Academic literature on the topic 'Multi-Mode non-Linear Schrödinger equation'
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Journal articles on the topic "Multi-Mode non-Linear Schrödinger equation"
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BOGOLUBOV, N. N., M. Yu. RASULOVA, and I. A. TISHABOEV. "QUANTUM DYNAMICS OF TWO-LEVEL ATOMS INTERACTING WITH AN ELECTROMAGNETIC FIELD." International Journal of Modern Physics B 28, no. 08 (February 24, 2014): 1450060. http://dx.doi.org/10.1142/s021797921450060x.
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We consider the dynamics of a system consisting of N two-level atoms interacting with a multi-mode cavity field. For the given system, the generalized kinetic equation (GKE) is obtained and conditions are given under which its solution is reduced to solution of a linear equation, and of the one-dimensional nonlinear Schrödinger equation, respectively.
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Misra, Shikha, Sanjay K. Mishra, and P. Brijesh. "Coaxial propagation of Laguerre–Gaussian (LG) and Gaussian beams in a plasma." Laser and Particle Beams 33, no. 1 (March 2015): 123–33. http://dx.doi.org/10.1017/s0263034615000142.
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AbstractThis paper investigates the non-linear coaxial (or coupled mode) propagation of Laguerre–Gaussian (LG) (in particular L01 mode) and Gaussian electromagnetic (em) beams in a homogeneous plasma characterized by ponderomotive and relativistic non-linearities. The formulation is based on numerical solution of non-linear Schrödinger wave equation under Jeffreys–Wentzel–Kramers–Brillouin approximation, followed by paraxial approach applicable in the vicinity of intensity maximum of the beams. A set of coupled differential equations for spot size (beam width) and phase evolution with space corresponding to coupled mode has been derived and numerically solved to determine the propagation dynamics. Using focusing equation a critical condition describing the self-trapped (i.e., spatial soliton) mode of laser beam propagation in the plasma has been discussed; as a consequence oscillatory focusing/defocusing of the beams in coupled mode propagation have been analyzed and presented graphically. As an important outcome, significant enhancement in the intensity of LG beam is noticed when it is coupled with the Gaussian mode.
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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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Zhu, Junyan, Jiang Cao, Chen Song, Bo Li, and Zhengsheng Han. "Numerical investigation on the convergence of self-consistent Schrödinger-Poisson equations in semiconductor device transport simulation." Nanotechnology 35, no. 31 (May 17, 2024): 315001. http://dx.doi.org/10.1088/1361-6528/ad4558.
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Abstract Semiconductor devices at the nanoscale with low-dimensional materials as channels exhibit quantum transport characteristics, thereby their electrical simulation relies on the self-consistent solution of the Schrödinger-Poisson equations. While the non-equilibrium Green’s function (NEGF) method is widely used for solving this quantum many-body problem, its high computational cost and convergence challenges with the Poisson equation significantly limit its applicability. In this study, we investigate the stability of the NEGF method coupled with various forms of the Poisson equation, encompassing linear, analytical nonlinear, and numerical nonlinear forms Our focus lies on simulating carbon nanotube field-effect transistors (CNTFETs) under two distinct doping scenarios: electrostatic doping and ion implantation doping. The numerical experiments reveal that nonlinear formulas outperform linear counterpart. The numerical one demonstrates superior stability, particularly evident under high bias and ion implantation doping conditions. Additionally, we investigate different approaches for presolving potential, leveraging solutions from the Laplace equation and a piecewise guessing method tailored to each doping mode. These methods effectively reduce the number of iterations required for convergence.
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Dabas, Bhawana, Jivesh Kaushal, Monika Rajput, and R. K. Sinha. "Study of Self Phase Modulation in Chalcogenide Glass Photonic Crystal Fiber." Applied Mechanics and Materials 110-116 (October 2011): 53–56. http://dx.doi.org/10.4028/www.scientific.net/amm.110-116.53.
