Narrow Results

We study numerically a prototype equation which arises generically as an envelope equation for a weakly inverted bifurcation associated to traveling waves: The complex quintic Ginzburg–Landau equation. We show six different stable localized structures including stationary pulses, moving pulses, stationary holes and moving holes, starting from localized initial conditions with periodic and Neumann boundary conditions.

In a microscopic quantum system one cannot perform a simultaneous measurement of particle and wave properties. This, however, may not be true for macroscopic quantum systems. As a demonstration, we propose to measure the local macroscopic current passed through two slits in a superconductor. According to the theory based on the linearized Ginzburg–Landau equation for the macroscopic pseudo wave function, the streamlines of the measured current should have the same form as particle trajectories in the Bohmian interpretation of quantum mechanics. By an explicit computation we find that the streamlines should show a characteristic wiggling, which is a consequence of quantum interference.

New boundary condition for the order parameter in the Ginzburg-Landau theory is applied to the case of CuO₂ planes which are the main structural elements responsible for superconductivity in high-T_c superconductors. It was found that the order parameter in these superconductors is significantly depressed in the CuO₂ planes. As a result, this boundary condition to the GL equations is found to limit the critical temperature of high-T_c superconductors. Thus, in order to increase T_c of cuprate superconductors, the number of CuO₂ planes that are within a short distance of each other in unit cell or insulating properties of the layers located in the vicinity to the CuO₂ planes should be increased.

This paper implements bifurcation method and the rational sine-Gordon expansion method to investigate the dynamical behavior of traveling wave solutions of a 2D complex Ginzburg–Landau equation. By varying the parameters, we obtained traveling wave solutions including the periodic wave solutions, solitary wave solution, kink and anti-kink wave solution and in addition by using the rational sine-Gordon expansion method, we determined bright and dark soliton which have a great contribution in the long distance telecommunication system.

The GL (Ginzburg–Landau) equation is a very important equation in superconductivity, which can be employed to explain many phenomena. In the present work, by using the expanded F-expansion method, some new exact Jacobi elliptic function solutions of the equation are obtained. More important is the phase factor of the solutions.

Ground states are studied by solving a modified Ginzburg–Landau model for mesoscopic metallic superconducting rings. It is found that surface effect related spin-orbit (SO) interaction can generate an effective orbital magnetic field of opposite orientations for spin-up and spin-down electrons which leads to spin-polarized states with opposite chirality. The quantum phase transition between the spin-polarized states and spin singlet superconducting states can occur by applying an external magnetic field normal to the ring-plane.

The purpose of the paper is to implement the collective variable method to investigate the generalized complex Ginzburg–Landau equation, which characterizes the kinetics of solitons in respect of pulse parameters for fiber optics. The statistical simulations of the interacting system of ordinary differential equations that reflect all the collective variables included in the pulse ansatz have been successfully carried out using a well-known numerical methodology, the fourth-order Runge–Kutta technique. The collective variable method is employed to plot the pulse variation characteristics as a function of propagation distance. The amplitude, temporal position, width, chirp, frequency, and phase of the pulse are all depicted against the propagated coordinate, where the width, phase of soliton, amplitude, and chirp all show a strong periodicity. The numerical dynamics of solitons have also been exhibited against varying values of pulse parameters to highlight differences in collective variables. Other key bits of the current investigation are also determined.

It is shown that pulses in the complete quintic one-dimensional Ginzburg–Landau equation with complex coefficients appear through a saddle-node bifurcation which is determined analytically through a suitable approximation of the explicit form of the pulses. The results are in excellent agreement with direct numerical simulations.

