Journal articles: '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

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Dias, João-Paulo. "Linear stability of shock profiles for a quasilinear Benney system in ℝ2 × ℝ+." Journal of Hyperbolic Differential Equations 17, no. 04 (December 2020): 797–807. http://dx.doi.org/10.1142/s0219891620500253.

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Following Dias et al. [Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws, J. Differential Equations 251 (2007) 555–563], we study the linearized stability of a pair [Formula: see text], where [Formula: see text] is a shock profile for a family of quasilinear hyperbolic conservation laws in [Formula: see text] coupled with a semilinear Schrödinger equation.

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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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Fagnola, Franco, and Carlos M. Mora. "Basic Properties of a Mean Field Laser Equation." Open Systems & Information Dynamics 26, no. 03 (September 2019): 1950015. http://dx.doi.org/10.1142/s123016121950015x.

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We study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schrödinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation.

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García-Muñoz, Juan D., Julio Cesar Pérez-Pedraza, and A. Raya. "Dirac materials in parallel electromagnetic fields generated by supersymmetry." Journal of Physics: Conference Series 2667, no. 1 (December 1, 2023): 012053. http://dx.doi.org/10.1088/1742-6596/2667/1/012053.

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Abstract In a Supersymetric Quantum Mechanics framework, the Dirac equation describing a Dirac material in the presence of electromagnetic fields is solved. Considering parallel static non-uniform electromagnetic fields, the Dirac equation is transformed into a two-dimensional system of equations. By means of variable separation, we can define one-dimensional eigenfunctions, which are solutions for two pairs of supersymmetric partner Schrödinger-like Hamiltonians. For Pöschl-Teller-like quantum potentials, we look for conditions that guarantee the existence of bound states, and determine an analytic zero-mode solution for the Dirac equation.

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Duffner, E., and A. Rieckers. "On the Global Quantum Dynamics of Multi- Lattice Systems with Non-linear Classical Effects." Zeitschrift für Naturforschung A 43, no. 6 (June 1, 1988): 521–32. http://dx.doi.org/10.1515/zna-1988-0602.

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Abstract The microscopic dynamics for a class of long range interacting multi-lattice quantum systems is constructed in the thermodynamical limit by means of operator algebraic concepts. By direct estimations the existence of the limiting Schrödinger dynamics is proven for a set of states, which comprises also globally non-equilibrium situations. The expectation values of the classical observables in the pure phase states are shown to satisfy a set of coupled non-linear differential equations. The limiting Heisenberg dynamics is derived as a W*-automorphism group in the partially universal von Neumann algebra which corresponds to the selected set of states; it is in general, however, not σ-weakly continuous in the time parameter.

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Gao, T., Z. Wang, and P. A. Milewski. "Nonlinear hydroelastic waves on a linear shear current at finite depth." Journal of Fluid Mechanics 876 (July 31, 2019): 55–86. http://dx.doi.org/10.1017/jfm.2019.528.

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This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

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Lill, Sascha, Lukas Nickel, and Roderich Tumulka. "Consistency Proof for Multi-time Schrödinger Equations with Particle Creation and Ultraviolet Cut-Off." Annales Henri Poincaré 22, no. 6 (April 1, 2021): 1887–936. http://dx.doi.org/10.1007/s00023-020-01009-w.

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AbstractFor multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

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HACKENBERG, PETER, and GOTTFRIED MANN. "Parallel weak envelope solitons in multi-ion plasmas." Journal of Plasma Physics 61, no. 4 (May 1999): 633–44. http://dx.doi.org/10.1017/s0022377899007631.

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Heavy ions frequently appear as minor components in space plasmas, for example as charged helium in the solar wind and heavy ions in the vicinity of comets. Both the different components of ions and the associated plasma waves are observed by extraterrestrial in situ measurements. These plasma waves appear as large-amplitude magnetic field fluctuations in space plasmas. They must be described appropriately by means of multifluid equations. Because of the nonlinear nature of these waves, we here investigate nonlinear waves in multi-ion plasmas. Solitary waves that can only exist in a magnetized bi-ion plasma are presented. We employ a perturbation theory at the linear solution of a left-hand circularly polarized, low-frequency (below the proton gyrofrequency) plasma wave and take only the first nonlinear terms into account. Thus the multifluid equations are reduced to a single equation of the type of a nonlinear Schrödinger equation. The derived soliton solution is valid for magnetic field amplitudes lower than 10% of the ambient unperturbed magnetic field. The solutions are discussed for plasma parameters that are typical of the solar wind. A density enhancement can be observed within the soliton, where the helium ion density is more enhanced than the proton density.

