Bäcklund transformation, analytic soliton solutions and numerical simulation for a (2+1)-dimensional complex Ginzburg–Landau equation in a nonlinear fiber

2017 ◽  
Vol 31 (28) ◽  
pp. 1750258
Author(s):  
Ming-Xiao Yu ◽  
Bo Tian ◽  
Jun Chai ◽  
Hui-Min Yin ◽  
Zhong Du
Keyword(s):  
Numerical Solutions ◽  
Landau Equation ◽  
Chromatic Dispersion ◽  
Soliton Solutions ◽  
Ginzburg Landau ◽  
Optical Filtering ◽  
Absolute Value ◽  
Ginzburg Landau Equation ◽  
Nonlinear Fiber ◽  
The Absolute

In this paper, we investigate a nonlinear fiber described by a (2[Formula: see text]+[Formula: see text]1)-dimensional complex Ginzburg–Landau equation with the chromatic dispersion, optical filtering, nonlinear and linear gain. Bäcklund transformation in the bilinear form is constructed. With the modified bilinear method, analytic soliton solutions are obtained. For the soliton, the amplitude can decrease or increase when the absolute value of the nonlinear or linear gain is enlarged, and the width can be compressed or amplified when the absolute value of the chromatic dispersion or optical filtering is enhanced. We study the stability of the numerical solutions numerically by applying the increasing amplitude, embedding the white noise and adding the Gaussian pulse to the initial values based on the analytic solutions, which shows that the numerical solutions are stable, not influenced by the finite initial perturbations.

scholarly journals Optical soliton solutions of the Ginzburg-Landau equation with conformable conformable derivative and Kerr law nonlinearity

2018 ◽  
Vol 65 (1) ◽  
pp. 73 ◽  
Author(s):  
Francisco Gomez ◽  
Behzad Ghanbari
Keyword(s):  
Rational Function ◽  
Function Method ◽  
Landau Equation ◽  
Soliton Solutions ◽  
Conformable Derivative ◽  
Ginzburg Landau ◽  
Kerr Law ◽  
Ginzburg Landau Equation ◽  
Kerr Law Nonlinearity ◽  
Exponential Rational Function Method

By using the generalized exponential rational function method we obtain new periodic and hyperbolic soliton solutions for the conformable Ginzburg-Landau equation with Kerr law nonlinearity. The conformable derivative was considered to obtain the exact solutions under constraint conditions. To determine the solution of the model, the method uses the generalization of the exponential rational function method. Numerical simulations are performed to confirm the efficiency of the proposed method.

scholarly journals Non-autonomous Ginzburg-Landau solitons using the He-Li mapping method

2020 ◽  
Vol 27 (4) ◽  
pp. e104
Author(s):  
Maximino Pérez Maldonado ◽  
Haret C. Rosu ◽  
Elizabeth Flores Garduño
Keyword(s):  
Landau Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Mapping Method ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

We find and discuss the non-autonomous soliton solutions in the case of variable nonlinearity and dispersion implied by the Ginzburg-Landau equation with variable coefficients. In this work we obtain non-autonomous Ginzburg-Landau solitons from the standard autonomous Ginzburg-Landau soliton solutions using a simplified version of the He-Li mapping. We find soliton pulses of both arbitrary and fixed amplitudes in terms of a function constrained by a single condition involving the nonlinearity and the dispersion of the medium. This is important because it can be used as a tool for the parametric manipulation of these non-autonomous solitons.

scholarly journals Spectral stability of pattern-forming fronts in the complex Ginzburg–Landau equation with a quenching mechanism

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 170-244
Author(s):  
Ryan Goh ◽  
Björn de Rijk
Keyword(s):  
Ground State ◽  
Intermediate State ◽  
Landau Equation ◽  
Superposition Principle ◽  
Spectral Stability ◽  
Ginzburg Landau ◽  
Invasion Speed ◽  
Ginzburg Landau Equation ◽  
The Absolute ◽  
Pattern Forming

Abstract We consider pattern-forming fronts in the complex Ginzburg–Landau equation with a traveling spatial heterogeneity which destabilises, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed c just below the linear invasion speed of the pattern-forming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearisation about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed c increases towards the linear invasion speed, the absolute spectrum stabilises with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectral stability of the front in L 2 ( R ) . The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivise the linear flow, and use Riemann surface unfolding in combination with a superposition principle to study the evolution of subspaces as solutions to the associated matrix Riccati differential equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati–Evans function, and can be located using winding number and parity arguments.