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In this paper, Self Phase Modulation (SPM) in chalcogenide As2Se3glass Photonic Crystal Fiber (PCF) is numerically studied by combining the fully vectorial effective index method (FVEIM) and Split Step Fourier Method (SSFM). The FVEIM is used to calculate the variation of effective refractive index of guided mode (neff), effective area (Aeff), dispersion and non-linear coefficient (γ) with wavelength for different designs of chalcogenide As2Se3PCF. The FVEIM solves the vector wave equations and SSFM solves non linear Schrödinger Equation (NLSE) for the different designing parameter of As2Se3PCF. In case of Self Phase Modulation (SPM), spectral width of the obtained output pulse at d/Λ=0.7 is 1.5 times greater than width of the output pulse obtained at d/L=0.3 using SSFM. Thus we can get the desired spectral broadening just by tailoring the design parameters of the PCF.
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Niedda, Jacopo, Luca Leuzzi, and Giacomo Gradenigo. "Intensity pseudo-localized phase in the glassy random laser." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 5 (May 1, 2023): 053302. http://dx.doi.org/10.1088/1742-5468/acd2c4.
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Abstract Evidence of an emergent pseudo-localized phase characterizing the low-temperature replica symmetry breaking phase of the complex disordered models for glassy light is provided in the mode-locked random laser model. A pseudo-localized phase corresponds to a state in which the intensity of light modes is neither equipartited among all modes nor strictly condensed on few of them. Such a hybrid phase, recently characterized as a finite size effect in other models, such as the discrete non-linear Schrödinger equation, in the low temperature phase of the glassy random laser appears to be robust in the limit of large size.
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Jahan, Sharmin, Rubaiya Khondoker Shikha, Abdul Mannan, and A. A. Mamun. "Modulational Instability of Ion-Acoustic Waves in Pair-Ion Plasma." Plasma 5, no. 1 (December 29, 2021): 1–11. http://dx.doi.org/10.3390/plasma5010001.
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The modulational instability (MI) of ion-acoustic waves (IAWs) is examined theoretically in a four-component plasma system containing inertialess electrons featuring a non-thermal, non-extensive distribution, iso-thermal positrons, and positively as well as negatively charged inertial ions. In this connection, a non-linear Schrödinger equation (NLSE), which dominates the conditions for MI associated with IAWs, is obtained by using the reductive perturbation method. The numerical analysis of the NLSE reveals that the increment in non-thermality leads to a more unstable state, whereas the enhancement in non-extensivity introduces a less unstable state. It also signifies the bright (dark) ion-acoustic (IA) envelope solitons mode in the unstable (stable) domain. The conditions for MI and its growth rate in the unstable regime of the IAWs are vigorously modified by the different plasma parameters (viz., non-thermal, non-extensive q-distributed electron, iso-thermal positron, the ion charge state, the mass of the ion and positron, non-thermal parameter α, the temperature of electron and positron, etc.). Our findings may supplement and add to prior research in non-thermal, non-extensive electrons and iso-thermal positrons that can co-exist with positive as well as negative inertial ions.
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Odegov, N. A., and I. S. Baleyev. "A NUMERICAL-ANALYTICAL METHOD FOR THE SYNTHESIS OF OPTIMAL IRREGULAR DWDM FREQUENCY PLANS." Proceedings of the O.S. Popov ОNAT 1, no. 2 (December 31, 2020): 70–81. http://dx.doi.org/10.33243/2518-7139-2020-1-2-70-81.
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The possibilities of increasing the throughput of fiber-optic transmission systems by using an uneven frequency grid are investigated. In this case, the bandwidth of each channel is selected so that the transmission rate is the same for all channels. In this work, both linear and some nonlinear effects are taken into account, leading to the distortion of the optical pulse. Simulation of nonlinear effects is based on a model in the form of a generalized nonlinear Schrödinger equation. The developed program provides modeling of linear and nonlinear distortions for the DWDM range (from 1460 to 1625 nm). The characteristics of different types of optical fiber are also provided. Non-linear effects are investigated for NZ DSF-type dispersion-shifted fiber. Differential equations are solved by the method of splitting according to physical factors. It is shown that for this type of fiber at distances of 100 km and more, a soliton transmission mode appears. In this case, the frequency band of the soliton regime can reach significant values (up to 5 THz) at typical lengths of the regeneration sections of the order of 100-300 km. A method for calculating the bandwidth of uneven frequency plans is proposed. This method has been tested for a 15 THz band. A specific example of calculations is given for the comparison base in the form of a uniform frequency plan with a single channel bandwidth of 50 GHz. It is shown that optimal non-uniform frequency plans can significantly increase the throughput of DWDM systems: in the given example, approximately 3 times. At the same time, the complexity of the equipment increases slightly.
