clock only. For elucidation of the physical meaning of the equation obtained in this way, various types of clocks are discussed. In addition, the corresponding equation for the density matrix and formulas for the mean values of operators are derived.

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11

Yan, Zhenya. "Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (April 28, 2013): 20120059. http://dx.doi.org/10.1098/rsta.2012.0059.

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The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg–de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross–Pitaevskii equation in Bose–Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail.

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12

Vougalter, Vitali, and Vitaly Volpert. "Solvability conditions for some non-Fredholm operators." Proceedings of the Edinburgh Mathematical Society 54, no. 1 (November 30, 2010): 249–71. http://dx.doi.org/10.1017/s0013091509000236.

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AbstractWe obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger-type operators. One of the main results of the paper concerns solvability conditions for the equation –Δu+V(x)u–au=fwhere a ≥ 0. The conditions are formulated in terms of orthogonality of the functionfto the solutions of the homogeneous adjoint equation.

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13

Mazepa, E. A., and D. K. Ryaboshlikova. "Symptotic behavior of solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifolds." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 1 (February 12, 2024): 35–49. http://dx.doi.org/10.26907/0021-3446-2024-1-35-49.

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The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on non-compact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular the Laplace-Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary non-compact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.

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14

Haouam, Ilyas. "The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space." Symmetry 11, no. 2 (February 14, 2019): 223. http://dx.doi.org/10.3390/sym11020223.

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The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only the non-commutativity in phases with a non-interacting fields case leads to the non-commutative Schrödinger equation; thereafter, considering the non-commutativity in phase and space with an external electromagnetic field thus leads to extract a phase-space non-commutative Schrödinger–Pauli equation; there, we examined the effect of the non-commutativity in phase-space on the non-relativistic limit of the DKP equation. However, with both Bopp–Shift linear transformation through the Heisenberg-like commutation relations, and the Moyal–Weyl product, we introduced the non-commutativity in phase and space.

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15

Anker, Jean-Philippe, Stefano Meda, Vittoria Pierfelice, Maria Vallarino, and Hong-Wei Zhang. "Schrödinger equation on non-compact symmetric spaces." Journal of Differential Equations 356 (May 2023): 163–87. http://dx.doi.org/10.1016/j.jde.2023.02.003.

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16

Popivanov, Petar, and Angela Slavova. "Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form." Mathematics 12, no. 7 (March 27, 2024): 1003. http://dx.doi.org/10.3390/math12071003.

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In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u=be−ax2, a<0, a,b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u=be−ax22, respectively, for the third-order PDE, u=AeiΦ, where the amplitude b and the phase function a are complex-valued functions, A>0, and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.

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17

Felmer, Patricio, and César Torres. "Non-linear Schrödinger equation with non-local regional diffusion." Calculus of Variations and Partial Differential Equations 54, no. 1 (September 23, 2014): 75–98. http://dx.doi.org/10.1007/s00526-014-0778-x.

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18

Mina, Mattia, David F. Mota, and Hans A. Winther. "SCALAR: an AMR code to simulate axion-like dark matter models." Astronomy & Astrophysics 641 (September 2020): A107. http://dx.doi.org/10.1051/0004-6361/201936272.

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We present a new code, SCALAR, based on the high-resolution hydrodynamics and N-body code RAMSES, to solve the Schrödinger equation on adaptive refined meshes. The code is intended to be used to simulate axion or fuzzy dark matter models where the evolution of the dark matter component is determined by a coupled Schrödinger-Poisson equation, but it can also be used as a stand-alone solver for both linear and non-linear Schrödinger equations with any given external potential. This paper describes the numerical implementation of our solver and presents tests to demonstrate how accurately it operates.

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19

Semina, I., V. Semin, F. Petruccione, and A. Barchielli. "Stochastic Schrödinger Equations for Markovian and non-Markovian Cases." Open Systems & Information Dynamics 21, no. 01n02 (March 12, 2014): 1440008. http://dx.doi.org/10.1142/s1230161214400083.

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Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples — the damped harmonic oscillator and a two-level atom with homodyne photodetection. We then consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markovian process proposed. Finally, an analytical approximation technique is tested with the help of the stochastic simulation in a model of a dissipative qubit.

