Two-dimensional Nonlinear Schrodinger Equation Research Articles

A hybrid variational method for beam propagation and interaction in a graded-index nonlinear waveguide

The hybrid variational method can be used to simplify multidimensional numerical simulations. We examine the applicability of this method to study the nonlinear propagation of a single beam and the interference of two spatial soliton beams in a graded-index optical waveguide governed by a two-dimensional nonlinear Schrodinger equation. We classified three distinct regimes of the dynamics that emerge during single Gaussian beam propagation, corresponding to beam decay, breather formation, and self-focusing. When two spatial soliton beams were set up to interfere in the waveguide, we identified the boundary where the nonlinear interaction during the interference led to excitation or self-focusing. These quite involved dynamics were used to test the predictive power of our variational models by comparing them with direct numerical simulations, obtaining good agreement.

Exact solutions of the two-dimensional nonlinear Schrodinger equation

The nonlinear Schrödinger equation as a classical model in physics is gradually becoming an urgent problem that needs to be studied. Obtaining an exact solution of the corresponding model can not only provide theoretical support for the experiment but also provide a basis for solving practical problems. In this work, we study the propagation of waves of the two-dimensional nonlinear Schrödinger equa- tion in nonlinear media with dispersion processes. The tan-cot function method is applied. This method is effective in solving nonlinear equations of mathematical physics. Various solutions in the form of periodic waves are obtained. Graphs of the solutions arepresented.

Fractionality-induced deformation in Airy and Hermit-Gaussian wavepackets traveling through the Gaussian wells

In this study, we investigate the transport properties of Airy and Hermit-Gaussian wavepackets traveling through a Gaussian potential well in the standard and fractional media. We use a split step Fourier method to solve the corresponding two-dimensional fractional nonlinear linear Schrodinger equation. We study the interference phenomenon, Huygens principle and trapping properties of the system in different situations. For this purpose, the position of the system has also selected inside the potential well to consider the wavepacket evolution. Then, the effect of the well and incident wavepackets parameters on the evolution the transport properties of employed wavepackets have properly been considered.

TWO-DIMENSIONAL NONLOCAL NONLINEAR SCHRODINGER EQUATION BASED ON THE ABLOWITZ-MUSSLIMANI SYMMETRY CONDITION

The nonlinear Schrodinger equation is a nonlinear partial differential equation and integrable equation that play an essential role in many branches of physics as nonrelativistic quantum mechanics, acoustics, and optics. In this work, motivated by the ideas of Ablowitz and Musslimani, we successfully obtain a two-dimensional nonlocal nonlinear Schrodinger equation where the nonlocality consists of reverse time fields as factors in the nonlinear terms. The nonlocal nonlinear Schrodinger equation admits a great number of good properties that the classical nonlinear Schrodinger equation possesses, e.g. PT-symmetric, admitting Lax-pair, and infinitely many conservation laws. We apply the Darboux transformation method to the two-dimensional nonlinear Schrodinger equation. The idea of this method is having a Lax representation, one can obtain various kinds of solutions of the Nth order with a spectral parameter. The exact solutions and graphical representation of obtained solutions are derived.

Numerical solutions of two-dimensional fractional Schrodinger equation

In this article, the authors proposed Chebyshev pseudospectral method for numerical solutions of two-dimensional nonlinear Schrodinger equation with fractional-order derivative in both time and space. Fractional-order partial differential equations are considered as generalizations of classical integer-order partial differential equations. The proposed method is established in both time and space to approximate the solutions. The Caputo fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to diagonally block system of nonlinear algebraic equations, which will be solved using Newton–Raphson iterative method. Some model examples of the equations, defined on a rectangular domain, have tested with various values of fractional order $$\alpha$$ and $$\beta$$. Moreover, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. For the proposed method, highly accurate numerical results are obtained which are compared with the analytical solution to confirm the accuracy and efficiency of the proposed method.