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REZNIK, G. M., V. ZEITLIN, and M. BEN JELLOUL. "Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model." Journal of Fluid Mechanics 445 (October 16, 2001): 93–120. http://dx.doi.org/10.1017/s002211200100550x.
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We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.
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Muhammad, Zahid, Ubaid Ullah Khalil, Anees Khan, Tanweer Ahmed, Waqas Khan, and Samra Naz. "Design Optimization of Fiber Laser for Generation of Femtosecond Optical Pulses." Scholars Journal of Physics, Mathematics and Statistics 11, no. 08 (August 30, 2024): 89–100. http://dx.doi.org/10.36347/sjpms.2024.v11i08.002.
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The precise coordination of dispersion management, temperature control, mode-locking mechanisms, and gain medium qualities are required in the design and optimization of fiber laser cavities for the generation of femtosecond pulses. The performance and capacities of femtosecond fiber lasers are being enhanced by developments in these fields, opening up new uses for them. The main focus of this research work was to design a lasing cavity for the generation of femtosecond optical pulses. So, we designed a laser cavity having six segments with a total length of 5.4 meters. The first segment is a 100-centimeter-long single mode fiber (SMF), the second one is an active fiber (Yb doped fiber) which is 40-centimeter long, and the third segment is a 70-centimeter-long SMF.A 130 cm free space region(cavity) makes up the fourth segment, which include a collimator, mirror, grating, half wave plate, quarter wave plate, isolator, and polarized beam splitter (PBS). Single-mode fibers of 80 cm and 120 cm in length comprises the fifth and sixth sections respectively. The calculated repetition rate of the laser cavity is 37.06 MHz.. We used the software "Ultrafast Pulse Propagator Version 3.0.0", created by Bilkent University in Ankara, Turkey, to accomplished this task. This application was initially created to examine fiber links, mode-locking, and fiber amplification. The physics of the code is based on the generalized non-linear Schrödinger equation, which includes high order dispersion, bandwidth, gain with restriction, saturation loss, and saturation absorption. For data visualization, this software uses FORTRAN code and MATLAB algorithms. The pulse width increased linearly from 1.2809 to 1.3227 Ps and the spectral width decreased linearly from 2.3841 to 2.2561 nm when the Yb doped fiber's length were changed between 5 and 50 cm. 94729 fs2 is the total dispersion from the 5.4 m long lasing cavity. In the end, we determined the pulses' repetition rate, which came out to be 37.0
Dissertations / Theses on the topic "Multi-Mode non-Linear Schrödinger equation"
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Nguyen, Tien Vinh. "Construction of dynamics with strongly interacting for non-linear dispersive PDE (Partial differential equation)." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLX024/document.