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CHEN, XUE-NONG, and Rong-Jue Wei. "Dynamic behaviour of a non-propagating soliton under a periodically modulated oscillation." Journal of Fluid Mechanics 259 (January 25, 1994): 291–303. http://dx.doi.org/10.1017/s0022112094000145.

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It has been found theoretically and experimentally that a non-propagating soliton in a small rectangular water tank manifests dynamic behaviour when subjected to a modulated oscillation. A modification of the cubic Schrödinger equation was generalized for this case and analysed by the inverse-scattering perturbation method. The problem was reduced to a lower-dimensional one, i.e. to a pair of first-order ordinary differential equations for the amplitude and phase of the soliton, which were solved numerically. It was found that the soliton executes multi-periodic and chaotic motions under the periodically modulated oscillation. Corresponding experiments were carried out and both qualitative and quantitative agreement was obtained for the phenomena predicted and the parameter ranges in which they occur.

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Ghizdovat, Vlad, Oana Rusu, Mihail Frasila, Cristina Marcela Rusu, Maricel Agop, and Decebal Vasincu. "Towards Multifractality through an Ernst-Type Potential in Complex Systems Dynamics." Entropy 25, no. 8 (July 31, 2023): 1149. http://dx.doi.org/10.3390/e25081149.

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Some possible correspondences between the Scale Relativity Theory and the Space–Time Theory can be established. Since both the multifractal Schrödinger equation from the Scale Relativity Theory and the General Relativity equations for a gravitational field with axial symmetry accept the same SL(2R)-type invariance, an Ernst-type potential (from General Relativity) and also a multi-fractal tensor (from Scale Relativity) are highlighted in the description of complex systems dynamics. In this way, a non-differentiable description of complex systems dynamics can become functional, even in the case of standard theories (General Relativity and Quantum Mechanics).

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TANG, BING, DE-JUN LI, KE HU, and YI TANG. "INTRINSIC LOCALIZED MODES IN QUANTUM FERROMAGNETIC ISING–HEISENBERG CHAINS WITH SINGLE-ION UNIAXIAL ANISOTROPY." International Journal of Modern Physics B 27, no. 25 (September 12, 2013): 1350139. http://dx.doi.org/10.1142/s0217979213501397.

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Based on the coherent-state method combined with the Dyson–Maleev representation of spin operators, the existence and properties of intrinsic localized spin-wave modes in quantum ferromagnetic Ising–Heisenberg chains with single-ion uniaxial anisotropy are investigated analytically in the semiclassical limit. With the help of the multiple-scale method combined with semidiscrete approximation, the equation of motion for the coherent-state amplitude is reduced to the nonlinear Schrödinger equation. It is found that, at the center of the Brillouin zone, a bright type intrinsic localized spin-wave mode can exist below the bottom of the linear spin-wave spectrum. Besides, we show that, at the boundary of the Brillouin zone, a dark type intrinsic localized spin-wave mode appears above the top of the linear spin-wave spectrum, which is different from the resonant nonpropagating kink mode.

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Mousavi Lajimi, S. Amir, and Michael I. Friswell. "Energy harvesting from a non-linear standing beam–mass system: Two- versus one-mode approximations." Journal of Intelligent Material Systems and Structures 28, no. 8 (September 27, 2016): 1010–22. http://dx.doi.org/10.1177/1045389x16667852.

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We investigate the effect of including the second mode of natural vibration on the computed response of a forced non-linear gravity-loaded beam–mass structure used for non-linear piezoelectric energy harvesting. Using the method of assumed-modes and Lagrange’s equations, we develop the discretized equations of generalized coordinates of the system including the electro-mechanical equation. The equation of motion is further simplified to find the single-mode approximation. The phase-portraits, time-histories, Poincaré sections, and frequency–response curves of the system are computed. It is shown that the number of mode shapes affects the response, and it is required to include higher modes to improve the analytical–computational results. The system shows distinct behavior varying from a linear single-frequency response to a multi-frequency chaotic response. The average power across the load resistor consequently shows a noticeable variation depending on the characteristics of the overall system response.

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Wang, Xiaoming, and Zhi-Qiang Wang. "Normalized multi-bump solutions for saturable Schrödinger equations." Advances in Nonlinear Analysis 9, no. 1 (December 14, 2019): 1259–77. http://dx.doi.org/10.1515/anona-2020-0054.