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20

Benseny, Albert, David Tena, and Xavier Oriols. "On the Classical Schrödinger Equation." Fluctuation and Noise Letters 15, no. 03 (September 2016): 1640011. http://dx.doi.org/10.1142/s0219477516400113.

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In this paper, the classical Schrödinger equation (CSE), which allows the study of classical dynamics in terms of wave functions, is analyzed theoretically and numerically. First, departing from classical (Newtonian) mechanics, and assuming an additional single-valued condition for the Hamilton’s principal function, the CSE is obtained. This additional assumption implies inherent non-classical features on the description of the dynamics obtained from the CSE: the trajectories do not cross in the configuration space. Second, departing from Bohmian mechanics and invoking the quantum-to-classical transition, the CSE is obtained in a natural way for the center of mass of a quantum system with a large number of identical particles. This quantum development imposes the condition of dealing with a narrow wave packet, which implicitly avoids the non-classical features mentioned above. We illustrate all the above points with numerical simulations of the classical and quantum Schrödinger equations for different systems.

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21

Larroche, O., M. Casanova, D. Pesme, and M. N. Bussac. "Soliton emission in the forced non-linear Schrödinger equation." Laser and Particle Beams 4, no. 3-4 (August 1986): 545–53. http://dx.doi.org/10.1017/s0263034600002226.

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The plasma waves generated by resonant absorption of light in the vicinity of the critical density of laser-produced plasmas, are modelled by a non-linear Schrödinger equation with additional terms accounting for the presence of a source and the inhomogeneity of the medium.We use an average lagrangian method to describe the behaviour of the solutions of this equation in the range of parameters where periodic soliton generation occurs. An iterating scheme describing the successive emission of solitons yields values for this range of parameters which are in reasonable agreement with those found from direct numerical simulations of the non-linear Schrödinger equation.

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22

Boulton, Lyonell, George Farmakis, and Beatrice Pelloni. "Beyond periodic revivals for linear dispersive PDEs." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2251 (July 2021): 20210241. http://dx.doi.org/10.1098/rspa.2021.0241.

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We study the phenomenon of revivals for the linear Schrödinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast to the case of the linear Schrödinger equation examined recently (which we develop further), we prove that, remarkably, the Airy equation does not generally exhibit revivals even for boundary conditions very close to periodic. We also describe a new, weaker form of revival phenomena, present in the case of certain Robin-type boundary conditions for the linear Schrödinger equation. In this weak revival, the dichotomy between the behaviour of the solution at rational and irrational times persists, but in contrast to the classical periodic case, the solution is not given by a finite superposition of copies of the initial condition.

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23

Zhao, Shiyin, Yufeng Zhang, and Jian Zhou. "Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations." Symmetry 14, no. 2 (February 17, 2022): 402. http://dx.doi.org/10.3390/sym14020402.

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By introducing a 3×3 matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral–nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the 3×3 matrix Lie algebra into a 2×2 matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral–nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schrödinger integrable system and a nonlocal nonlinear Schrödinger equation, which indicates that we unite the KdV equation and the nonlinear Schrödinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.

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24

CAZENAVE, THIERRY, FLÁVIO DICKSTEIN, and FRED B. WEISSLER. "NON-REGULARITY IN HÖLDER AND SOBOLEV SPACES OF SOLUTIONS TO THE SEMILINEAR HEAT AND SCHRÖDINGER EQUATIONS." Nagoya Mathematical Journal 226 (September 9, 2016): 44–70. http://dx.doi.org/10.1017/nmj.2016.35.

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In this paper, we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $f(u)=\unicode[STIX]{x1D706}|u|^{\unicode[STIX]{x1D6FC}}u$. We show that low regularity of $f$ (i.e., $\unicode[STIX]{x1D6FC}>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_{t}=f(w)$. This yields, in particular, an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields ill-posedness results for the nonlinear Schrödinger equation in certain $H^{s}$-spaces, which depend on the smallness of $\unicode[STIX]{x1D6FC}$ rather than the scaling properties of the equation. Our second method is to consider the semilinear equation as a perturbation of the linear equation via Duhamel’s formula. This yields, in particular, that if $\unicode[STIX]{x1D6FC}$ is sufficiently small and $N$ is sufficiently large, then the nonlinear heat equation is ill-posed in $H^{s}(\mathbb{R}^{N})$ for all $s\geqslant 0$.

