Downstream and upstream influence in river meandering. Part 2. Planimetric development

2001 ◽  
Vol 438 ◽  
pp. 213-230 ◽  
Author(s):  
G. SEMINARA ◽  
G. ZOLEZZI ◽  
M. TUBINO ◽  
D. ZARDI
Keyword(s):  
Evolution Equation ◽  
Numerical Solutions ◽  
Landau Equation ◽  
Fourier Series Expansion ◽  
Equilibrium Solutions ◽  
Ginzburg Landau ◽  
Small Perturbations ◽  
Channel Pattern ◽  
River Meandering ◽  
Spatial Modulations

The exact solution of the problem of river morphodynamics derived in Part 1 is employed to formulate and solve the problem of planimetric evolution of river meanders. A nonlinear integrodifferential evolution equation in intrinsic coordinates is derived. An exact periodic solution of such an equation is then obtained in terms of a modified Fourier series expansion such that the wavenumbers of the various Fourier modes are time dependent. The amplitudes of the Fourier modes and their wavenumbers satisfy a nonlinear system of coupled ordinary differential equations of the Landau type. Solutions of this system display the occurrence of two possible scenarios. In the sub-resonant regime, i.e. when the aspect ratio of the channel is smaller than the resonant value, meandering evolves according to the classical picture: a periodic train of small-amplitude sine-generated meanders migrating downstream evolve into the classical, upstream skewed, train of meanders of Kinoshita type. Evolution displays all the experimentally observed features: the meander growth rate increases up to a maximum and then decreases, while the migration speed decreases monotonically. No equilibrium solutions are found. In the super-resonant regime the picture is essentially reversed: downstream skewing develops while meanders migrate upstream.Numerical solutions of the planimetric evolution equation are obtained for the case when the initial channel pattern exhibits random small perturbations of the straight configuration. Under these conditions, the evolution displays the typical features of solutions of the Ginzburg–Landau equation, in particular, the occurrence of spatial modulations of the meandering pattern which organizes itself in the form of wavegroups. Furthermore, multiple loops develop in the advanced stage of meander growth.

scholarly journals The interaction of free Rossby waves with semi-transparent equatorial waveguide – wave-mean flow interaction

2009 ◽  
Vol 16 (3) ◽  
pp. 381-392 ◽  
Author(s):  
G. M. Reznik ◽  
V. Zeitlin
Keyword(s):  
Rossby Waves ◽  
Landau Equation ◽  
Spatial Modulation ◽  
Equilibrium Solutions ◽  
Mean Flow ◽  
Ginzburg Landau ◽  
Baroclinic Waves ◽  
Ginzburg Landau Equation ◽  
The Mean ◽  
Mean Current

Abstract. Nonlinear interactions of the barotropic Rossby waves propagating across the equator with trapped baroclinic Rossby or Yanai modes and mean zonal flow are studied within the two-layer model of the atmosphere, or the ocean. It is shown that the equatorial waveguide with a mean current acts as a resonator and responds to barotropic waves with certain wavenumbers by making the trapped baroclinic modes grow. At the same time the equatorial waveguide produces the barotropic response which, via nonlinear interaction with the mean equatorial current and with the trapped waves, leads to the saturation of the growing modes. The excited baroclinic waves can reach significant amplitudes depending on the magnitude of the mean current. In the absence of spatial modulation the nonlinear saturation of thus excited waves is described by forced Landau-type equation with one or two attracting equilibrium solutions. In the latter case the spatial modulation of the baroclinic waves is expected to lead to the formation of characteristic domain-wall defects. The evolution of the envelopes of the trapped Rossby waves is governed by driven Ginzburg-Landau equation, while the envelopes of the Yanai waves obey the "first-order" forced Ginzburg-Landau equation. The envelopes of short baroclinic Rossby waves obey the damped-driven nonlinear Schrodinger equation well studied in the literature.

Resolving weak internal layer interactions for the Ginzburg–Landau equation

1994 ◽  
Vol 5 (4) ◽  
pp. 495-523 ◽  
Author(s):  
Luis G. Reyna ◽  
Michael J. Ward
Keyword(s):  
Steady State ◽  
Landau Equation ◽  
Spatial Dimension ◽  
Internal Layer ◽  
Equilibrium Solutions ◽  
Singular Perturbation Method ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Mass Constraint ◽  
Unconstrained Problem

The internal layer behaviour, in one spatial dimension, associated with two classes of Ginzbug–Landau equation with double-well nonlinearities and small diffusivities is investigated. The problems that are examined are the Ginzburg–Landau equation with and without a constant mass constraint. For the constrained problem, steady-state internal layer solutions are constructed using a formal projection method. This method is also used to derive a differential-algebraic system describing the slow dynamics of the constrained internal layer motion. The dynamics of a two-layer evolution is studied in detail. For the unconstrained problem, a nonlinear WKB-type transformation is introduced that magnifies exponentially weak layer interactions and leads to well-conditioned steady problems. A conventional singular perturbation method, without the need for exponential asymptotics, is used on the resulting transformed problem as an alternative method to construct equilibrium solutions and metastable patterns. Exponentially sensitive steady-state internal layer solutions as well as a one-layer evolution are computed accurately using the transformed problem.

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

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 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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