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Abstract In this paper, we are concerned with the existence of multi-bump solutions for a class of semiclassical saturable Schrödinger equations with an density function: $$\begin{array}{} \displaystyle -{\it\Delta} v +{\it\Gamma} \frac{I(\varepsilon x) + v^2}{1+I(\varepsilon x) +v^2} v =\lambda v,\, x\in{{\mathbb{R}}^{2}}. \end{array}$$ We prove that, with the density function being radially symmetric, for given integer k ≥ 2 there exist a family of non-radial, k-bump type normalized solutions (i.e., with the L2 constraint) which concentrate at the global maximum points of density functions when ε → 0+. The proof is based on a variational method in particular on a convexity technique and the concentration-compactness method.

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Hanapi, Mohd Syafiq M., Abdel-Baset M. A. Ibrahim, Rafael Julius, and Hichem Eleuch. "Quantum Kerr nonlinear coupler: analytical versus phase-space method." Canadian Journal of Physics 99, no. 9 (September 2021): 832–40. http://dx.doi.org/10.1139/cjp-2020-0389.

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The generation of squeezed states of light in a two-mode Kerr nonlinear directional coupler (NLDC) was investigated using two different methods in quantum mechanics. First, the analytical method, a Heisenberg-picture-based method where the operators are evolving in time but the state vectors are time-independent. In this method, an analytical solution to the coupled Heisenberg equations of motion for the propagating modes was proposed based on the Baker–Hausdorff (BH) formula. Second, the phase space method, a Schrödinger-picture-based method in which the operators are constant and the density matrix evolves in time. In this method, the quantum mechanical master equation of the density matrix was converted to a corresponding classical Fokker–Planck (FP) equation in positive-P representation. Then, the FP equation was converted to a set of stochastic differential equations using Ito rules. The strengths and weaknesses of each method are discussed. Good agreement between both methods was achieved, especially at early evolution stages and lower values of linear coupling coefficient. On one hand, the analytical method seems insensitive to higher values of nonlinear coupling coefficients. Nevertheless, it demonstrated better numerical stability. On the other hand, the solution of the stochastic equations resulting from the phase space method is numerically expensive as it requires averaging over thousands of trajectories. Besides, numerically unstable trajectories appear with positive-P representation at higher values of nonlinearity.

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ESFANDYARI-KALEJAHI, A., I. KOURAKIS, and M. AKBARI-MOGHANJOUGHI. "Nonlinear modulation of ion-acoustic waves in two-electron-temperature plasmas." Journal of Plasma Physics 76, no. 2 (February 4, 2010): 169–81. http://dx.doi.org/10.1017/s0022377810000024.

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AbstractThe amplitude modulation of ion-acoustic waves is investigated in a plasma consisting of adiabatic warm ions, and two different populations of thermal electrons at different temperatures. The fluid equations are reduced to nonlinear Schrödinger equation by employing a multi-scale perturbation technique. A linear stability analysis for the wave packet amplitude reveals that long wavelengths are always stable, while modulational instability sets in for shorter wavelengths. It is shown that increasing the value of the hot-to-cold electron temperature ratio (μ), for a given value of the hot-to-cold electron density ratio (ν), favors instability. The role of the ion temperature is also discussed. In the limiting case ν = 0 (or ν → ∞), which correspond(s) to an ordinary (single) electron-ion plasma, the results of previous works are recovered.

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WANG, ZHONGCHENG, and YONGMING DAI. "A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION." International Journal of Modern Physics C 14, no. 08 (October 2003): 1087–105. http://dx.doi.org/10.1142/s0129183103005194.

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A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

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Zhang, Yanwei, Xiong You, and Yonglei Fang. "Exponentially fitted multi-derivative linear methods for the resonant state of the Schrödinger equation." Journal of Mathematical Chemistry 55, no. 1 (September 9, 2016): 223–37. http://dx.doi.org/10.1007/s10910-016-0683-y.

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Zhang, Yanwei, Yonglei Fang, Xiong You, and Guangde Liu. "Trigonometrically-fitted multi-derivative linear methods for the resonant state of the Schrödinger equation." Journal of Mathematical Chemistry 56, no. 4 (December 21, 2017): 1250–61. http://dx.doi.org/10.1007/s10910-017-0851-8.

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Cao, Daomin, Shanfa Lai, and Weilin Yu. "Multi-peak solutions to the Schrödinger equations coupled with a neutral scalar field." Journal of Mathematical Physics 64, no. 2 (February 1, 2023): 021510. http://dx.doi.org/10.1063/5.0121726.

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In this paper, we consider the problem of Schrödinger equation coupled with a neutral scalar field. By constructing solutions with multiple peaks, we prove that the number of non-radial solutions of this problem goes to infinity as the Maxwell coupling constant tends to infinity. The Chern–Simons limit of those solutions is also discussed.

