General Nonlinear Schrodinger Research Articles

A local differential quadrature method for the generalized nonlinear Schrödinger (GNLS) equation

A local differential quadrature method based on Fourier series expansion numerically solves the generalized nonlinear Schrodinger equation. For time integration, a Runge-Kutta fourth-order method is used. Matrix stability analysis is used to examine the method's stability. Three test problems involving the motion of a single solitary wave, the interaction of two solitary waves, and a solution that blows up in finite time, respectively, demonstrate the accuracy and efficiency of the provided method. Finally, the numerical results obtained from the presented method are compared with the exact solution and those obtained in earlier works available in the literature.

A Vector General Nonlinear Schrödinger Equation with $$(m+n)$$ Components

A vector general nonlinear Schrodinger equation with $$(m+n)$$ components is proposed, which is a new integrable generalization of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrodinger equation. Resorting to the Riccati equations associated with the Lax pair and the gauge transformations between the Lax pairs, a general N-fold Darboux transformation of the vector general nonlinear Schrodinger equation with $$(m+n)$$ components is constructed, which can be reduced directly to the classical N-fold Darboux transformation and the generalized Darboux transformation without taking limits. As an illustrative example, some exact solutions of the two-component general nonlinear Schrodinger equation are obtained by using the general Darboux transformation, including a first-order rogue-wave solution, a fourth-order rogue-wave solution, a breather solution, a breather–rogue-wave interaction, two solitons and the fission of a breather into two solitons. It is a very interesting phenomenon that, for all $$M>0$$, there exists a rogue-wave solution for the two-component general nonlinear Schrodinger equation such that the amplitude of the rogue wave is M times higher than its background wave.

负曲率空芯光纤对飞秒超短脉冲光的压缩研究

In order to compress femtosecond ultrashort pulses, a novel fiber called negative-curvature hollow-core fiber, is investigated and is used to compress the output of a Ti:Sapphire laser. Firstly, a hollow-core fiber with circular tubes cladding is introduced, and the loss parameter is calculated with the Finite Element Method and the calculated results are compared with the experimental results. Following this, the evolution of an ultrashort pulse along the fiber is simulated in the General Nonlinear Schrodinger Equation. At last the compression experiment using NC-HCF is conducted. The 160 fs output of a Ti:Sapphire laser is coupled into the fiber with high-pressure Argon. An output of 84 fs is achieved, which is the result of the balance between anomalous dispersion and self-phase modulation in the fiber. The experimental results matched the simulations. This novel fiber, which has the advantages of a high damage threshold, low and adjustable dispersion and nonlinear coefficients, is a promising material in the field of ultrafast optics.

Breathers, solitons and rogue waves of the quintic nonlinear Schrödinger equation on various backgrounds

We investigate the generation of breathers, solitons, and rogue waves of the quintic nonlinear Schrodinger equation (QNLSE) on uniform and elliptical backgrounds. The QNLSE is the general nonlinear Schrodinger equation that includes all terms up to the fifth-order dispersion. We use Darboux transformation to construct initial conditions for the dynamical generation of localized high-intensity optical waves. The condition for the breather-to-soliton conversion is provided with the analysis of soliton intensity profiles. We discover a new class of higher-order solutions in which Jacobi elliptic functions are set as background seed solutions of the QNLSE. We also introduce a method for generating a new class of rogue waves—the periodic rogue waves—based on the matching of the periodicity of higher-order breathers with the periodicity of the background dnoidal wave.

Darboux transformation of a new generalized nonlinear Schrödinger equation: soliton solutions, breather solutions, and rogue wave solutions

In this paper, a new generalized nonlinear Schrodinger (GNLS) equation is investigated by Darboux matrix method. Firstly, the n-fold Darboux transformation (DT) of the GNLS equation is constructed. Then, the soliton solutions, breather solutions, and rogue wave solutions of the GNLS equation are studied based on the DT by choosing different seed solutions. Furthermore, the dynamic features of these solutions are explicitly delineated through some figures with the help of Maple software.

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal

A general nonlocal nonlinear Schrodinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a $$\beta $$ -plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.

The general mixed nonlinear Schrödinger equation: Darboux transformation, rogue wave solutions, and modulation instability

In this paper, the Darboux transformation method has been successfully applied to a general mixed nonlinear Schrodinger equation and some rogue wave solutions are proposed. First of all, the determinant representation of an n-fold DT is given explicitly. Then starting with a periodic seed solution, we obtain some rogue wave solutions of the general mixed nonlinear Schrodinger equation through iteration of a generalized DT. Second, the three-dimensional images and density profiles of the rogue waves are plotted to show the structures of these rogue wave solutions. Finally, we give evidence for the connection between the occurrence of rogue wave solutions and the modulation instability.

Painlevé-integrability and explicit solutions of the general two-coupled nonlinear Schrödinger system in the optical fiber communications

A general two-coupled nonlinear Schrodinger system is investigated with symbolic computation. The system is regarded to be a more general model than other coupled nonlinear Schrodinger systems since its coefficients of the self-phase modulation, cross-phase modulation, and four-wave mixing terms are arbitrary. Painleve-integrability associated with the system is examined by means of Painleve test. As a result, Painleve–Backlund transformation is constructed with truncating the Laurent series at the constant level term. In addition, a family of explicit solutions corresponding to the vacuum solutions are derived through the Painleve–Backlund transformation.

