Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation

Robert Conte (ENS Cachan); Tuen-Wai Ng (The University of Hong Kong)

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Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation

Robert Conte (ENS Cachan), Tuen-Wai Ng (The University of Hong Kong)

Pattern Formation and Solitons · Submitted 13 Apr 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim On the example of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to overcome the difficulties arising in two such methods: (i) the criterium that the sum of residues of an elliptic solution should be zero, (ii) the construction of a first order differential equation admitting the given equation as a differential consequence (subequation method).

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arXiv ID1204.3028 (version 1)

Pith IDLVJ6UML6, Pith's permanent record of this paper and its review

CitationRobert Conte (ENS Cachan), Tuen-Wai Ng (The University of Hong Kong). "Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation." arXiv:1204.3028 (2012). https://pith.science/paper/LVJ6UML6

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.27 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.25 · mentions code or data availability 0.02.

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In evolution equations for a complex amplitude, the phase obeys a much more intricate equation than the amplitude. Nevertheless, general methods should be applicable to both variables. On the example of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to overcome the difficulties arising in two such methods: (i) the criterium that the sum of residues of an elliptic solution should be zero, (ii) the construction of a first order differential equation admitting the given equation as a differential consequence (subequation method).

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In evolution equations for a complex amplitude, the phase obeys a much more intricate equation than the amplitude.	background 1.00	0.07	0.03	asserted 1.00
s1	Nevertheless, general methods should be applicable to both variables.	background 0.88	0.06	0.07	hedged possibility 0.60
s2	On the example of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to overcome the difficulties arising in two such methods: (i) the criterium that the sum of residues of an elliptic solution should be zero, (ii) the construction of a first order differential equation admitting the given equation as a differential consequence (subequation method).	contribution 0.90	0.86	0.46	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.67 · evidence from proof or derivation 0.99.

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Original abstract

In evolution equations for a complex amplitude, the phase obeys a much more intricate equation than the amplitude. Nevertheless, general methods should be applicable to both variables. On the example of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to overcome the difficulties arising in two such methods: (i) the criterium that the sum of residues of an elliptic solution should be zero, (ii) the construction of a first order differential equation admitting the given equation as a differential consequence (subequation method).

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Methods for exact solutions of nonlinear ordinary differential equations\
nlin.SI 2025-03 conditional novelty 4.0 of 10

A review that turns Eremenko's theorem and 19th-century elliptic-function theory into an exhaustive search method for meromorphic ODE solutions, with new elliptic and kink examples.
Singularity methods for meromorphic solutions of differential equations
nlin.SI 2018-06 unverdicted novelty 2.0 of 10

A tutorial presenting singularity-analysis methods that classify all meromorphic solutions of autonomous algebraic ODEs, demonstrated on the Kuramoto-Sivashinsky and complex Ginzburg-Landau equations.

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4 arXiv papers cite this in 9 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 8 sentences without a confident label

Uses its method or tool 1 paper

In order to detect a CGL5 elliptic solution, the criterium of residues needs to be applied not only to $M$ but also [ref] to $(M^(k))^j$, $(j,k)=(0,2),(0,3),(0,4),(1,2)$, it shows that $M$ must possess four poles and the parameters must obey four real constraints.

All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations · CGL5 elliptic solution · 0.82

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Uses the method or tool of 2 papers

Its successive steps are [ref] : * Find the analytic structure of movable singularities (e.g., 4 families of simple poles, 2 of double poles).

Elliptic general analytic solutions · Method to determine all elliptic and degenerate elliptic solutions · 0.92
In section [ref], we indicate how to correctly apply a second method (the subequation method [ref] ) to $M$ and $ψ$, in order to build a first order ODE sharing elliptic or degenerate elliptic solutions with the above third order ODEs ([ref]) and ([ref]).

nlin.PS/0302051 · Introduction. Modulus vs. phase in amplitude equations · 0.96

Cites as background 1 paper

For a summary of results on CGL5, see the reviews [ref] .

The World of the Complex Ginzburg-Landau Equation · Introduction. Modulus vs. phase in amplitude equations · 1.00

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