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Stalin, S., R. Ramakrishnan, and M. Lakshmanan. "Nondegenerate Bright Solitons in Coupled Nonlinear Schrödinger Systems: Recent Developments on Optical Vector Solitons." Photonics 8, no. 7 (July 5, 2021): 258. http://dx.doi.org/10.3390/photonics8070258.

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Nonlinear dynamics of an optical pulse or a beam continue to be one of the active areas of research in the field of optical solitons. Especially, in multi-mode fibers or fiber arrays and photorefractive materials, the vector solitons display rich nonlinear phenomena. Due to their fascinating and intriguing novel properties, the theory of optical vector solitons has been developed considerably both from theoretical and experimental points of view leading to soliton-based promising potential applications. Mathematically, the dynamics of vector solitons can be understood from the framework of the coupled nonlinear Schrödinger (CNLS) family of equations. In the recent past, many types of vector solitons have been identified both in the integrable and non-integrable CNLS framework. In this article, we review some of the recent progress in understanding the dynamics of the so called nondegenerate vector bright solitons in nonlinear optics, where the fundamental soliton can have more than one propagation constant. We address this theme by considering the integrable two coupled nonlinear Schrödinger family of equations, namely the Manakov system, mixed 2-CNLS system (or focusing-defocusing CNLS system), coherently coupled nonlinear Schrödinger (CCNLS) system, generalized coupled nonlinear Schrödinger (GCNLS) system and two-component long-wave short-wave resonance interaction (LSRI) system. In these models, we discuss the existence of nondegenerate vector solitons and their associated novel multi-hump geometrical profile nature by deriving their analytical forms through the Hirota bilinear method. Then we reveal the novel collision properties of the nondegenerate solitons in the Manakov system as an example. The asymptotic analysis shows that the nondegenerate solitons, in general, undergo three types of elastic collisions without any energy redistribution among the modes. Furthermore, we show that the energy sharing collision exhibiting vector solitons arises as a special case of the newly reported nondegenerate vector solitons. Finally, we point out the possible further developments in this subject and potential applications.

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Paldor, Nathan, Yair De-Leon, and Ofer Shamir. "Planetary (Rossby) waves and inertia–gravity (Poincaré) waves in a barotropic ocean over a sphere." Journal of Fluid Mechanics 726 (May 30, 2013): 123–36. http://dx.doi.org/10.1017/jfm.2013.219.

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AbstractThe construction of approximate Schrödinger eigenvalue equations for planetary (Rossby) waves and for inertia–gravity (Poincaré) waves on an ocean-covered rotating sphere yields highly accurate estimates of the phase speeds and meridional variation of these waves. The results are applicable to fast rotating spheres such as Earth where the speed of barotropic gravity waves is smaller than twice the tangential speed on the equator of the rotating sphere. The implication of these new results is that the phase speed of Rossby waves in a barotropic ocean that covers an Earth-like planet is independent of the speed of gravity waves for sufficiently large zonal wavenumber and (meridional) mode number. For Poincaré waves our results demonstrate that the dispersion relation is linear, (so the waves are non-dispersive and the phase speed is independent of the wavenumber), except when the zonal wavenumber and the (meridional) mode number are both near 1.

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Cai, Hao, Feng-Ming Liu, and Nian-Ning Huang. "Dark Multi-Soliton Solution of the Nonlinear Schrödinger Equation with Non-Vanishing Boundary." International Journal of Theoretical Physics 44, no. 2 (February 2005): 255–65. http://dx.doi.org/10.1007/s10773-005-1691-z.

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Chen, Li-Qun, Hu Ding, and C. W. Lim. "Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification." Shock and Vibration 19, no. 4 (2012): 527–43. http://dx.doi.org/10.1155/2012/948459.

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Transverse non-linear vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about a constant mean speed. The material time derivative is used in the viscoelastic constitutive relation. The transverse motion can be governed by a non-linear partial-differential equation or a non-linear integro-partial-differential equation. The method of multiple scales is applied to the governing equations to determine steady-state responses. It is confirmed that the mode uninvolved in the resonance has no effect on the steady-state response. The differential quadrature schemes are developed to verify results via the method of multiple scales. It is demonstrated that the straight equilibrium configuration becomes unstable and a stable steady-state emerges when the axial speed variation frequency is close to twice any linear natural frequency. The results derived for two governing equations are qualitatively the same, but quantitatively different. Numerical simulations are presented to examine the effects of the mean speed and the variation of the amplitude of the axial speed, the dynamic viscosity, the non-linear coefficients, and the boundary constraint stiffness on the instability interval and the steady-state response amplitude.