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25

Goubet, Olivier, and Wided Kechiche. "Uniform attractor for non-autonomous nonlinear Schrödinger equation." Communications on Pure and Applied Analysis 10, no. 2 (December 2010): 639–51. http://dx.doi.org/10.3934/cpaa.2011.10.639.

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26

Ferkous, N., A. Bounames, and M. Maamache. "Time-dependent Schrödinger equation with non-central potentials." Physica Scripta 88, no. 3 (August 7, 2013): 035001. http://dx.doi.org/10.1088/0031-8949/88/03/035001.

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27

Tang, X. H. "NON-NEHARI MANIFOLD METHOD FOR SUPERLINEAR SCHRÖDINGER EQUATION." Taiwanese Journal of Mathematics 18, no. 6 (November 2014): 1957–79. http://dx.doi.org/10.11650/tjm.18.2014.3541.

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28

Mosley, Shaun N. "Non-dispersive wavepacket solutions of the Schrödinger equation." Journal of Physics A: Mathematical and Theoretical 41, no. 26 (June 11, 2008): 265305. http://dx.doi.org/10.1088/1751-8113/41/26/265305.

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29

Burq, Nicolas, and Maciej Zworski. "Instability for the Semiclassical Non-linear Schrödinger Equation." Communications in Mathematical Physics 260, no. 1 (August 2, 2005): 45–58. http://dx.doi.org/10.1007/s00220-005-1402-x.

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30

Chiappinelli, R., and D. E. Edmunds. "Eigenvalue asymptotics and a non-linear Schrödinger equation." Israel Journal of Mathematics 81, no. 1-2 (February 1993): 179–92. http://dx.doi.org/10.1007/bf02761304.

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31

Joubert-Doriol, Loïc, Ilya G. Ryabinkin, and Artur F. Izmaylov. "Non-stochastic matrix Schrödinger equation for open systems." Journal of Chemical Physics 141, no. 23 (December 21, 2014): 234112. http://dx.doi.org/10.1063/1.4903829.

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32

Ván, Peter, and Tamás Fülöp. "Weakly non-local fluid mechanics: the Schrödinger equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2066 (December 9, 2005): 541–57. http://dx.doi.org/10.1098/rspa.2005.1588.

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A weakly non-local extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit, the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of the Schrödinger equation (stochastic, Fisher information, exact uncertainty) is clarified.

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Cole, Eric A. B., Tobias Boettcher, and Christopher M. Snowden. "Two-dimensional Modelling of HEMTs Using Multigrids with Quantum Correction." VLSI Design 8, no. 1-4 (January 1, 1998): 29–34. http://dx.doi.org/10.1155/1998/61608.

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The two-dimensional multi-layered HEMT is modelled isothermally by solving the Poisson and current continuity equations consistently with the Schrödinger equation. A multigrid method is used on the Poisson and current continuity equations while the electron density is calculated at each level by solving the Schrödinger equation in onedimensional slices perpendicular to the layer structure. A correction factor is introduced which enables relatively accurate solutions to be obtained using a low number of eigensolutions. A novel method for discretising the current density which can be generalised to the non-isothermal case is described. Results are illustrated using a two layer AlGaAs-GaAs HEMT.

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Dinh, Van Duong. "Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions." Journal of Hyperbolic Differential Equations 18, no. 01 (March 2021): 1–28. http://dx.doi.org/10.1142/s0219891621500016.

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We consider a class of [Formula: see text]-supercritical inhomogeneous nonlinear Schrödinger equations in two dimensions [Formula: see text] where [Formula: see text] and [Formula: see text]. Using a new approach of Arora et al. [Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 148 (2020) 1653–1663], we show the energy scattering for the equation with radially symmetric initial data. In the focusing case, our result extends the one of Farah and Guzmán [Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc. (N.S.) 51 (2020) 449–512] to the whole range of [Formula: see text] where the local well-posedness is available. In the defocusing case, our result extends the one in [V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ. 19(2) (2019) 411–434], where the energy scattering for non-radial initial data was established in dimensions [Formula: see text].

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35

Fernández, Francisco M. "Comment on “`Striped' rectangular rigid box with Hermitian and non-Hermitian PT symmetric potentials” [J. Math. Phys. 62, 102102 (2021)]." Journal of Mathematical Physics 63, no. 3 (March 1, 2022): 034101. http://dx.doi.org/10.1063/5.0082599.

