He’s variational iteration method for the modified equal width equation

Variational iteration method is introduced to solve the modified equal width equation. This method provides remarkable accuracy in comparison with the analytical solution. Three conservation quantities are reported. Numerical results demonstrate that this method is a promising and powerful tool for solving the modified equal width equation.     

Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.     

Application of He’s variational iteration method to Helmholtz equation

In this article, we implement a new analytical technique, He’s variational iteration method for solving the linear Helmholtz partial differential equation. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via the variational theory. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the boundary/initial conditions. The results compare well with those obtained by the Adomian’s decomposition method.     

Some continuous dependence results on the complex Ginzburg–Landau equation

Continuous dependence on a modelling parameter are established for solutions to a problem for a complex Ginzburg–Landau equation. We establish continuous dependence on the coefficient of the cubic term, and also on the coefficient of the term multiplying the Laplacian. Copyright 2003 John Wiley & Sons, Ltd.     

Continuous dependence on modelling for a complex Ginzburg–Landau equation with complex coefficients

Continuous dependence on a modelling parameter is established for solutions of a problem for a complex Ginzburg–Landau equation. A homogenizing boundary condition is also used to discuss the continuous dependence results. We derive a priori estimates that indicate that solutions depend continuously on a parameter in the governing differential equation. Copyright © 2004 John Wiley & Sons, Ltd.     

Exact solutions of the one-dimensional generalized modified complex Ginzburg–Landau equation

The one-dimensional (1D) generalized modified complex Ginzburg–Landau (MCGL) equation for the traveling wave systems is analytically studied. Exact solutions of this equation are obtained using a method which combines the Painlevé test for integrability in the formalism of Weiss–Tabor–Carnevale and Hirota technique of bilinearization. We show that pulses, fronts, periodic unbounded waves, sources, sinks and solution as collision between two fronts are the important coherent structures that organize much of the dynamical properties of these traveling wave systems. The degeneracies of the 1D generalized MCGL equation are examined as well as several of their solutions. These degeneracies include two important equations: the 1D generalized modified Schrödinger equation and the 1D generalized real modified Ginzburg–Landau equation. We obtain that the one parameter family of traveling localized source solutions called “Nozaki–Bekki holes” become a subfamily of the dark soliton solutions in the 1D generalized modified Schrödinger limit.     

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

A high‐order finite difference method for the two‐dimensional complex Ginzburg–Landau equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergent order in maximum norm is two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme is proposed. Some numerical examples are given to confirm the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 876–899, 2015     

A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

The numerical solution for the one‐dimensional complex fractional Ginzburg–Landau equation is considered and a linearized high‐order accurate difference scheme is derived. The fractional centered difference formula, combining the compact technique, is applied to discretize fractional Laplacian, while Crank–Nicolson/leap‐frog scheme is used to deal with the temporal discretization. A rigorous analysis of the difference scheme is carried out by the discrete energy method. It is proved that the difference scheme is uniquely solvable and unconditionally convergent, in discrete maximum norm, with the convergence order of two in time and four in space, respectively. Numerical simulations are given to show the efficiency and accuracy of the scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 105–124, 2017     

Limit behavior of global attractors for the complex Ginzburg–Landau equation on infinite lattices

In this work, the authors first show the existence of global attractors for the following lattice complex Ginzburg–Landau equation:

and for the following lattice Schrödinger equation:

Then they prove that the solutions of the lattice complex Ginzburg–Landau equation converge to that of the lattice Schrödinger equation as ε→0+. Also they prove the upper semicontinuity of as ε→0+ in the sense that .     

On a non-stationary Ginzburg–Landau superconductivity model

In this paper, we study a non-stationary superconductivity model derived from Ginzburg–Landau macroscopic theory. By using gauge invariance and studying a linear problem with curl boundary conditions, we obtain the existence of solutions. The solution is unique in the sense of gauge equivalence.     

Optimal control problem of a generalized Ginzburg–Landau model equation in population problems

In this paper, we consider the problem for distributed optimal control of the generalized Ginzburg–Landu model equation in population. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved, and the optimality system is established. Copyright © 2013 John Wiley & Sons, Ltd.     

Application of He’s homotopy perturbation method for Laplace transform

In this paper, an application of He’s homotopy perturbation method is proposed to compute Laplace transform. The results reveal that the method is very effective and simple.     

Difference methods for computing the Ginzburg‐Landau equation in two dimensions

In this article, three difference schemes of the Ginzburg‐Landau Equation in two dimensions are presented. In the three schemes, the nonlinear term is discretized such that nonlinear iteration is not needed in computation. The plane wave solution of the equation is studied and the truncation errors of the three schemes are obtained. The three schemes are unconditionally stable. The stability of the two difference schemes is proved by induction method and the time‐splitting method is analysized by linearized analysis. The algebraic multigrid method is used to solve the three large linear systems of the schemes. At last, we compute the plane wave solution and some dynamics of the equation. The numerical results demonstrate that our schemes are reliable and efficient. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011py; 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011     

Convergence analysis of a linearized Crank–Nicolson scheme for the two‐dimensional complex Ginzburg–Landau equation

A linearized Crank–Nicolson‐type scheme is proposed for the two‐dimensional complex Ginzburg–Landau equation. The scheme is proved to be unconditionally convergent in the L² ‐norm by the discrete energy method. The convergence order is \begin{align*}\mathcal{O}(\tau^2+h_1^2+h^2_2)\end{align*}, where τ is the temporal grid size and h₁,h₂ are spatial grid sizes in the x ‐ and y ‐directions, respectively. A numerical example is presented to support the theoretical result. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the Validity of the degenerate Ginzburg—Landau equation

The Ginzburg–Landau equation which describes nonlinear modulation of the amplitude of the basic pattern does not give a good approximation when the Landau constant (which describes the influence of the nonlinearity) is small. In this paper a derivation of the so-called degenerate (or generalized) Ginzburg–Landau (dGL)-equation is given. It turns out that one can understand the dGL-equation as an example of a normal form of a co-dimension two bifurcation for parabolic PDEs. The main body of the paper is devoted to the proof of the validity of the dGL as an equation whose solution approximate the solution of the original problem. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Vortex analysis of the periodic Ginzburg–Landau model

We study the vortices of energy minimizers in the London limit for the Ginzburg–Landau model with periodic boundary conditions. For applied fields well below the second critical field we are able to describe the location and number of vortices. Many of the results presented appeared in [H. Aydi, Doctoral Dissertation, Université Paris-XII, 2004], others are new.     

Exact solutions of generalized Zakharov and Ginzburg–Landau equations

By using the homogeneous balance principle, the exact solutions of the generalized Zakharov equations and generalized Ginzburg–Landau equation are obtained with the aid of a set of subsidiary higher-order ordinary differential equations (sub-equations for short).     

Analysis of some finite difference schemes for two‐dimensional Ginzburg‐Landau equation

We study the rate of convergence of some finite difference schemes to solve the two‐dimensional Ginzburg‐Landau equation. Avoiding the difficulty in estimating the numerical solutions in uniform norm, we prove that all the schemes are of the second‐order convergence in L₂ norm by an induction argument. The unique solvability, stability, and an iterative algorithm are also discussed. A numerical example shows the correction of the theoretical analysis.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1340‐1363, 2011     

Long time behaviour for generalized complex Ginzburg–Landau equation

In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL)

u_t=ρu−Δφ(u)−(1+iγ)Δu−νΔ²u−(1+iμ)|u|^2σu+αλ₁(|u|²u)+β(λ₂)|u|²