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Zolotaryuk, Alexander V., and Yaroslav Zolotaryuk. "Conditions for realizing one-point interactions from a multi-layer structure model." Journal of Physics A: Mathematical and Theoretical 55, no. 8 (February 1, 2022): 085201. http://dx.doi.org/10.1088/1751-8121/ac4a1f.

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Abstract A heterostructure composed of N parallel homogeneous layers is studied in the limit as their widths l 1, …, l N shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schrödinger equation is given by the strengths V 1, …, V N as functions of l 1, …, l N , respectively. The key point is the derivation of the conditions on the functions V 1(l 1), …, V N (l N ) for realizing a family of one-point interactions as l 1, …, l N tend to zero along available paths in the N-dimensional space. The existence of equations for a squeezed structure, the solution of which determines the system parameter values, under which the non-zero tunneling of quantum particles through a multi-layer structure occurs, is shown to exist and depend on the paths. This tunneling appears as a result of an appropriate cancellation of divergences.

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BALA, PARVEEN, TARSEM SINGH GILL, and HARVINDER KAUR. "Ion-acoustic envelope excitations in multispecies plasma with non-thermally distributed electrons." Journal of Plasma Physics 78, no. 3 (February 6, 2012): 265–78. http://dx.doi.org/10.1017/s0022377812000013.

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AbstractBy using the standard reductive perturbation technique, a nonlinear Schrödinger equation is derived to study the stability of finite amplitude ion-acoustic waves in an unmagnetized plasma consisting of warm positive and negative ions and non-thermal electrons. The effect of relative temperature of positive and negative ions, their charge and mass ratios, density ratios and non-thermally distributed electrons on modulational instability of fast and slow ion-acoustic mode is investigated. It is found that these parameters significantly change the domain of modulation instability. Both, envelope and dark solitons appear in different regions of parameter space.

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Xu, Xice, Yang Lu, and Xufeng Wu. "Dynamic Modeling and Control Law Design of a Fuel-electric Hybrid Multi-rotor UAV." International Journal of Micro Air Vehicles 14 (January 2022): 175682932210789. http://dx.doi.org/10.1177/17568293221078925.

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In this paper, the design of control law for a new concept fuel-electric hybrid multi-rotor UAV with lift/attitude control separation is investigated. The remarkable feature of the UAV is that it has a large proportion of fuel weight. Firstly, based on the quasi-coordinate Lagrangian equation and sloshing equivalent model using the multi-mass-spring analogy, the non-linear dynamic model of the UAV considering the fuel slosh dynamics is established. Compared with the existing multi-rotor modeling method, it is more intuitive and accurate to describe the non-linear coupling process of sloshing and UAV's motion degrees of freedom. Secondly, the attitude control law is designed based on the finite-time sliding mode observer and cascaded continuous sliding mode controller to eliminate the adverse effects of fuel sloshing and mass changing, and only using the measurable angles. Furthermore, aiming at the problem of power redundancy of the altitude channel, a memoryless non-linear altitude authority assignment controller based on vertical acceleration is proposed for improving the control performance. Finally, the simulation results of the waypoint flight illustrate the feasibility and effectiveness of the proposed control strategy.

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Caruso, Enrico, David Esseni, Elena Gnani, Daniel Lizzit, Pierpaolo Palestri, Alessandro Pin, Francesco Maria Puglisi, Luca Selmi, and Nicolò Zagni. "Modeling Nanoscale III–V Channel MOSFETs with the Self-Consistent Multi-Valley/Multi-Subband Monte Carlo Approach." Electronics 10, no. 20 (October 12, 2021): 2472. http://dx.doi.org/10.3390/electronics10202472.

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We describe the multi-valley/multi-subband Monte Carlo (MV–MSMC) approach to model nanoscale MOSFETs featuring III–V semiconductors as channel material. This approach describes carrier quantization normal to the channel direction, solving the Schrödinger equation while off-equilibrium transport is captured by the multi-valley/multi-subband Boltzmann transport equation. In this paper, we outline a methodology to include quantum effects along the transport direction (namely, source-to-drain tunneling) and provide model verification by comparison with Non-Equilibrium Green’s Function results for nanoscale MOSFETs with InAs and InGaAs channels. It is then shown how to use the MV–MSMC to calibrate a Technology Computer Aided Design (TCAD) simulation deck based on the drift–diffusion model that allows much faster simulations and opens the doors to variability studies in III–V channel MOSFETs.

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ISAR, A., A. SANDULESCU, H. SCUTARU, E. STEFANESCU, and W. SCHEID. "OPEN QUANTUM SYSTEMS." International Journal of Modern Physics E 03, no. 02 (June 1994): 635–714. http://dx.doi.org/10.1142/s0218301394000164.

