Analytic expressions of hydrothermal waves

Conte, Robert; Musette, Micheline

doi:10.1016/s0034-4877(01)80010-0

Cited by 12 publications

(10 citation statements)

References 28 publications

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“…(1) a N -th order algebraic ODE (6), N ≥ 2, (2) a Laurent series representing its general analytic solution, (3) a first order algebraic ODE sharing its general solution with (6).…”

Section: Methods To Obtain the First Order Autonomous Subequationmentioning

confidence: 99%

“…For two coupled CGL3 equations, see analytic results in Ref. 6 . (2) The Kuramoto and Sivashinsky (KS) equation, ϕ t + νϕ xxxx + bϕ xxx + µϕ xx + ϕϕ x = 0, ν = 0,…”

Section: Introductionmentioning

confidence: 99%

“…In section 5, we review the consequences of the assumption of singlevaluedness for a solution of the ODE (6), and present an algorithm to implement them. In section 6, we present the consequences of the assumption of meromorphy for a solution of (6). The last section 7 states the open problems.…”

Section: Introductionmentioning

confidence: 99%

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Integration of Partially Integrable Equations

Conte¹

2006

Waves and Stability in Continuous Media

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Most evolution equations are partially integrable and, in order to explicitly integrate all possible cases, there exist several methods of complex analysis, but none is optimal. The theory of Nevanlinna and Wiman-Valiron on the growth of the meromorphic solutions gives predictions and bounds, but it is not constructive and restricted to meromorphic solutions. The Painlevé approach via the a priori singularities of the solutions gives no bounds but it is often (not always) constructive. It seems that an adequate combination of the two methods could yield much more output in terms of explicit (i.e. closed form) analytic solutions. We review this question, mainly taking as an example the chaotic equation of Kuramoto and Sivashinsky νu ′′′ + bu ′′ + µu ′ + u 2 /2 + A = 0, ν = 0, with (ν, b, µ, A) constants.

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“…(1) a N -th order algebraic ODE (6), N ≥ 2, (2) a Laurent series representing its general analytic solution, (3) a first order algebraic ODE sharing its general solution with (6).…”

Section: Methods To Obtain the First Order Autonomous Subequationmentioning

confidence: 99%

“…For two coupled CGL3 equations, see analytic results in Ref. 6 . (2) The Kuramoto and Sivashinsky (KS) equation, ϕ t + νϕ xxxx + bϕ xxx + µϕ xx + ϕϕ x = 0, ν = 0,…”

Section: Introductionmentioning

confidence: 99%

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Integration of Partially Integrable Equations

Conte¹

2006

Waves and Stability in Continuous Media

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“…The first one contains the unphysical reduction U 2 = U 1 , while the second one describes a truly coupled behaviour. In both cases, just like in [2], the eight Fuchs indices are 1, 0, 0 (respectively corresponding to the arbitrary functions ϕ 0 , a 1 , a 2 ) and five irrational values, therefore there is no no-log condition to compute.…”

Section: Case G = 0 and τ =mentioning

confidence: 99%

Analytic study of a coupled Kerr-SBS system

Conte

Gandarias

2017

Communications in Nonlinear Science and Numerical Simulation

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In order to describe the coupling between the Kerr nonlinearity and the stimulated Brillouin scattering, Mauger et alii recently proposed a system of partial differential equations in three complex amplitudes. We perform here its analytic study by two methods. The first method is to investigate the structure of singularities, in order to possibly find closed form singlevalued solutions obeying this structure. The second method is to look at the infinitesimal symmetries of the system in order to build reductions to a lesser number of independent variables. Our overall conclusion is that the structure of singularities is too intricate to obtain closed form solutions by the usual methods. One of our results is the proof of the nonexistence of traveling waves.Comment: 16 pages, no figure. To appear, Communications in nonlinear science and numerical simulatio

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“…This number of arbitrary constants, smaller than three, can be computed either from singularity analysis [4,7], or from topological arguments [19], and it is equal to one, namely the origin ξ 0 of ξ. The question is therefore to find the unique particular solution of (4),…”

Section: Below)mentioning

confidence: 99%

Analytic solitary waves of nonintegrable equations

Musette¹,

Conte²

2003

Physica D: Nonlinear Phenomena

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A major drawback of most methods to find analytic expressions for solitary waves is the a priori restriction to a given class of expressions. To overcome this difficulty, we present a new method, applicable to a wide class of autonomous equations, which builds as an intermediate information the first order autonomous ODE satisfied by the solitary wave. We discuss its application to the cubic complex one-dimensional Ginzburg-Landau equation, and conclude to the elliptic nature of the yet unknown most general solitary wave.

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Analytic expressions of hydrothermal waves

Cited by 12 publications

References 28 publications

Integration of Partially Integrable Equations

Integration of Partially Integrable Equations

Analytic study of a coupled Kerr-SBS system

Analytic solitary waves of nonintegrable equations

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