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We show that the two-dimensional quantum-mechanical model proposed by Kulkarni and Pathak [J. Math. Phys. 62, 102102 (2021)] is separable. Consequently, instead of solving a two-dimensional Schrödinger equation, it is sufficient to solve two one-dimensional eigenvalue equations, one of which is exactly solvable. The solution to the remaining equation can be given in terms of a transcendental equation.

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36

Delgado, Rafael L., Sebastian Steinbeißer, Michael Strickland, and Johannes H. Weber. "QuantumFDTD - A computational framework for the relativistic Schrödinger equation." EPJ Web of Conferences 274 (2022): 04004. http://dx.doi.org/10.1051/epjconf/202227404004.

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We extend the publicly available quantumfdtd code. It was originally intended for solving the time-independent three-dimensional Schrödinger equation via the finite-difference time-domain (FDTD) method and for extracting the ground, first, and second excited states. We (a) include the case of the relativistic Schrödinger equation and (b) add two optimized FFT-based kinetic energy terms for the non-relativistic case. All the three new kinetic terms are computed using Fast Fourier Transform (FFT).We release the resulting code as version 3 of quantumfdtd. Finally, the code now supports arbitrary external filebased potentials and the option to project out distinct parity eigenstates from the solutions. Our goal is quark models used for phenomenological descriptions of QCD bound states, described by the three-dimensional Schrödinger equation. However, we target any field where solving either the non-relativistic or the relativistic three-dimensional Schrödinger equation is required.

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37

Gürbüz, Nevin. "Moving non-null curves according to Bishop frame in Minkowski 3-space." International Journal of Geometric Methods in Modern Physics 12, no. 05 (May 2015): 1550052. http://dx.doi.org/10.1142/s0219887815500528.

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In this paper, we introduce three new transformations and establish connections between moving non-null curves and soliton equations according to Bishop frame in Minkowski 3-space. Later we find formulas for differentials of these three new transformations associated with the nonlinear heat system and repulsive type nonlinear Schrödinger equation.

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38

Losev, A. G., and V. V. Filatov. "Liouville type theorems for solutions of semilinear equations on non-compact Riemannian manifolds." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 4 (December 2021): 629–39. http://dx.doi.org/10.35634/vm210407.

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It is proved that the Liouville function associated with the semilinear equation $\Delta u -g(x,u)=0$ is identical to zero if and only if there is only a trivial bounded solution of the semilinear equation on non-compact Riemannian manifolds. This result generalizes the corresponding result of S.A. Korolkov for the case of the stationary Schrödinger equation $ \Delta u-q (x) u = 0$. The concept of the capacity of a compact set associated with the stationary Schrödinger equation is also introduced and it is proved that if the capacity of any compact set is equal to zero, then the Liouville function is identically zero.

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39

Pashaev, Oktay K., and Jyh-Hao Lee. "Black holes and solitons of the quantized dispersionless NLS and DNLS equations." ANZIAM Journal 44, no. 1 (July 2002): 73–81. http://dx.doi.org/10.1017/s1446181100007926.

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AbstractThe classical dynamics of non-relativistic particles are described by the Schrödinger wave equation, perturbed by quantum potential nonlinearity. Quantization of this dispersionless equation, implemented by deformation of the potential strength, recovers the standard Schrödinger equation. In addition, the classically forbidden region corresponds to the Planck constant analytically continued to pure imaginary values. We apply the same procedure to the NLS and DNLS equations, constructing first the corresponding dispersionless limits and then adding quantum deformations. All these deformations admit the Lax representation as well as the Hirota bilinear form. In the classically forbidden region we find soliton resonances and black hole phenomena. For deformed DNLS the chiral solitons with single event horizon and resonance dynamics are constructed.

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40

Mihálka, Zsuzsanna É., Ádám Margócsy, Ágnes Szabados, and Péter R. Surján. "On the variational principle for the non-linear Schrödinger equation." Journal of Mathematical Chemistry 58, no. 1 (November 23, 2019): 340–51. http://dx.doi.org/10.1007/s10910-019-01082-5.