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The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber P representation, the antinormal ordering Q representation, and the Wigner W representation. The density matrix is represented via a generating function, which is obtained by solving a timedependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. The solution of the master equation in the Weyl-Wigner-Moyal representation is of Gaussian type if the initial form of the Wigner function is taken to be a Gaussian corresponding (for example) to a coherent wavefunction. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator, are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow’s transitions with energy conservation, additional transitions with energy loss are found. The tunneling spectrum is obtained as a function of the barrier characteristics. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment of the atomic observables are obtained. With these equations, some characteristics of the laser radiation absorption spectrum and optical bistability are described.

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Verheest, Frank, and B. Buti. "Parallel solitary Alfvén waves in warm multi-species beam-plasma systems. Part 1." Journal of Plasma Physics 47, no. 1 (February 1992): 15–24. http://dx.doi.org/10.1017/s0022377800024041.

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A self-consistent reductive perturbation analysis for parallel-propagating magnetohydrodynamic waves in warm multi-species plasmas, in which different constituents can have differing equilibrium drifts, leads to a derivative nonlinear Schrödinger equation for the wave magnetic field. Soliton solutions are discussed, including applications to plasmas with two ion species. Such solitons are larger (in amplitude) and wider than in the non-streaming and/or cold-plasma case, other parameters being equal.

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Veklenko, Boris A. "Coherent Stimulated Radiation, Coherent Spontaneous Radiation and Coupled Photon Pairs in Non-Uniform Thermally Excited Gaseous Media: The Theory." Issue 01-2022, no. 01-2022 (February 2022): 4–11. http://dx.doi.org/10.33383/2021-099.

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It is demonstrated that presence of stimulated radiation processes in gaseous media makes it possible to generate rays inversely reflected from interfaces of thermally excited gaseous media and the processes of left-hand refraction of the rays having passed through these media. These processes are caused by availability of coupled photonic states and stimulated radiation in media the properties of which differ from those of stimulated radiation in vacuum. Estimation of these processes is based on exclusion of the procedure of forced breaking of quantum correlators and correct consideration of high-order correlators. Along with Schrödinger’s and Heisenberg’s representations, this article uses also the less-known Γ- representation, which allows us to reduce a multi-particle problem of quantum electrodynamics to a single-particle problem within the an infinite-dimensional space. The equations formed in Г- representation are mathematically simpler and may be studied using the standard Wick’s theorem rather than its approximated thermodynamic variant suggested by T.A. Matsubara. Therefore it is possible to keep all possible solutions of the initial system of equations in mind. Of course, approximation methods are used for explicit solution of these equations. Using Feynman diagrams instead of the Dyson equation for quantum correlators, it is possible to find an equation defining the complete density matrix of a photonic sub-system.

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Broadbridge, Philip, and Kathryn Deutscher. "Solution of Non-Autonomous Schrödinger Equation for Quantized de Sitter Klein-Gordon Oscillator Modes Undergoing Attraction-Repulsion Transition." Symmetry 12, no. 6 (June 3, 2020): 943. http://dx.doi.org/10.3390/sym12060943.

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For a scalar field in an exponentially expanding universe, constituent modes of elementary excitation become unstable consecutively at shorter wavelength. After canonical quantization, a Bogoliubov transformation reduces the minimally coupled scalar field to independent 1D modes of two inequivalent types, leading eventually to a cosmological partitioning of energy. Due to accelerated expansion of the coupled space-time, each underlying mode transits from an attractive oscillator with discrete energy spectrum to a repulsive unit with continuous unbounded energy spectrum. The underlying non-autonomous Schrödinger equation is solved here as the wave function evolves through the attraction-repulsion transition and ceases to oscillate.

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Sergeyev, Sergey V. "Fast and slowly evolving vector solitons in mode-locked fibre lasers." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, no. 2027 (October 28, 2014): 20140006. http://dx.doi.org/10.1098/rsta.2014.0006.

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We report on a new vector model of an erbium-doped fibre laser mode locked with carbon nanotubes. This model goes beyond the limitations of the previously used models based on either coupled nonlinear Schrödinger or Ginzburg–Landau equations. Unlike the previous models, it accounts for the vector nature of the interaction between an optical field and an erbium-doped active medium, slow relaxation dynamics of erbium ions, linear birefringence in a fibre, linear and circular birefringence of a laser cavity caused by in-cavity polarization controller and light-induced anisotropy caused by elliptically polarized pump field. Interplay of aforementioned factors changes coherent coupling of two polarization modes at a long time scale and so results in a new family of vector solitons (VSs) with fast and slowly evolving states of polarization. The observed VSs can be of interest in secure communications, trapping and manipulation of atoms and nanoparticles, control of magnetization in data storage devices and many other areas.