The study of nonlinear phenomena associated with physical phenomena is a hot topic in the present era. The fundamental aim of this paper is to find the iterative solution for generalized quintic complex Ginzburg–Landau (GCGL) equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus. We consider the projected equations by incorporating the Caputo fractional operator and investigate two examples for different initial values to present the efficiency and applicability of the FNDM. We presented the nature of the obtained results defined in three distinct cases and illustrated with the help of surfaces and contour plots for the particular value with respect to fractional order. Moreover, to present the accuracy and capture the nature of the obtained results, we present plots with different fractional order, and these plots show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The present investigation confirms the efficiency and applicability of the considered method and fractional operators while analyzing phenomena in science and technology.

The main novelty of this paper lies in five aspects. (1) Our study on the fractional-order derivative stochastic Ginzburg–Landau equation (FODSGLE) has resulted in numerous exact solutions using Jacobian elliptic functions (JEFs). These solutions offer valuable insights into complex physical phenomena and have practical implications across various fields. (2) By examining the effect of noise on FODSGLE solutions, we found consistent behavior regardless of the form of fractional derivatives. As noise intensity increases, the solutions deteriorate and tend toward zero. This study contributes to the existing literature by providing new insights and surpassing previous efforts in understanding the dynamics of FODSGLE. (3) Our research investigates how fractional order affects noise in solutions of the FODSGLE. Surprisingly, we found that changes in the fractional order ($\alpha$) have minimal influence on the system when the noise intensity ($\sigma$) is fixed. This indicates an aspect that has not been addressed extensively in existing literatures. (4) We compare different types of fractional derivatives within the context of the FODSGLE, presenting a novel contribution, while keeping the noise intensity or fractional order fixed. Our results demonstrate that the CFOD closely aligns with the MFOD, while displaying more differences with the solutions obtained using the BFOD. This comparison sheds light on an aspect that has not been thoroughly investigated in previous literatures. (5) This paper delves into the phase portraits of the FODSGLE and investigates the associated sensitivity and chaotic behaviors. Notably, previous studies on the stochastic Ginzburg–Landau equation (SGLE) have not extensively examined this aspect. By analyzing the phase portraits, we gain valuable insights into the dynamics and stability of FODSGLE solutions, uncovering intricate behaviors within the system. Our exploration of sensitivity and chaotic behaviors adds another dimension to understanding the (FODSGLE), laying a foundation for further research in this domain.

The complex Ginzburg–Landau equation is solved numerically and three types of pulsating pulses are obtained: plain pulsating, erupting, and creeping solitons. We discuss the main characteristics of these pulses and demonstrate the possibility of converting them into fixed-shape pulses under the influence of some higher-order effects.

This paper deals with the Wong–Zakai approximations and random attractors for stochastic Ginzburg–Landau equations with a white noise. We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation. In addition, when the stochastic Ginzburg–Landau equation is driven by an additive white noise, we establish the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero.

This paper is devoted to prove analyticity of stable invariant manifold in a neighbourhood of an unstable steady-state solution for Ginzburg-Landau equation defined in a bounded domain of dimension not more than three. This investigation is made for possible applications in stabilization theory for semilinear parabolic equation.

Some time ago, DeWitt-Morette¹et al. discussed the problem of the propagation of radiation or particles in the presence of a wedge. Their treatment includes path integral solutions of the wedge problem with Dirichlet or Neumann boundary conditions for various situations. Recently, the superconducting phase in a wedge has gained increasing interest. The linearized Ginzburg-Landau equation for the order parameter is formally similar to the Schrödinger equation. But the conditions of no normal current at the boundary pose very specific problems, which are discussed in the present paper.

New boundary condition for the order parameter in the Ginzburg-Landau theory is applied to the case of CuO₂ planes which are the main structural elements responsible for superconductivity in high-T_c superconductors. It was found that the order parameter in these superconductors is significantly depressed in the CuO₂ planes. As a result, this boundary condition to the GL equations is found to limit the critical temperature of high-T_c superconductors. Thus, in order to increase T_c of cuprate superconductors, the number of CuO₂ planes that are within a short distance of each other in unit cell or insulating properties of the layers located in the vicinity to the CuO₂ planes should be increased.