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Cette thèse est consacrée à l’étude des propriétés dynamiques des solutions de type soliton d'équations aux dérivées partielles (EDP) dispersives non linéaires. `A travers des exemples-type de telles équations, l'équation de Schrödinger non-linéaire (NLS), l'équation de Korteweg-de Vries généralisée (gKdV) et le système de Schrödinger, on traite du comportement des solutions convergeant en temps grand vers des sommes de solitons (multi-solitons). Dans un premier temps, nous montrons que dans une configuration symétrique, avec des interactions fortes, le comportement de séparation des solitons logarithmique en temps est universel à la fois dans le cas sous-critique et sur-critique pour (NLS). Ensuite, en adaptant les techniques précédentes à l'équation (gKdV), nous prouvons un résultat similaire de l'existence de multi-solitons avec distance relative logarithmique; pour (gKdV), les solitons sont répulsifs dans le cas sous-critique et attractifs dans le cas sur-critique. Finalement, nous identifions un nouveau régime de distance logarithmique où les solitons sont non-symétriques pour le système de Schrödinger non-intégrable; une telle solution n'existe pas dans le cas intégrable pour le système et pour (NLS)This thesis deals with long time dynamics of soliton solutions for nonlinear dispersive partial differential equation (PDE). Through typical examples of such equations, the nonlinear Schrödinger equation (NLS), the generalized Korteweg-de Vries equation (gKdV) and the coupled system of Schrödinger, we study the behavior of solutions, when time goes to infinity, towards sums of solitons (multi-solitons). First, we show that in the symmetric setting, with strong interactions, the behavior of logarithmic separation in time between solitons is universal in both subcritical and supercritical case. Next, adapting previous techniques to (gKdV) equation, we prove a similar result of existence of multi-solitons with logarithmic relative distance; for (gKdV), the solitons are repulsive in the subcritical case and attractive in the supercritical case. Finally, we identify a new logarithmic regime where the solitons are non-symmetric for the non-integrable coupled system of Schrödinger; such solution does not exist in the integrable case for the system and for (NLS)
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Gaudillat, Valentine. "Étude du mélange à quatre ondes sensible à la phase dans les fibres faiblement multimodes." Electronic Thesis or Diss., Université de Rennes (2023-....), 2024. http://www.theses.fr/2024URENS028.
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Au cours des dernières années, le besoin en débit des télécommunications a considérablement augmenté. Pour maintenir une avance significative, il est essentiel d’améliorer les réseaux existants et de développer de nouvelles infrastructures plus performantes. Ainsi, les réseaux du futur pourraient être constitués de fibre faiblement multimode afin d’augmenter le nombre de canaux indépendants dans une même fibre. Il faudrait alors transférer les fonctions optiques déjà démontrées dans les réseaux actuels telles que la conversion de fréquence ou la régénération de phase. Cette thèse étudie numériquement et expérimentalement le mélange à quatre ondes sensible et insensible à la phase dans les fibres faiblement multimodes. Les simulations présentées dans cette thèse sont basées sur l’équation non-linéaire de Schrödinger multimode implémentée par une méthode de split-step Fourier. Les simulations ont démontré que la régénération de phase intra- ou inter-modale serait possible. Expérimentalement, la fibre utilisée n’a pas permis de mettre en œuvre du mélange à quatre ondes suffisamment efficace pour réaliser cette fonction optique. Cependant, pour la première fois à notre connaissance, nous avons démontré expérimentalement du mélange à quatre ondes sensible à la phase dans les modes LP01 et LP11 d’une fibre faiblement multimodeIn recent years, the demand for bandwidth in telecommunications has significantly increased. To maintain a considerable lead, it is essential to improve existing networks and develop more efficient new infrastructures. Consequently, the networks of the future could be composed of few-mode fibers to increase the number of independent channels within the same fiber. It would then be necessary transferring optical functions, already demonstrated in current networks such as frequency conversion or phase regeneration. This thesis studies both numerically and experimentally phase-sensitive and phase-insensitive four-wave mixing in few-mode fibers. The simulations presented in this thesis are based on the multimode nonlinear Schrödinger equation implemented by a split-step Fourier method. The simulations have shown that intra- or inter-modal phase regeneration could be possible. Experimentally, the fiber used did not allow efficient implementation of four-wave mixing to perform this optical function. However, for the first time to our knowledge, we have experimentally demonstrated phase-sensitive four-wave mixing in the LP01 and LP11 modes of a few-mode fiber
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Sun, Ruoci. "Comportement en grand temps et intégrabilité de certaines équations dispersives sur l'espace de Hardy Long time behavior of the NLS-Szegö equation Traveling waves of the quintic focusing NLS-Szegö equation Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds." Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASS111.