Improving the accuracy for numerical solutions of the Ginzburg - Landau equation

2020 ◽  
pp. 45-57
Author(s):  
Виктор Иванович Паасонен ◽  
Михаил Петрович Федорук
Keyword(s):  
Boundary Conditions ◽  
Fiber Optics ◽  
Numerical Solutions ◽  
Landau Equation ◽  
Simultaneous Increase ◽  
Ginzburg Landau ◽  
Actual Problem ◽  
Order Of Accuracy ◽  
Additional Boundary Conditions ◽  
Ginzburg Landau Equation

Решение актуальной задачи повышения порядка точности разностных методов решения задач нелинейной волоконной оптики выше четвертого путем непосредственного построения сложных схем на расширенных шаблонах сопряжено с усложнением матрицы системы и с затруднениями в постановке дополнительных граничных условий. Кроме того, при таком подходе не происходит одновременное повышение точности также и по эволюционной переменной. В данной работе рассматривается альтернативный путь - применение экстраполяции Ричардсона, которая сводится к построению подходящих линейных комбинаций решений на различных сетках. Этот способ позволяет повышать порядок точности по обеим переменным, избегая при этом проблем с усложнением шаблонов, постановкой дополнительных граничных условий и реализацией алгоритмов. Как средство дополнительного улучшения точности наряду с простыми (однократными) поправками исследуются также двойные поправки на основе экстраполяции Ричардсона. Методика протестирована на нескольких точных решениях уравнения Гинзбурга - Ландау Increasing the order of accuracy for difference methods is an actual problem in nonlinear fiber optics. Computations, which use higher than the fourth order of accuracy by the direct construction of complex circuits on extended templates pose the complication of the system matrix and difficulties in setting additional boundary conditions. In addition, with this approach, there is no simultaneous increase in accuracy for the evolutionary variable. In this paper, we consider an alternative way, namely, application of the Richardson extrapolation, which reduces to construction of suitable linear combinations for solutions on various grids. This method allows improving the order of accuracy for both variables, while avoiding problems associated with the complication of templates, implementation of algorithms and setting additional boundary conditions. Double corrections are also considered to further improve accuracy. The technique was tested on exact solutions of the Ginzburg - Landau equation

Exact homoclinic wave and soliton solutions for the 2D Ginzburg–Landau equation

2008 ◽  
Vol 372 (17) ◽  
pp. 3010-3014 ◽  
Author(s):  
Zhengde Dai ◽  
Zitian Li ◽  
Zhenjiang Liu ◽  
Donglong Li
Keyword(s):  
Landau Equation ◽  
Soliton Solutions ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

Dark soliton solutions of the cubic-quintic complex Ginzburg–Landau equation with high-order terms and potential barriers in normal-dispersion fiber lasers

2021 ◽  
Author(s):  
Marco A. Viscarra ◽  
Deterlino Urzagasti
Keyword(s):  
Optical Fibers ◽  
Landau Equation ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Two Dimensions ◽  
Ginzburg Landau ◽  
Normal Dispersion ◽  
Homogeneous Solutions ◽  
Dark Solitons ◽  
Ginzburg Landau Equation

In this paper, we numerically study dark solitons in normal-dispersion optical fibers described by the cubic-quintic complex Ginzburg–Landau equation. The effects of the third-order dispersion, self-steepening, stimulated Raman dispersion, and external potentials are also considered. The existence, chaotic content and interactions of these objects are analyzed, as well as the tunneling through a potential barrier and the formation of dark breathers aside from dark solitons in two dimensions and their mutual interactions as well as with periodic potentials. Furthermore, the homogeneous solutions of the model and the conditions for their stability are also analytically obtained.

scholarly journals NUMERICAL SOLUTIONS OF A VECTOR GINZBURG–LANDAU EQUATION WITH A TRIPLE-WELL POTENTIAL

2003 ◽  
Vol 13 (11) ◽  
pp. 3295-3306 ◽  
Author(s):  
JOHN M. NEUBERGER ◽  
DENNIS R. RICE ◽  
JAMES W. SWIFT
Keyword(s):  
Small Parameter ◽  
Triple Junction ◽  
Morse Index ◽  
Numerical Solutions ◽  
Landau Equation ◽  
Gradient Algorithm ◽  
Existence Proof ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Index 2

We numerically compute solutions to the vector Ginzburg–Landau equation with a triple-well potential. We use the Galerkin Newton Gradient Algorithm of Neuberger and Swift and bifurcation techniques to find solutions. With a small parameter, we find a Morse index 2 triple junction solution. This is the solution for which Flores, Padilla and Tonegawa gave an existence proof. We classify all of the solutions guaranteed to exist by the Equivariant Branching Lemma at the first bifurcation points of the trivial solutions. Guided by the symmetry analysis, we numerically compute the solution branches.

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