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AbstractWhile variation of the energy functional yields the Schrödinger equation in the usual, linear case, no such statement can be formulated in the general nonlinear situation when the Hamiltonian depends on its eigenvector. In this latter case, as we illustrate by sample numerical calculations, the points of the energy expectation value hypersurface where the eigenvalue equation is satisfied separate from those where the energy is stationary. We show that the variation of the energy at the eigensolution is determined by a generalized Hellmann–Feynman theorem. Functionals, other than the energy, can, however be constructed, that result the nonlinear Schrödinger equation upon setting their variation zero. The second centralized moment of the Hamiltonian is one example.

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41

Elton, Daniel M. "Decay rates at infinity for solutions to periodic Schrödinger equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 3 (January 30, 2019): 1113–26. http://dx.doi.org/10.1017/prm.2018.87.

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AbstractWe consider the equation Δu = Vu in the half-space ${\open R}_ + ^d $, d ⩾ 2 where V has certain periodicity properties. In particular, we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation Δu = Vu is studied as part of a broader class of elliptic evolution equations.

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42

Vinita and Santanu Saha Ray. "Optical soliton group invariant solutions by optimal system of Lie subalgebra with conservation laws of the resonance nonlinear Schrödinger equation." Modern Physics Letters B 34, no. 35 (August 25, 2020): 2050402. http://dx.doi.org/10.1142/s0217984920504023.

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In this article, the resonance nonlinear Schrödinger equation is studied, which elucidates the propagation of one-dimensional long magnetoacoustic waves in a cold plasma, dynamic of solitons and Madelung fluids in various nonlinear systems. The Lie symmetry analysis is used to achieve the invariant solution and similarity reduction of the resonance nonlinear Schrödinger equation. The infinitesimal generators, symmetry groups, commutator table and adjoint table have been obtained by the aid of invariance criterion of Lie symmetry. Also, one-dimensional system of subalgebra is constructed with the help of adjoint representation of a Lie group on its Lie algebra. By one-dimensional optimal subalgebra, the main equations are reduced to ordinary differential equations and their invariant solutions are provided. The general conservation theorem has been used to establish a set of non-local and non-trivial conservation laws.

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43

JOVANOVIĆ, D., R. FEDELE, F. TANJIA, S. DE NICOLA, and M. BELIĆ. "Coherent quantum hollow beam creation in a plasma wakefield accelerator." Journal of Plasma Physics 79, no. 4 (February 21, 2013): 397–403. http://dx.doi.org/10.1017/s0022377813000111.

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AbstractA theoretical investigation of the propagation of a relativistic electron (or positron) particle beam in an overdense magnetoactive plasma is carried out within a fluid plasma model, taking into account the individual quantum properties of beam particles. It is demonstrated that the collective character of the particle beam manifests mostly through the self-consistent macroscopic plasma wakefield created by the charge and the current densities of the beam. The transverse dynamics of the beam–plasma system is governed by the Schrödinger equation for a single-particle wavefunction derived under the Hartree mean field approximation, coupled with a Poisson-like equation for the wake potential. These two coupled equations are subsequently reduced to a nonlinear, non-local Schrödinger equation and solved in a strongly non-local regime. An approximate Glauber solution is found analytically in the form of a Hermite–Gauss ring soliton. Such non-stationary (‘breathing’ and ‘wiggling’) coherent structure may be parametrically unstable and the corresponding growth rates are estimated analytically.

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Bruce, S. A. "The Schrödinger Equation and Negative Energies." Zeitschrift für Naturforschung A 73, no. 12 (November 27, 2018): 1129–35. http://dx.doi.org/10.1515/zna-2018-0321.

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AbstractIt is known that there is no room for anti-particles within the Schrödinger regime in quantum mechanics. In this article, we derive a (non-relativistic) Schrödinger-like wave equation for a spin-$1/2$ free particle in 3 + 1 space-time dimensions, which includes both positive- and negative-energy eigenstates. We show that, under minimal interactions, this equation is invariant under $\mathcal{P}\mathcal{T}$ and 𝒞 discrete symmetries. An immediate consequence of this is that the particle exhibits Zitterbewegung (‘trembling motion’), which arises from the interference of positive- and negative-energy wave function components.

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Feng, Lili, Fajun Yu, and Li Li. "Darboux Transformation for Coupled Non-Linear Schrödinger Equation and Its Breather Solutions." Zeitschrift für Naturforschung A 72, no. 1 (January 1, 2017): 9–15. http://dx.doi.org/10.1515/zna-2016-0342.

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AbstractStarting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics.