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On s'intéresse dans cette thèse à trois modèles d'équations hamiltoniennes dispersives non linéaires : l'équation de Schrödinger cubique défocalisante sur le cercle, filtrée par le projecteur de Szegö, qui enlève tous les modes de Fourier strictement négatifs (NLS--Szegö cubique), l'équation de Schrödinger quintique focalisante filtrée par le projecteur de Szegö sur la droite (NLS--Szegö quintique) et l'équation de Benjamin--Ono (BO) sur la droite. Comme pour les deux modèles précédents, l'équation de BO peut encore s'écrire sous la forme d'une équation de Schrödinger quadratique filtrée par le projecteur de Szegö. Ces trois modèles nous donnent l'occasion d'étudier les propriétés qualitatives de certaines ondes progressives, le phénomène de croissance des normes de Sobolev, le phénomène de diffusion non linéaire et certaines propriétés d'intégrabilité de systèmes dynamiques hamiltoniens. Le but de cette thèse est de comprendre l'influence des opérateurs de Szegö (non locaux) sur les équations de type Schrödinger, et d'adapter les outils liés à l'espace de Hardy sur le cercle et sur la droite. On applique aussi la méthode de forme normale de Birkhoff, l'argument de concentration--compacité, qui est précisé à travers le théorème de d'ecomposition en profils, et la transformée spectrale inverse pour résoudre ces problèmes. Dans le troisième modèle, la théorie de l'intégrabilité permet de faire le lien avec certains aspects algébriques et géométriquesWe are interested in three non linear dispersive Hamiltonian equations: the defocusing cubic Schrödinger equation filtered by the Szegö projector on the torus that cancels every negative Fourier modes, leading to the cubic NLS--Szegö equation on the torus; the focusing quintic Schrödinger equation, which is filtered by the Szegö projector on the line, leading to the quintic NLS--Szegö equation on the line and the Benjamin--Ono (BO) equation on the line. Similarly to the other two models, the BO equation on the line can be written as a quadratic Schrödinger-type equation that is filtered by the Szegö projector on the line. These three models allow us to study their qualitative properties of some traveling waves, the phenomenon of the growth of Sobolev norms, the phenomenon of non linear scattering and some properties about the complete integrability of Hamiltonian dynamical systems. The goal of this thesis is to investigate the influence of the Szegö projector on some one-dimensional Schrödinger-type equations and to adapt the tools of the Hardy space on the torus and on the line. We also use the Birkhoff normal form transform, the concentration--compactness argument, refined as the profile decomposition theorem, and the inverse spectral transform in order to solve these problems. In the third model, the integrability theory allows to establish the connection with some algebraic and geometric aspects
Books on the topic "Multi-Mode non-Linear Schrödinger equation"
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Levin, Frank S. Quantum Boxes, Stringed Instruments. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198808275.003.0008.
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Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.
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Horing, Norman J. Morgenstern. Superfluidity and Superconductivity. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0013.
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Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function<ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.
Conference papers on the topic "Multi-Mode non-Linear Schrödinger equation"
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Paré, C., M. Florjanczyk, and P. A. Bélanger. "Variational model of soliton interaction in two-mode fibers." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1990. http://dx.doi.org/10.1364/oam.1990.thy20.
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All-optical switching in a nonlinear directional coupler and the evolution of the state of polarization in a nonlinear biréfringent fiber represent two important practical problems of propagation in a two-mode, nonlinear dispersive system. Both cases are described by a nonintegrable system of coupled nonlinear Schrödinger equations. On the basis of numerical simulations,1 it has already been shown that solitons present important advantages over differently shaped pulses. To improve our physical insight into the dynamics resulting from the complex interplay between nonlinearity (self- and cross-phase modulation), dispersion, and linear coupling, we have developed an approximate all-analytical model of soliton interaction. An extension of a variational method2 has been used, and the results can be recast in a suggestive potential-well picture. Besides improved estimates of optical switching powers, a symmetry-breaking instability can easily be anticipated. A detailed comparison with numerical simulations, showing a good overall agreement between exact and approximate results, will be presented.
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Kaplan, A. E. "Bistable Optical Solitons." In Optical Bistability. Washington, D.C.: Optica Publishing Group, 1988. http://dx.doi.org/10.1364/obi.1988.wb.3.
