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"Schrodinger equation."
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Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN
We use the physics-informed neural network to solve a variety of femtosecond optical soliton solutions of the high-order nonlinear Schrödinger equation, including one-soliton solution, two-soliton solution, rogue wave solution, W-soliton solution and M-soliton solution. The prediction error for one-soliton, W-soliton and M-soliton is smaller. As the prediction distance increases, the prediction error will gradually increase. The unknown physical parameters of the high-order nonlinear Schrödinger equation are studied by using rogue wave solutions as data sets. The neural network is optimized from three aspects including the number of layers of the neural network, the number of neurons, and the sampling points. Compared with previous research, our error is greatly reduced. This is not a replacement for the traditional numerical method, but hopefully to open up new ideas.
Journal Article
The Schrödinger-Newton equation and its foundations
The necessity of quantising the gravitational field is still subject to an open debate. In this paper we compare the approach of quantum gravity, with that of a fundamentally semi-classical theory of gravity, in the weak-field non-relativistic limit. We show that, while in the former case the Schrödinger equation stays linear, in the latter case one ends up with the so-called Schrödinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schrödinger-Newton equation does not describe the collapse of the wave-function, although it was initially proposed for exactly this purpose. Together with the standard collapse postulate, fundamentally semi-classical gravity gives rise to superluminal signalling. A consistent fundamentally semi-classical theory of gravity can therefore only be achieved together with a suitable prescription of the wave-function collapse. We further discuss, how collapse models avoid such superluminal signalling and compare the nonlinearities appearing in these models with those in the Schrödinger-Newton equation.
Journal Article
Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results
The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schrödinger equation (NLSE) including fractional one-dimensional or two-dimensional diffraction and cubic or cubic-quintic nonlinear terms, as well as linear potentials. The fractional diffraction is represented by fractional-order spatial derivatives of the Riesz type, defined in terms of the direct and inverse Fourier transform. In this form, it can be realized by spatial-domain light propagation in optical setups with a specially devised combination of mirrors, lenses, and phase masks. The results presented in the article were chiefly obtained in a numerical form. Some analytical findings are included too, in particular, for fast moving solitons and the results produced by the variational approximation. Moreover, dissipative solitons are briefly considered, which are governed by the fractional complex Ginzburg–Landau equation.
Journal Article
Bifurcation analysis, chaotic behaviors, sensitivity analysis, and soliton solutions of a generalized Schrödinger equation
The main goal of the present study is to conduct a deeper investigation into a generalized Schrödinger equation describing the propagation of optical pulses in media. To this end, the dynamical system of the governing equation is derived using the Galilean transformation, and its bifurcation is carried out using the theory of the planar dynamical system. By considering a perturbed term in the resulting dynamical system, the existence of chaotic behaviors of the generalized Schrödinger equation is investigated by giving some two- and three-dimensional phase portraits. Additionally, the sensitivity analysis of the dynamical system is accomplished using the Runge–Kutta method validating that small changes in initial conditions do not affect the stability of the solution very much. In the end, several bright and dark solitons to the governing model are constructed using the method of the planar dynamical system. The results of the current paper show that bright and dark solitons can be effectively controlled in terms of their width and height.
Journal Article
Effective amplification of optical solitons in high power transmission systems
We study the effective amplification of optical solitons in high power transmission systems based on a nonlinear Schrödinger equation with variable coefficients. Three soliton solutions to this model are constructed analytically. We analyze the characteristics of three soliton interaction and achieve the effective amplification of solitons. We analyze the influence of parallel and nonparallel propagation of optical solitons on optical soliton amplification. The conclusion of this study is beneficial to the propagation and amplification of optical solitons in high-power transmission system, so as to effectively improve the transmission distance of the system.
Journal Article
Semiclassical standing waves with clustering peaks, for nonlinear Schrödinger equations
We study the following singularly perturbed problem
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Controlling effect of vector and scalar crossed double-Ma breathers in a partially nonlocal nonlinear medium with a linear potential
We follow our interest in a nonautonomous (2+1)-dimensional coupled nonlinear Schrödinger equation with partially nonlocal nonlinear effect and a linear potential, and get a relational expression mapping nonautonomous equation into autonomous one. Further applying the Darboux method, we find affluent vector and scalar solutions, including the crossed double-Ma breather solution. Regulating values of initial width, initial chirp and diffraction parameters so that the maximal value of accumulated time changes to compare with values of peak positions, we actualize the controlling effect of vector and scalar crossed double-Ma breathers including the complete shape, crest shape and nascent shape excitations in different linear potentials.
Journal Article
ERROR ESTIMATES OF A REGULARIZED FINITE DIFFERENCE METHOD FOR THE LOGARITHMIC SCHRÖDINGER EQUATION
We present a regularized finite difference method for the logarithmic Schrödinger equation (LogSE) and establish its error bound. Due to the blowup of the logarithmic nonlinearity, i.e., In ρ → ∞ when ρ → 0⁺ with ρ = |u|² being the density and u being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the roundoff error and to avoid blowup, a regularized LogSE (RLogSE) is proposed with a small regularization parameter 0 < e « 1 and linear convergence is established between the solutions of RLogSE and LogSE in term of . Then a semi-implicit finite difference method is presented for discretizing the RLogSE and error estimates are established in terms of the mesh size h and time step as well as the small regularization parameter . Finally numerical results are reported to illustrate our error bounds.
Journal Article
Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality
Hierarchies of Peregrine solution and breather solution are derived in a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation with partial nonlocality. Based on these solutions, we study the control of the excitation of Peregrine solution and breather solution in different planes. In particular, the localized Peregrine solution and breather solution are firstly reported in two-dimensional space. It is expected that our analysis and results may give new insight into higher-dimensional localized rogue waves in nonlocal media.
Journal Article