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Yapanmış, Burak Emre, and Süleyman Murat Bağdatlı. "Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam." Zeitschrift für Naturforschung A 77, no. 4 (December 23, 2021): 305–21. http://dx.doi.org/10.1515/zna-2021-0300.

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Abstract In this present work, linear and non-linear vibration of multi-supported nanobeams, which are a fundamental part of the nano-electromechanical systems, is examined. To the best of the researchers’ knowledge, there is no study performed into multi-supported nanobeam in the literature. The governing equations of the system are obtained by dint of the Hamilton principle and solved via the perturbation technique which is divided linear and non-linear parts of the main equations. The natural frequencies and mode shapes are calculated from the linear problem. The non-linear natural frequencies and amplitude-phase modulation graphs are obtained from the non-linear equation. All equations are written in generalized form, and 3, 4 and 5 supported nanobeams are investigated in detail. The nonlocal coefficient, support number and position and end condition types are focused on. The three to one internal resonance cases are also investigated. It is occurred that the clamped-end conditions shift right in the hardening behaviour graphs more than the simply supported condition. Moreover, it is shown that the supported numbers play a significant role in natural frequency.

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Ying, Zu-Guang, and Yi-Qing Ni. "Vibration Localization and Anti-Localization of Nonlinear Multi-Support Beams with Support Periodicity Defect." Symmetry 13, no. 12 (November 23, 2021): 2234. http://dx.doi.org/10.3390/sym13122234.

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A response analysis method for nonlinear beams with spatial distribution parameters and non-periodic supports was developed. The proposed method is implemented in four steps: first, the nonlinear partial differential equation of the beams is transformed into linear partial differential equations with space-varying parameters by using a perturbation method; second, the space-varying parameters are separated into a periodic part and a non-periodic part describing the periodicity defect, and the linear partial differential equations are separated into equations for the periodic and non-periodic parts; third, the equations are converted into ordinary differential equations with multiple modes coupling by using the Galerkin method; fourth, the equations are solved by using a harmonic balance method to obtain vibration responses, which are used to discover dynamic characteristics including the amplitude–frequency relation and spatial mode. The proposed method considers multiple vibration modes in the response analysis of nonlinear non-periodic structures and accounts for mode-coupling effects resulting from structural nonlinearity and parametric non-periodicity. Thus, it can handle nonlinear non-periodic structures with a high parameter-varying wave in wide frequency vibration. In numerical studies, a nonlinear beam with non-periodic supports (resulting in non-periodic distribution parameters or periodicity defect) under harmonic excitations was explored using the proposed method, which revealed some new dynamic response characteristics of this kind of structure and the influences of non-periodic parameters. The characteristics include remarkable variation in frequency response and spatial mode, and in particular, vibration localization and anti-localization. The results have potential applications in vibration control and the support damage detection of nonlinear structures with non-periodic supports.

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Xia, Wei, Xinghong Zhao, Datao Li, and Shengping Shen. "Nonlinear flutter response of pre-heated functionally graded panels." International Journal of Computational Materials Science and Engineering 07, no. 01n02 (June 2018): 1850012. http://dx.doi.org/10.1142/s2047684118500124.

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The thermo-mechanical behavior of functionally graded (FG) panels is investigated in supersonic air flow. The three-node triangular element based on the Mindlin plate theory is employed to account for the transverse shear strains, and the von-Karman nonlinear strain–displacement relation is utilized considering the geometric nonlinearity. The effective material properties of the FG material are assumed to vary through the thickness according to simple power law distribution. The aeroelastic equation is established using the first-order piston theory, the linear rule of mixture and the principle of virtual work. A multi-mode approach is utilized to form the reduced-order model. Nonlinear flutter response is obtained by solving the reduced-order aeroelastic equation in time using the Runge–Kutta fourth-order method. Numerical simulation reveals that the multi-mode reduced-order model has a good convergence property. By using the 24-mode model the variation of flutter amplitude with dimensionless dynamic pressure, and the route of nonlinear flutter response from simple harmonic limit cycle oscillation (LCO) to non-harmonic periodic oscillation are examined.

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Niu, Yahui, Shuying Tian, and Pingping Yang. "Local uniqueness of multi-peak solutions to a class of Schrödinger equations with competing potential." Journal of Mathematical Physics 64, no. 3 (March 1, 2023): 031509. http://dx.doi.org/10.1063/5.0134220.