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We showed recently [1] that a single soliton solution of the highly-nonlinear Schrödinger equation becomes multi-stable for a certain class of nonlinearities. This implies that more than one amplitude profile and propagation constant of a single soliton may exist for the same total power carried by the soliton. The multistable solitons exist only if the nonlinear component of the susceptibility as a function of intensity is either changing its sign or its derivative has a sufficiently sharp peak. This can result in such effects as bistable (or multistable, in general) self-trapping of light in nonlinear media as well as bistable propagation of soliton pulses in nonlinear optical fibers. Both of these processes may be described by the same nonlinear Schrödinger equation with a nonlinear term in the form Ef(|E|2) where f(|E|2) is an arbitrary function of the field intensity |E|2 with f(0) = 0. The so called cubic nonlinear Schrödinger equation with f(|E|2) proportional |E|2, corresponds to Kerr-nonlinearity in optical propagation and does not give rise to bistable solitons). By evaluating the propagation constant of a solitary solution δ as a function of its total power P for each given f(|E|)2), we found a class of nonlinear functions f(I) for which the dependence δ(P) becomes multivalued. We conjectured earlier [1] that stable (unstable) solitary solutions are those with dδ/dP > 0 (dδ/dP < 0). The first computer results by Enns and Rangnekar [2] supported this conjecture for a certain non-Kerr nonlinearity. In our further collaborative research [3], we discovered that depending on the nonlinear function f(I), the stable solitary solutions (dδ/dP > 0) fall into two classes: “weak” solitons (stable only against sufficiently small perturbation), and “robust” solitons (stable against any perturbation, in particular, against collisions with other solitons). It was shown [3] that while the condition dδ/dP > 0 is sufficient for the “weak” stability, some extra conditions imposed on the function f(I) warrant existence of “robust” solitons (one of them e.g. is f(I)/I2 = o(1) as I → ∞; stability against collapsing behavior). It was shown most recently by Enns [4] that at least for some of the nonlinear functions of (|E|2) giving rise to bistable solitons, the nonlinear Schrödinger equation passes the so called Painleve test and is therefore completely integrable. This confirms that the “robust” solitary solutions are indeed solitons. The propagation of single pulses in nonlinear fibers with an appropriate nonlinearity may provide the first known opportunity to obtain a temporal (dynamic) bistability as contrasted to the known kinds of optical bistability which are so far based on the equilibrium regimes.
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Hoang, Van-Hung, and Uwe Thumm. "Strong-field-driven dissociation dynamics in CO2+." In Frontiers in Optics. Washington, D.C.: Optica Publishing Group, 2023. http://dx.doi.org/10.1364/fio.2023.jtu5a.79.
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We investigate strong-field XUV-IR pump-probe dissociative ionization of CO2 by solving the Schrödinger equation for the nuclear motion in full dimensionality on the lowest five coupled Oppenheimer (BO) potential-energy surfaces. Applying a multi-configurational self-consistent-field quantum-chemistry code to calculate ab initio non-BO, laser dipole, and spin-orbit couplings between adiabatic electronic states, we provide kinetic energy release (KER) spectra for the O(3Pg) + CO+( X2Σ+) and O+(4Su) + CO(X1Σ+) dissociation channels and their branching ratio. Our KER spectra identify the vibrational excitations of CO+ fragments along a dominant 3ω dissociation paths. Mediated by the nuclear dynamics near the conical intersection of the CO2+ A2∏u and B2Σu+ states, we reproduce a core-hole oscillation period 115 fs, in good agreement with the experiment of Timmers et al. [1]. In addition, we find 62 fs oscil-lations in the CO+ fragmentation channel due to quantum beats between specific vibrational and electronic CO2+ states.
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De Rossi, Alfredo, Claudio Conti, and Stefano Trillo. "Stability criterion and multistability of Kerr-like gap solitons." In Nonlinear Guided Waves and Their Applications. Washington, D.C.: Optica Publishing Group, 1998. http://dx.doi.org/10.1364/nlgw.1998.ntha.5.
