Complex Nonlinear Schrodinger Research Articles

Modulational Collapsing and Adjusting with External Potential in a Complex Nonlinear Plasma

Modulational collapse of a wave pulse in a complex nonlinear plasma described by a complex nonlinear Schrodinger equation (CNSE) including the effects of viscous heating and nonlinear damping, and the adjusting by an external potential are investigated. Theoretical and numerical results reveal that the original pulse first suffers modulational instability, the growth rate of the modulational instability increases with the increase of viscous heating, but it is not sensitive to the nonlinear damping. The instability induces the self-similar collapsing, then the fields rapidly transform into a turbulent state with short-wavelength modes. The results are explained in terms of the evolution of the total energy, which can become invariant during the evolution even though the system is nonconservative. The external potential can delay the self-defocusing process of the pulse. Those results can apply to the phenomena described by the CNSE.

More general families of exact solitary wave solutions of the nonlinear Schrödinger equation with their applications in nonlinear optics

In this article we analytically studied the complex nonlinear Schrodinger equation with Kerr law nonlinearity using the auxiliary equation mapping method, as a result, we found a series of more general and new families of exact solutions, which are more powerful in the development of soliton dynamics, quantum plasma, adiabatic parameter dynamics, biomedical problems, fluid dynamics, industrial studies, nonlinear optics and many other fields. The calculations demonstrate that this method is more reliable, straightforward and effective to analytically study other nonlinear complicated physical problems modeled by complex nonlinear partial differential equations arising in mathematical physics, hydrodynamics, fluid mechanics, mathematical biology, plasma physics, engineering disciplines, chemistry and many other natural sciences. We have also expressed our solutions graphically with the help of Mathematica 10.4 to physically understand the behavior of different shapes of solutions including kink-type, anti-kink-type, half-bright and dark solitons.

Nonlinear properties and stabilities of polaritonic crystals beyond the low-excitation-density limit

Coherent properties of a two dimensional spatially periodic structure - polaritonic crystal (PolC) formed by trapped two-level atoms in an optical cavity array interacting with a light field, are analyzed. By considering the wave function overlapping both for photonic and atomic states, a cubic-quintic complex nonlinear Schrodinger equation (CNLSE) is derived for the dynamics of coupled atom-light states - wave function of low branch (LB) polaritons, associated with PolC in the continuous limit. The variational approach predicts that a stable ground state wave function of PolC exists but is accompanied by an oscillating width. For a negative scattering length, the wave function collapses in the presence of a small quintic nonlinearity appear due to a three body polariton interaction. Studying non-equilibrium (dissipative) dynamics of polaritons with adiabatic approximation we have shown that the collapse of PolC wave function can be prevented even in the presence of small decaying of a number of polariton particles.

Relativistic Burgers and nonlinear SchrÖdinger equations

We construct relativistic complex Burgers-Schrodinger and nonlinear Schrodinger equations. In the nonrelativistic limit, they reduce to the standard Burgers and nonlinear Schrodinger equations and are integrable through all orders of relativistic corrections.

On a complex Duffing system with random excitation

Abstract In this paper, we consider a complex Duffing system subjected to nonstationary random excitation of the form, z ¨ ( t ) + 2 ω ξ z ˙ ( t ) + ω 2 z + ϵ z ( t ) | z ( t ) | 2 = α F ( t ) , where z(t) is a complex function, α = 1 + i, i denotes the imaginary unit, ω, ξ represent natural frequency and damping coefficient respectively, ϵ is the small perturbation parameter and nonlinearity strength, and F(t) is a random function. This equation with F(t) = 0 has connection to the complex nonlinear Schrodinger equation which appears in many important fields of physics. The truncated Wiener–Hermite expansion is applied to derive the deterministic integro-differential equations. These equations have been solved by the small parameter perturbation approach to describe the root mean square response. The approximate solution moments for the original systems has been obtained analytically. Figures are presented to show the effect of the nonlinearity strength and the damping coefficients, respectively.

Small-signal analysis of cross-phase modulation instability in lossy fibres

Based on the small signal analysis theory and split-step Fourier method, the complex nonlinear Schrodinger equation (NLSE) with fibre loss can be solved. This procedure is also adapted to the NLSE with the high order dispersion terms (β3, β4). Because the fibre loss is taken into account, experiment demonstrates that the cross-phase modulation instability gain spectrum is more similar to represent the actual systems. The fibre loss decreases the modulation instability (MI) gain and has an effect on SNR which is caused by the cross-phase modulation instability. The MI intensity fluctuation caused by the nonlinear and dispersion effects is directly derived. As a result, the initial stage of MI can be described, and the whole characteristic of MI can also be discussed in this way.

Solution of nonlinear wave equations of the complex quintic Ginzburg-Landau and nonlinear Schrodinger type

In this letter, we present a new method to solve the nonlinear wave equations of the complex quintic Ginzburg-Landau and nonlinear Schrodinger type for arbitrarily large coefficients. The key point is to take the features of each term and to separate the equation into two parts, a linear part and a nonlinear one. The former is analyzed by means of the Fourier transform in the complex field. The latter is solved by modulus-phase formulation leading to an hyperbolic system highly connected to the nonlinear systems treated in fluid dynamics. This will allow the analysis of the difficult cases that appear when the nonlinear higher order terms (that give rise to phenomena such as shock formation) are considered.

Envelope Modulational Instability in a Nonlinear Dissipative Transmission Line

A bi-inductance LC circuit is considered, and the envelope modulation is reduced to the complex nonlinear Schrodinger (CNLS) equation. The Benjamin–Feir (or modulational, as it is sometimes called) instability for the CNLS equation is investigated to lead the Lange–Newell criterion.

Dynamics of Fluctuating Bose–Einstein Condensates

We present a generalized Gross–Pitaevskii equation that describes the dissipative dynamics of a trapped partially Bose-condensed gas. It takes the form of a complex nonlinear Schrodinger equation with noise. We consider an approximation to this Langevin field equation that preserves the correct equilibrium for both the condensed and the noncondensed parts of the gas. We then use this formalism to describe the reversible formation of a one-dimensional Bose condensate, and compare with recent experiments. In addition, we determine the frequencies and the damping of collective modes in this case.