Steady solutions of the Kuramoto-Sivashinsky equation for small wave speed

Jones, James William; Troy, William C.; MacGillivary, A.D

doi:10.1016/0022-0396(92)90143-b

Cited by 13 publications

(17 citation statements)

References 8 publications

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“…For this system, an unfolding of (3) which has been widely studied (see [Mic86], [JTM92], [KT76], [RMT97]), there is a rigorous proof of an asymptotic formula of the heteroclinic splitting given in [RMT97]. The proof, which falls in the context of singular perturbation theory, draws heavily on the fact that the Michelson system comes from a third-order differential equation.…”

Section: Some Comments About the Singular Case P = −2mentioning

confidence: 99%

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Baldomá

Seara

2006

J Nonlinear Sci

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Summary. In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in R 3 . This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit between two critical points along the z axis. We prove that, generically, when the whole family is considered, this heteroclinic connection is destroyed. Moreover, we give an asymptotic formula of the distance between the stable and unstable manifolds when they meet the plane z = 0. This distance is exponentially small with respect to the unfolding parameter, and the main term is a suitable version of the Melnikov integral given in terms of the Borel transform of some function depending on the higher-order terms of the family. The results are obtained in a perturbative setting that does not cover the generic unfoldings of the Hopf singularity, which can be obtained as a singular limit of the considered family. To deal with this singular case, other techniques are needed. The reason to study the breakdown of the heteroclinic orbit is that it can lead to the birth of some homoclinic connection to one of the critical points in the unfoldings of the Hopf-zero singularity, producing what is known as a Shilnikov bifurcation.

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Section: Some Comments About the Singular Case P = −2mentioning

confidence: 99%

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Baldomá

Seara

2006

J Nonlinear Sci

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“…We are therefore led to the bounded existence interval of the solution to (11). Lemma 2.3 is also utilized to prove (7), (8). We show (8); the proof of (7) is handled similarly.…”

Section: By Virtue Of Yoe(x)=−`v(y(x))mentioning

confidence: 95%

“…We remark that Toland [17] already made an elementary but ingenious approach to proving the nonexistence of monotonic solutions of (1), (2) if l \ 2/9. Expanding on the existence of solutions to (1), (2), we recall that Jones et al [8] exhibit that there is an odd periodic solution for all small l > 0. We again refer to [15], where the existence of a unique monotonic solution to (1), (2) is shown with l=1/2 on the half interval {0 [ x < .}…”

Section: F(y) Ymentioning

confidence: 98%

“…In [19], the same method is applied to conclude the nonexistence of monotonic solutions; that is, there is no solution to (1), (2) with l % 1/2 such that yOE(x) > 0 for all x ¥ R and y(x) Q ± 1 as x Q ± ., respectively. The nonexistence of monotonic global solutions is later extended in [8] to hold for all l > 0. See also [15].…”

Section: F(y) Ymentioning

confidence: 99%

“…See also [15]. The method of proof in [8] should be compared with the one [1] given for a similar third-order ODE…”

Section: F(y) Ymentioning

confidence: 99%

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Remarks on Third-Order ODEs Relevant to the Kuramoto–Sivashinsky Equation

Ishimura

2002

Journal of Differential Equations

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dedicated to professor hiroko morimoto on her jubileeWe are interested in the third-order ordinary differential equations (ODEs) which are related to the Kuramoto-Sivashinsky equation. So-called steady solutions of the Kuramoto-Sivashinsky equation are known to admit several types; for example, bounded global solutions or periodic solutions. We show that, in addition to these, there exist solutions which blow up on bounded intervals. Moreover, for certain classes of these ODEs, the nonexistence of nontrivial bounded entire solutions is exhibited.

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A note on the nonexistence of monotonic steady solutions of the Kuramoto-Sivashinsky equation

Ishimura

Nakamura

2000

Ann. Univ. Ferrara

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Steady solutions of the Kuramoto-Sivashinsky equation for small wave speed

Cited by 13 publications

References 8 publications

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Remarks on Third-Order ODEs Relevant to the Kuramoto–Sivashinsky Equation

A note on the nonexistence of monotonic steady solutions of the Kuramoto-Sivashinsky equation

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