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In this paper, we consider the nonlinear Schrödinger equations. Let [Formula: see text]. Under some conditions on [Formula: see text], we show the local uniqueness of positive multi-peak solutions concentrating near k( k ≥ 2) distinct non-degenerate critical points of [Formula: see text] by using the local Pohozaev identity. We generalize Cao–Li–Luo’s results to the competing potential cases and show how these two potentials impact the uniqueness of concentrated solutions.

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Yong, Xuelin, Yajing Fan, Yehui Huang, Wen-Xiu Ma, and Jing Tian. "Darboux transformation and solitons for an integrable nonautonomous nonlinear integro-differential Schrödinger equation." Modern Physics Letters B 31, no. 30 (October 26, 2017): 1750276. http://dx.doi.org/10.1142/s0217984917502761.

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By modifying the scheme for an isospectral problem, the non-isospectral Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy is constructed via allowing the time varying spectrum. In this paper, we consider an integrable nonautonomous nonlinear integro-differential Schrödinger equation discussed before in “Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation” [Y. J. Zhang, D. Zhao and H. G. Luo, Ann. Phys. 350 (2014) 112]. We first analyze the integrability conditions and identify the model. Second, we modify the existing Darboux transformation (DT) for such a non-isospectral problem. Third, the nonautonomous soliton solutions are obtained via the resulting DT and basic properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. In the process, a technique by selecting appropriate spectral parameters instead of the variable inhomogeneities is employed to realize a different type of one-soliton management. Several novel optical solitons are constructed and their features are shown by some specific figures. In addition, four kinds of the special localized two-soliton solutions are obtained. The solitonic excitations localized both in space and time, which exhibit the feature of the so-called rogue waves but with a zero background, are discussed.

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Le Coz, Stefan, Dong Li, and Tai-Peng Tsai. "Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 6 (November 23, 2015): 1251–82. http://dx.doi.org/10.1017/s030821051500030x.

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We study infinite soliton trains solutions of nonlinear Schrödinger equations, i.e. solutions behaving as the sum of infinitely many solitary waves at large time. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighbourhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

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Xue, Danting, Ruigang Zhang, Quansheng Liu, and Zhaodong Ding. "Instability of Liquid Film with Odd Viscosity over a Non-Uniformly Heated and Corrugated Substrate." Nanomaterials 13, no. 19 (September 28, 2023): 2660. http://dx.doi.org/10.3390/nano13192660.

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The effect of odd viscosity on the instability of liquid film along a wavy inclined bottom with linear temperature variation is investigated. By utilizing the long-wave approximation, the non-linear evolution equation of the free surface is derived. By applying the normal mode method, the linear instability of thin film flow is investigated. With the help of multi-scale analysis methods, the weakly non-linear instability of thin film flow is also investigated. The results reveal that the Marangoni effect caused by non-uniform temperature distribution promotes the instability of the liquid film, while the odd viscosity has a stabilizing effect. In addition, for a positive local inclination angle θ, an increase in bottom steepness ζ inhibits the instability of the liquid film flow. In contrast, with a negative local inclination angle θ, increased bottom steepness ζ promotes the instability of the liquid film flow. The results of the temporal linear instability analysis and the weakly non-linear instability analysis have been substantiated through numerical simulations of the non-linear evolution equations.

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Ibarra-Villalon, H. E., O. Pottiez, A. Gómez-Vieyra, J. P. Lauterio-Cruz, and Y. E. Bracamontes-Rodriguez. "Numerical study of polarization evolution governed by linear birefringence, twist-induced circular birefringence and nonlinear birefringence in a single-mode optical fiber." Journal of Optics 23, no. 12 (October 26, 2021): 123501. http://dx.doi.org/10.1088/2040-8986/ac2eaa.

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Abstract This tutorial presents a numerical analysis of continuous wave and ultrashort pulse evolution through a twisted single-mode optical fiber, modeled by the nonlinear Schrödinger equation. In this model, the polarization evolutions of the continuous wave and the pulse profile are studied by the changes in ellipticity and ellipse rotation, which are driven by the inherent linear birefringence of the optical fiber, the induced nonlinear birefringence due to the centrosymmetric response of the fiber and the induced circular birefringence due to a uniform twist applied along the fiber. In particular, the role of each birefringence effect is studied in detail. As a result, it is pointed out that a large uniform twist rate allows viewing the optical fiber as an isotropic waveguide that preserves ellipticity. On the other hand, a saturable absorber mechanism based on a linear polarizer and the ellipse rotation in a twisted fiber, which introduces a nonlinear transmission characteristic that is part of the principles of operation of the mode-locked fiber lasers, is analyzed in order to illustrate the applicability of this numerical study.

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Journal articles on the topic 'Multi-Mode non-Linear Schrödinger equation'

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