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46

MA, LI, XIANFA SONG, and LIN ZHAO. "ON GLOBAL ROUGH SOLUTIONS TO A NON-LINEAR SCHRÖDINGER SYSTEM." Glasgow Mathematical Journal 51, no. 3 (September 2009): 499–511. http://dx.doi.org/10.1017/s0017089509005138.

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AbstractThe non-linear Schrödinger systems arise from many important physical branches. In this paper, employing the ‘I-method’, we prove the global-in-time well-posedness for a coupled non-linear Schrödinger system in Hs(n) when n = 2, s > 4/7 and n = 3, s > 5/6, respectively, which extends the results of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao (Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math Res. Lett. 9, 2002, 659–682) to the system.

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HO, CHOON-LIN, and YUTAKA HOSOTANI. "ANYON EQUATION ON A TORUS." International Journal of Modern Physics A 07, no. 23 (September 20, 1992): 5797–831. http://dx.doi.org/10.1142/s0217751x92002647.

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Starting from the quantum field theory of nonrelativistic matter on a torus interacting with Chern-Simons gauge fields, we derive the Schrödinger equation for an anyon system. The nonintegrable phases of the Wilson line integrals on a torus play an essential role. In addition to generating degenerate vacua, they enter in the definition of a many-body Schrödinger wave function in quantum mechanics, which can be defined as a regular function of the coordinates of anyons. It obeys a non-Abelian representation of the braid group algebra, being related to Einarsson’s wave function by a singular gauge transformation.

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48

SCHULZE-HALBERG, AXEL. "EXACT WAVE FUNCTIONS AND ENERGIES OF A NON-RELATIVISTIC FREE QUANTUM PARTICLE ON THE SURFACE OF A DEGENERATE TORUS." Modern Physics Letters A 19, no. 23 (July 30, 2004): 1759–66. http://dx.doi.org/10.1142/s0217732304014938.

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We study the non-relativistic Schrödinger equation for a free quantum particle constrained to the surface of a degenerate torus, parametrized by its polar and azimuthal angle. On restricting to wave functions that depend on the polar angle only, the Schrödinger equation becomes exactly-solvable. We compute its physical solutions (continuous, normalizable and 2π-periodic) and the associated energies in closed form.

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Duo, Siwei, and Yanzhi Zhang. "Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well." Communications in Computational Physics 18, no. 2 (July 30, 2015): 321–50. http://dx.doi.org/10.4208/cicp.300414.120215a.

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AbstractIn this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.

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Sakhabutdinov, Airat Zh, Vladimir I. Anfinogentov, Oleg G. Morozov, Vladimir A. Burdin, Anton V. Bourdine, Artem A. Kuznetsov, Dmitry V. Ivanov, Vladimir A. Ivanov, Maria I. Ryabova, and Vladimir V. Ovchinnikov. "Numerical Method for Coupled Nonlinear Schrödinger Equations in Few-Mode Fiber." Fibers 9, no. 1 (January 2, 2021): 1. http://dx.doi.org/10.3390/fib9010001.

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This paper discusses novel approaches to the numerical integration of the coupled nonlinear Schrödinger equations system for few-mode wave propagation. The wave propagation assumes the propagation of up to nine modes of light in an optical fiber. In this case, the light propagation is described by the non-linear coupled Schrödinger equation system, where propagation of each mode is described by own Schrödinger equation with other modes’ interactions. In this case, the coupled nonlinear Schrödinger equation system (CNSES) solving becomes increasingly complex, because each mode affects the propagation of other modes. The suggested solution is based on the direct numerical integration approach, which is based on a finite-difference integration scheme. The well-known explicit finite-difference integration scheme approach fails due to the non-stability of the computing scheme. Owing to this, here we use the combined explicit/implicit finite-difference integration scheme, which is based on the implicit Crank–Nicolson finite-difference scheme. It ensures the stability of the computing scheme. Moreover, this approach allows separating the whole equation system on the independent equation system for each wave mode at each integration step. Additionally, the algorithm of numerical solution refining at each step and the integration method with automatic integration step selection are used. The suggested approach has a higher performance (resolution)—up to three times or more in comparison with the split-step Fourier method—since there is no need to produce direct and inverse Fourier transforms at each integration step. The key advantage of the developed approach is the calculation of any number of modes propagated in the fiber.

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