Chebyshev pseudospectral approximation of two dimensional fractional Schrodinger equation on a convex and rectangular domain

In this article, the authors report the Chebyshev pseudospectral method for solving twodimensional nonlinear Schrodinger equation with fractional order derivative in time and space both. The modified Riemann-Liouville fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to a diagonally block system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. The proposed methods have shown error analysis without any dependency on time and space step restrictions. Some model examples of the equations, defined on a convex and rectangular domain, have tested with various values of fractional order <i>α</i> and <i>β</i>. Moreover, numerical solutions are demonstrated to justify the theoretical results.

A conservative numerical method for the fractional nonlinear Schrödinger equation in two dimensions

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrodinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grunwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrodinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

Solitons generated by self-organization in bismuth germanium oxide single crystals during the interaction with laser beam

We present here the experimental, theoretical, and numerical investigations of Kerr solitons generated by self-organization in black and yellow high quality bismuth germanium oxide (Bi12GeO20) single crystals. A picosecond laser beam of increasing power induces competing cubic and quintic nonlinearities. The numerical evolution of two-dimensional complex cubic-quintic nonlinear Schrodinger equation with measured values of nonlinearities shows the compensation of diffraction by competing cubic and quintic nonlinearities of opposite sign, i.e., the self-generation and stable propagation of solitons. Experiments as well as numerical simulations show higher nonlinearity in the black Bi12GeO20 than in the more transparent yellow one.

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrodinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrodinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.

Numerical analysis of two local conservative methods for two-dimensional nonlinear Schr&amp;ouml;dinger equation

In this paper, we propose two local conservative methods for solving two-dimensional nonlinearSchrodinger equation. Without consideration of the boundary conditions, they can preserve corresponding local energy and momentum conservation laws exactly at arbitrary spatial-temporal regions. Meanwhile, the charge, global energy and global momentum conservation laws can be conserved under suitable boundary conditions.Numerical analysis, including conservative properties and error estimation analysis, are investigated.Furthermore, a similar multi-symplectic Preissman scheme is constructed as comparison.Numerical experiments show the advantages of the proposed methods during long-time numerical simulations and validate the analysis.

New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity

A shifted Jacobi collocation method in two stages is constructed and used to numerically solve nonlinear Schrodinger equations (NLSEs) with a Kerr law nonlinearity, subject to initial-boundary conditions. An expansion in a series of spatial shifted Jacobi polynomials with temporal coefficients for the approximate solution is considered. The first stage, collocation at the shifted Jacobi Gauss-Lobatto (SJ-GL) nodes, is applied for a spatial discretization; its spatial derivatives occur in the NLSE with a treatment of the boundary conditions. This in all will produce a system of ordinary differential equations (SODEs) for the coefficients. The second stage is to collocate at the shifted Jacobi Gauss-Radau (SJ-GR-C) nodes in the temporal discretization to reduce the SODEs to a system of algebraic equations which is solved by an iterative method. Both stages can be extended to solve the two-dimensional NLSEs. Numerical examples are carried out to confirm the spectral accuracy and the efficiency of the proposed algorithms.

Wave–vortex interactions in the nonlinear Schrödinger equation

This is a theoretical study of wave–vortex interaction effects in the two-dimensional nonlinear Schrodinger equation, which is a useful conceptual model for the limiting dynamics of superfluid quantum condensates at zero temperature. The particular wave–vortex interaction effects are associated with the scattering and refraction of small-scale linear waves by the straining flows induced by quantized point vortices and, crucially, with the concomitant nonlinear back-reaction, the remote recoil, that these scattered waves exert on the vortices. Our detailed model is a narrow, slowly varying wavetrain of small-amplitude waves refracted by one or two vortices. Weak interactions are studied using a suitable perturbation method in which the nonlinear recoil force on the vortex then arises at second order in wave amplitude, and is computed in terms of a Magnus-type force expression for both finite and infinite wavetrains. In the case of an infinite wavetrain, an explicit asymptotic formula for the scattering angle is also derived and cross-checked against numerical ray tracing. Finally, under suitable conditions a wavetrain can be so strongly refracted that it collapses all the way onto a zero-size point vortex. This is a strong wave–vortex interaction by definition. The conditions for such a collapse are derived and the validity of ray tracing theory during the singular collapse is investigated.