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It is well-known that periodic Bragg structures induce gaps in the linear dispersion relation where the propagation is forbidden. Thanks to the nonlinearity, the e.m. energy can be localized at these frequencies in the form of gap solitons [1-5]. Their most intriguiging feature is the possibility to travel with velocities much slower than the light velocity, or remarkably even at zero velocity (stationary trapping) in the limit case. Propagation of slow solitonic envelopes was recently demonstrated experimentally operating near Bragg wavelength of a fiber grating [5]. However no experimental data in the region of zero (or extremely small) velocities has been reported to date. To address the observability of stationary localization, a fundamental prerequesite to be fullfilled is the stability of the gap solitons. In spite of its importance the stability problem for gap solitons was left practically unaddressed, except for few numerical simulations. Here our purpose is to derive an analytical stability criterion, starting from the usual coupled-mode formulation of the propagation [1]. This involves derivatives of the invariants similarly to a well-known criterion for equations with second-order dispersion (i.e., nonlinear Schrödinger type [6]), recently extended to quadratic solitons [7,8]. Specifically, we consider the Lorentz-invariant Hamiltonian equations which rule the propagation of forward (+) and backward (−) envelopes u ± at Bragg or gap-center carrier frequency with cubic nonlinearity [1,2] where H is the conserved Hamiltonian of Eqs. (1).
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Yim, Solomon C., Alfred R. Osborne, and Ali Mohtat. "Nonlinear Ocean Wave Models and Laboratory Simulation of High Seastates and Rogue Waves." In ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/omae2017-62706.
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With the increasing demand for marine structures, including ships and wave energy devices, to operate in energetic, high seastates, the need for modeling and simulation of nonlinear ocean wave fields in large-scale wave basins is becoming essential. In response to this demand, a number of large-scale wave basins have been placed into operation over the years and larger and more sophisticated new ones are under planning and construction. In this article, the current state of practice and technical issues in modeling and simulation of high seastate ocean waves are summarized. A novel methodology for quantitative evaluation of the suitability of competing linear and nonlinear wave theories for a given wave field with multi-spatial measurements is presented. Preliminary results of an on-going study on wave modeling and analysis of measured data from a wave simulation performance study of the Oregon State University directional wave basin, using nonlinear wave theory (e.g. the nonlinear Schrödinger equation), nonlinear Fourier analysis and inference to the existence of rogue waves, are presented. Suggestions on future development of nonlinear wavemaker theories and numerical modeling and simulation of large-scale wave basin nonlinear wave generation are proposed. The article concludes with some observations and remarks on the importance of using an appropriate wave theory to determine the existence of nonlinear coherence structures, including breathers and rogue waves.
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Petela, G., and K. K. Botros. "Magnetic Bearing Control of Flexible Shaft Vibrations Based on Multiaccess Velocity-Displacement Feedback." In ASME 1993 International Gas Turbine and Aeroengine Congress and Exposition. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/93-gt-294.
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A model of the forced vibrations of a flexible, asymmetric and unbalanced shaft, supported by two magnetic bearings is derived to simulate the effect of different schemes of active control on shaft dynamic behaviour. Simulation results were compared for several cases of single and multi-access bearing controls, rigid-body-mode only and rigid with flexible mode control, and linear and non-linear bearing responses. It is shown that the multi-access bearing response calculated from the known equation of the stable ROCL (Reduced Order Closed Loop) and based on the direct velocity-displacement feedback, provided the most precise shift in critical frequencies and also reasonable suppression of shaft vibration amplitudes. The non-linear bearing design was also briefly discussed. The stability analysis showed that stability limits were influenced by more parameters in this case, but no particular advantages were observed in suppression of the vibration amplitudes as compared to the linear case.
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Xu, Chao, Pinaki Pal, Xiao Ren, Sibendu Som, Magnus Sjöberg, Noah Van Dam, Yunchao Wu, Tianfeng Lu, and Matthew McNenly. "Numerical Investigation of Fuel Property Effects on Mixed-Mode Combustion in a Spark-Ignition Engine." In ASME 2019 Internal Combustion Engine Division Fall Technical Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/icef2019-7265.