On Nth-order rogue wave solution to the generalized nonlinear Schrödinger equation

Abstract In this Letter, the generalized nonlinear Schrodinger (GNLS) equation is investigated by Darboux matrix method. A generalized Darboux transformation (DT) of the GNLS equation is constructed with the help of the gauge transformation for an Ablowitz–Kaup–Newell–Segur (AKNS) type GNLS spectral problem, from which a unified formula of N th-order rogue wave solution to the GNLS equation is given. In particular, the first and second-order rogue wave solutions to the GNLS equation are explicitly illustrated through some figures.

The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions

The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.

Spectral variations in supercontinuum pulse propagation

In this paper, we have used the general nonlinear Schrodinger equation (GNLSE) to study the spectral evolution of a short light pulse propagating through an optical fiber. The equation includes the main factors affecting its propagation through the fiber, involving the linear and nonlinear dispersive characteristics. We find the asymmetry feature of the spectrum due to the Raman Effect. The first order group velocity dispersion sets in after a finite propagation length. We have systematically studied the evolution of the pulse through a nonlinear medium.

Propagation of Gaussian beam in an absorbing waveguide

Propagation of a Gaussian beam in an absorbing waveguide is analyzed for cubic–quintic and saturable media taking into account both linear and nonlinear absorption. A “collective variable approach” technique, based on trial functions, is used to solve the general nonlinear Schrodinger equation. In the absence of losses, we construct a diagram which defines regions of oscillatory and diffractive beam propagation for both types of media, and also a diagram that compares bistable behavior in such media. We show that if the linear and nonlinear absorption coefficients are small, the behavior of the oscillations of the beam width on propagation allows one to distinguish between cubic–quintic and saturable media. By reversing the sign of the linear absorption, we analyze the behavior of the beam propagation in media with gain and nonlinear absorption. In cubic–quintic media, the energy reaches a plateau for certain ratios of gain to losses, whereas for saturable media, the energy increases throughout the beam propagation.

Quintic B-spline collocation method for the generalized nonlinear Schrödinger equation

Abstract In this study, the generalized nonlinear Schrodinger (GNLS) equation is solved numerically by the quintic B-spline collocation finite element method. In the method, Crank–Nicolson scheme is used for the time integration, and the space variable is discretized by means of quintic B-spline functions. Finally, we investigate properties of the numerically computed solutions of the GNLS equation; in particular we study the generation of solitary waves, interaction of solitons and blow up.

Bifurcations and the stability of the surface envelope solitons for a finite-depth fluid

The dynamics of the quasi-monochromatic surface gravitational waves in a finite-depth fluid is studied for the case where the product of the wavenumber by the depth of the fluid is close to the critical value kcrh ≈ 1.363. Within the framework of the Hamiltonian formalism, the general nonlinear Schrodinger equation is derived. In contrast to the classical nonlinear Schrodinger equation, this equation involves the gradient terms to the four-wave interaction, as well as the six-wave interaction. This equation is used to analyze the modulation instability of the monochromatic waves, as well as the bifurcations of the soliton solutions and their stability. It is shown that the solitons are stable and unstable to finite perturbations for focusing and defocusing nonlinearities, respectively.

Solutions of Two Kinds of Non-Isospectral GeneralizedNonlinear Schrödinger Equation Related to Bose–Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger (GNLS) equations,which are two important models of nonlinear excitations of matter wavesin Bose–Einstein condensates, are studied. Two novel transformationsare constructed such that these two GNLS equations are transformed tothe well-known nonlinear Schrödinger (NLS) equation, which is anisospectral equation. Therefore, once one solution of the NLS equationis provided, we can immediately obtain one solution for two GNLSequations by these transformations. Thus it is unnecessary to solvethese two non-isospectral GNLS equations directly. Soliton solutions andperiodic solutions are obtained for them by two transformations fromthe corresponding solutions of the NLS equation, which are generated byDarboux transformation.

Envelope exact solutions for the generalized nonlinear Schrödinger equation with a source

In this letter, the generalized nonlinear Schrodinger (GNLS) equation, phase-locked a source κ ei[χ(ξ)−ωt]: , is investigated. Firstly, we reduce this equation to a second-order non-homogeneous nonlinear ordinary differential equation via a plane wave transformation and some constraint conditions. Then we use some fractional transformations to study exact solutions in obtaining the GNLS equation. As a consequence, many types of exact solutions are deduced such as envelope rational solutions, envelope periodic wave solutions, envelope solitary wave solutions and envelope doubly periodic solutions. Similarly, the corresponding exact solutions can also be obtained for the Hirota-type GNLS equation with a source and their combined equation.

From Ginzburg-Landau to Gross-Pitaevskii

In this note we consider the Gross-Pitaevskii equation iϕ t +Δϕ+ϕ(1−∣ϕ∣2)=0, where ϕ is a complex-valued function defined on ℝN × ℝ, and study the following 2-parameters family of solitary waves: ϕ(x, t)=e iωt v(x 1−ct, x′), where $\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}(\omega, c)\in {\Op R}^2, {\upsilon} \in L_{\rm loc}^3 ({\Op R}^N, {\bf C})$ and x′ denotes the vector of the last N−1 variables in ℝ N . We prove that every distribution solution ϕ, of the considered form, satisfies the following universal (and sharp) L ∞-bound: $$\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}\Vert\varphi\Vert_{L^\infty({\Op R}^N\times{\Op R})}^2 \le \max\bigg\{0, 1-\omega+{c^2\over 4}\bigg\}.$$ This bound has two consequences. The first one is that ϕ is smooth and the second one is that a solution φ≢0 exists, if and only if $1-\omega+{c^2\over 4} > 0$ . We also prove a non-existence result for some solitary waves having finite energy. Some more general nonlinear Schrodinger equations are considered in the third and last section. The proof of our theorems is based on previous results of the author ([7]) concerning the Ginzburg-Landau system of equations in ℝ N .