A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation

Abstract In this paper, an efficient numerical technique is developed to approximate the solution of two-dimensional cubic nonlinear Schrodinger equations. The method is based on the nonsymmetric radial basis function collocation method (Kansa's method), within an operator Newton algorithm. In the proposed process, three-dimensional radial basis functions (especially, three-dimensional Multiquadrics (MQ) and Inverse multiquadrics (IMQ) functions) are used as the basis functions. For solving the resulting nonlinear system, an algorithm based on the Newton approach is constructed and applied. In the multilevel Newton algorithm, to overcome the instability of the standard methods for solving the resulting ill-conditioned system an interesting and efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-conditioned system. Finally, the presented method is used for solving some examples of the governing problem. The comparison between the obtained numerical solutions and the exact solutions demonstrates the reliability, accuracy and efficiency of this method.

Exact Solutions of two-Dimensional Nonlinear Schrödinger Equations with External Potentials

In this paper, exact solutions of two-dimensional nonlinear Schrodinger equation with kerr, saturable and quintic type of nonlinearities are studied by means of the Homotopy analysis method (HAM). Linear stability properties of these solutions are investigated by the linearized eigenvalue problem. We also investigate nonlinear stability properties of the exact solutions obtained by HAM by direct simulations.

Two-dimensional nonlinear Schrödinger equation with random radial data

We study radial solutions of a certain two-dimensional nonlinear Schrodinger (NLS) equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schrodinger equation with Lp estimates of Laguerre functions, we are able to prove an almost-sure global well-posedness result and the invariance of the Gibbs measure. We also discuss an application to the NLS equation without harmonic potential.

Scattering for the cubic Klein–Gordon equation in two space dimensions

We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the defocusing case, solutions are global and have finite global $L^4_{t,x}$ spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state. These results rely on analogous statements for the two-dimensional cubic nonlinear Schrodinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional. It was previously shown by Nakanishi that spacetime bounds for Klein--Gordon equations imply the same for nonlinear Schrodinger equations.

Two-dimensional dissipative solitons supported by localized gain

We show that the balance between localized gain and nonlinear cubic dissipation in the two-dimensional nonlinear Schrödinger equation allows for the existence of stable localized modes that we identify as solitons. Such modes exist only when the gain is strong enough and the energy flow exceeds certain threshold value. Above the critical value of the gain, symmetry breaking occurs and asymmetric dissipative solitons emerge.

Nonlinear effects of self-action of waves in 2D coupled ferromagnetic structures

Magnetostatic wave self-action effects in a 2D structure consisting of two ferromagnetic films have been studied using a numerical solution to the system of two-dimensional nonlinear Schrodinger equations. Main features of these effects have been analyzed in comparison with similar effects in a 2D structure based on a single film. The applicability of coupled structures to control the formation of 2D nonlinear wave beams and magnetostatic wave packets has been considered.

Two-dimensional solitons in irregular lattice systems

We compute and study localized nonlinear modes (solitons) in the semi-infinite gap of the focusing two-dimensional nonlinear Schrodinger (NLS) equation with various irregular lattice-type potentials. The potentials are characterized by large variations from periodicity, such as vacancy defects, edge dislocations, and a quasicrystal structure. We use a spectral fixed-point computational scheme to obtain the solitons. The eigenvalue dependence of the soliton power indicates parameter regions of self-focusing instability; we compare these results with direct numerical simulations of the NLS equation. We show that in the general case, solitons on local lattice maximums collapse. Furthermore, we show that the Nth-order quasicrystal solitons approach Bessel solitons in the large-N limit.

Existence and stability of a solution blowing up on a sphere for an L2-supercritical nonlinear Schrödinger equation

We consider the quintic two-dimensional focusing nonlinear Schrodinger equation iut=−Δu−|u|4u which is L2-supercritical. Even though the existence of finite-time blow-up solutions in the energy space H1 is known, very little is understood concerning the singularity formation. Numerics suggest the existence of a stable blow-up dynamic corresponding to a self-similar blowup at one point in space. We prove the existence of a different type of dynamic and exhibit an open set among the H1-radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere. This is the first result of an explicit description of a blow-up dynamic in the L2-supercritical setting