New type of travelling wave solutions

Schneider, К. R.; Shchepakina, Elena; Sobolev, Vladimir

doi:10.1002/mma.404

Cited by 20 publications

(2 citation statements)

References 16 publications

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“…In addition, we study the stability in time of the steady states of (1.1); in other words, we are interested in canard-like phenomena in the context of PDEs. Among the few studies in this direction, we mention canard travelling waves, which have been analysed in a moving frame as homoclinic connections with a canard segment [10][11][12], shock-like structures in PDEs which can also be interpreted in terms of canards [13,14] and canard-related bifurcation delay in reactiondiffusion PDEs [15,16]. In recent work, we classified for the first time spatio-temporal canards in an infinite-dimensional dynamical system arising in neuroscience applications, using an interfacial dynamics approach [17].…”

Section: Introductionmentioning

confidence: 99%

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

et al. 2017

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A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial dynamical system.

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Section: Introductionmentioning

confidence: 99%

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

et al. 2017

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“…This approach was extended to combustion models with distributed parameters. It should be noted that such an approach was used in [21] to describe canard traveling waves. and then to draw conclusions about the qualitative behavior of the full system (30) for sufficiently small ε.…”

Section: Resultsmentioning

confidence: 99%

Canards and critical behavior in autocatalytic combustion models

Gorelov

Shchepakina²,

Sobolev³

2006

J Eng Math

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The basic elements of the theory of slow invariant manifolds and canard phenomena of singularly perturbed nonlinear differential equations in the context of thermal-explosion problems are outlined. The mathematical results are applied to the investigation of the critical phenomena in autocatalytic combustion models described by singularly perturbed differential equations with lumped and distributed parameters. Critical regimes are modeled by canards (one-dimensional stable-unstable slow invariant manifolds). The geometric approach in combination with asymptotic and numeric methods permits to explain the strong parametric sensitivity and to obtain asymptotic representations of the critical conditions of self-ignition.

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On the (in)stability of quasi-static paths of smooth systems: definitions and sufficient conditions

Martins¹,

Rebrova²,

Sobolev³

2006

Math. Meth. Appl. Sci.

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Communicated by W. Spr o ig SUMMARYA concept of stability of quasi-static paths is discussed in this paper that takes into account the existence of fast (dynamic) and slow (quasi-static) time scales in the evolution of many mechanical systems. The proposed concept is essentially a continuity property with respect to the smallness of the initial perturbations (as in Lyapunov stability) and the smallness of the quasi-static loading rate (that plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Several examples illustrate the relevance of the deÿnitions. Su cient conditions for attractiveness or for instability of quasi-static paths of smooth systems are proved.

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New type of travelling wave solutions

Cited by 20 publications

References 16 publications

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

Canards and critical behavior in autocatalytic combustion models

On the (in)stability of quasi-static paths of smooth systems: definitions and sufficient conditions

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