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Abstract In the present study, mixed-mode combustion of an E30 fuel in a direct-injection spark-ignition engine is numerically investigated at a fuel-lean operating condition using multidimensional computational fluid dynamics (CFD). A fuel surrogate matching Research Octane Number (RON) and Motor Octane Number (MON) of E30 is first developed using neural network based non-linear regression model. To enable efficient 3D engine simulations, a 164-species skeletal reaction mechanism incorporating NOx chemistry is reduced from a detailed chemical kinetic model. A hybrid approach that incorporates the G-equation model for tracking turbulent flame front, and the multi-zone well-stirred reactor model for predicting auto-ignition in the end gas, is employed to account for turbulent combustion interactions in the engine cylinder. Predicted in-cylinder pressure and heat release rate traces agree well with experimental measurements. The proposed modelling approach also captures moderated cyclic variability. Two different types of combustion cycles, corresponding to purely deflagrative and mixed-mode combustion, are observed. In contrast to the purely deflagrative cycles, mixed-mode combustion cycles feature early flame propagation followed by end-gas auto-ignition, leading to two distinctive peaks in heat release rate traces. The positive correlation between mixed-mode combustion cycles and early flame propagation is well captured by simulations. With the validated numerical setup, effects of NOx chemistry on mixed-mode combustion predictions are investigated. NOx chemistry is found to promote auto-ignition through residual gas recirculation, while the deflagrative flame propagation phase remains largely unaffected. Local sensitivity analysis is then performed to understand effects of physical and chemical properties of the fuel, i.e., heat of evaporation (HoV) and laminar flame speed (SL). An increased HoV tends to suppress end-gas auto-ignition due to increased vaporization cooling, while the impact of HoV on flame propagation is insignificant. In contrast, an increased SL is found to significantly promote both flame propagation and auto-ignition. The promoting effect of SL on auto-ignition is not a direct chemical effect; it is rather caused by an advancement of the combustion phasing, which increases compression heating of the end gas.
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Dall'Aqua, Marcelo J., Emilio J. R. Coutinho, Eduardo Gildin, Zhenyu Guo, Hardik Zalavadia, and Sathish Sankaran. "Guided Deep Learning Manifold Linearization of Porous Media Flow Equations." In SPE Reservoir Simulation Conference. SPE, 2023. http://dx.doi.org/10.2118/212204-ms.
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Abstract Integrated reservoir studies for performance prediction and decision-making processes are computationally expensive. In this paper, we develop a novel linearization approach to reduce the computational burden of intensive reservoir simulation execution. We achieve this by introducing two novel components: (1) augment the state-space to yield a bi-linear system, and (2) an autoencoder based on a deep neural network to linearize physics reservoir equations in a reduced manifold employing a Koopman operator. Recognizing that reservoir simulators execute expensive Newton-Raphson iterations after each timestep to solve the nonlinearities of the physical model, we propose "lifting" the physics to a more amenable manifold where the model behaves close to a linear system, similar to the Koopman theory, thus avoiding the iteration step. We use autoencoder deep neural networks with specific loss functions and structure to transform the nonlinear equation and frame it as a bilinear system with constant matrices over time. In such a way, it forces the states (pressures and saturations) to evolve in time by simple matrix multiplications in the lifted manifold. We also adopt a "guided" training approach: our training process is performed in three steps: we initially train the autoencoder, then we use a "conventional" MOR (Dynamic Mode Decomposition) as an initializer for the final full training when we use reservoir knowledge to improve and to lead the results to physically meaningful output. Many simulation studies exhibit extremely nonlinear and multi-scale behavior, which can be difficult to model and control. Koopman operators can be shown to represent any dynamical system through linear dynamics. We applied this new framework to a two-dimensional two-phase (oil and water) reservoir subject to a waterflooding plan with three wells (one injector and two producers) with speed ups around 100 times faster and accuracy in the order of 1-3 percent on the pressure and saturations predictions. It is worthwhile noting that this method is a non-intrusive data-driven method since it does not need access to the reservoir simulation internal structure; thus, it is easily applied to commercial reservoir simulators and is also extendable to other studies. In addition, an extra benefit of this framework is to enable the plethora of well-developed tools for MOR of linear systems. This is the first work that utilizes the Koopman operator for linearizing the system with controls to the author's knowledge. As with any ROM method, this can be directly applied to a well-control optimization problem and well-placement studies with low computational cost in the prediction step and good accuracy.