Stability of Travelling Waves

Sandstede, Björn

doi:10.1016/s1874-575x(02)80039-x

Cited by 352 publications

(421 citation statements)

References 138 publications

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“…0 < min nÞ0 n ), then the y-equation does not, to the leading order, depend on . Then, the distance between the oscillons behaves like in the stationary case [5][6][7]13]: at 0 Þ 0 stable bound states are formed near 0 y þ 10 ¼ ð2k þ 1Þ, independently of the value of the phase difference . Possible phase synchronization effects appear on a much longer time scale and are governed by nonzero harmonics.…”

Section: We Look For the Solution Of Eq (4) In The Form Of Two Intermentioning

confidence: 95%

“…By performing asymptotic expansions, similar to what is done for stationary solitons [4][5][6][7]13], we obtain the leading order approximation for the oscillon interaction equation:…”

Section: We Look For the Solution Of Eq (4) In The Form Of Two Intermentioning

confidence: 99%

“…It is seen from this figure that when the oscillon separation is sufficiently small, we have only in-phase and antiphase stable bound states, which is typical for Eq. (6) with N ¼ 1 and BC cosð 2N À 1N Þ > 0. However, at larger oscillon separations, stable bound states with the phase difference around =2 appear and the phase portrait becomes consistent with Eq.…”

Section: We Look For the Solution Of Eq (4) In The Form Of Two Intermentioning

confidence: 99%

“…Exact analytical expressions for dissipative solitons are rarely available, so qualitative methods become especially important in their study. An interesting problem which can be treated by qualitative methods is the interaction of dissipative solitons [2][3][4][5][6][7][8]. While most of the studies here were focused on the case of stationary solitons, in this Letter we analyze the interaction of dissipative solitons which oscillate in time.…”

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confidence: 99%

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Long-Range Interaction and Synchronization of Oscillating Dissipative Solitons

2012

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We study the interaction of well-separated oscillating localized structures (oscillons). We show that oscillons emit weakly decaying dispersive waves, which lead to the formation of bound states due to harmonic synchronization. We also show that in optical applications the Andronov-Hopf bifurcation of stationary localized structures leads to a drastic increase in their interaction strength. DOI: 10.1103/PhysRevLett.108.263906 PACS numbers: 42.65.Sf, 47.54.Àr The investigation of localized structures arising in physical systems of various nature is an important subject of nonlinear science. Lately a lot of attention has been paid to the so-called dissipative solitons [1]. Their formation requires a balance of energy gain and dissipation, which makes the dissipative solitons more stable to perturbations and, therefore, more attractive for practical applications (e.g., for optical information processing) than the classical solitons of integrable Hamiltonian equations. Exact analytical expressions for dissipative solitons are rarely available, so qualitative methods become especially important in their study. An interesting problem which can be treated by qualitative methods is the interaction of dissipative solitons [2][3][4][5][6][7][8]. While most of the studies here were focused on the case of stationary solitons, in this Letter we analyze the interaction of dissipative solitons which oscillate in time.It is well known that a stationary soliton can exhibit instabilities that lead to various dynamical regimes. One of the simplest and most frequently encountered between these instabilities is the Andronov-Hopf (AH) bifurcation resulting in undamped pulsations of the soliton's parameters, such as amplitude, width, etc. [1,[8][9][10]. Here, we show that the transition from stationary to an oscillating soliton (oscillon) leads to the formation of various new types of multisoliton bound states. In particular, the AH bifurcation of stationary optical pulses results in a considerable increase of their interaction strength.Although the approach we use is general, to illustrate the enhancement of the solitons' interaction, we consider a specific model equation (Lugiato-Lefever model [11])which describes formation of transverse patterns in Kerr cavity [10] or ''temporal cavity solitons'' in fibers [12]. Here, a is the field envelope, is the cavity decay rate, is the cavity detuning, and p is the external coherent pumping. Spatial filtering (or, in time domain, gain dispersion) is typically quite small in optical applications.The soliton in Eq. (1) is asymptotic to a nonzero stationary value a s ðpÞ. The existence region of the soliton is presented in the bifurcation diagram in Fig. 1. As the pumping parameter p increases above the critical value p AH , the soliton undergoes an AH bifurcation [10]. After this bifurcation, the soliton starts to oscillate, and the oscillations in the tail can be interpreted as a radiation of evanescent waves, see Fig. 2 for an illustration.In order to find the dispersion relation which de...

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Section: We Look For the Solution Of Eq (4) In The Form Of Two Intermentioning

confidence: 95%

“…By performing asymptotic expansions, similar to what is done for stationary solitons [4][5][6][7]13], we obtain the leading order approximation for the oscillon interaction equation:…”

Section: We Look For the Solution Of Eq (4) In The Form Of Two Intermentioning

confidence: 99%

Section: We Look For the Solution Of Eq (4) In The Form Of Two Intermentioning

confidence: 99%

mentioning

confidence: 99%

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Long-Range Interaction and Synchronization of Oscillating Dissipative Solitons

2012

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“…A more advanced approach is that of nonlinear, or Lyapunov, stability, which means that in a properly defined functional space there exists a class of initial conditions which tend to the travelling wave solution as time goes to infinity. A modern review of stability of travelling waves with an emphasis on dynamical systems approaches can be found in Sandstede [27].…”

Section: The Stability Of Travelling Wavesmentioning

confidence: 99%

Front propagation in a phase field model with phase-dependent heat absorption

Blyuss

Ashwin

Wright

et al. 2006

Physica D: Nonlinear Phenomena

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We present a model for the spatio-temporal behaviour of films exposed to radiative heating, where the film can change reversibly between amorphous (glassy) and crystalline states. Such phase change materials are used extensively in read-write optical disk technology.In cases where the heat absorption of the crystal phase is less than that in the amorphous state we find that there is a bi-stability of the phases. We investigate the spatial behaviours that are a consequence of this property and use a phase field model for the spatio-temporal dynamics in which the phase variable is coupled to a suitable temperature field. It is shown that travelling wave solutions of the system are possible and, depending on the precise system parameters, these waves can take a range of forms and velocities. Some examples of possible dynamical behaviours are discussed and we show, in particular, that the waves may collide and annihilate. The longitudinal and transverse stability of the travelling waves are examined using an Evans function method which suggests that the fronts are stable structures.

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Dynamics of Patterns in Nonlinear Equivariant PDEs

Beyn

Thümmler

2009

GAMM-Mitteilungen

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Many solutions of nonlinear time dependent partial differential equations show particular spatio-temporal patterns, such as traveling waves in one space dimension or spiral and scroll waves in higher space dimensions. The purpose of this paper is to review some recent progress on the analytical and the numerical treatment of such patterns. Particular emphasis is put on symmetries and on the dynamical systems viewpoint that goes beyond existence, uniqueness and numerical simulation of solutions for single initial value problems. The nonlinear asymptotic stability of dynamic patterns is discussed and a numerical approach (the freezing method) is presented that allows to compute co-moving frames in which solutions converging to the patterns become stationary. The results are related to the theory of relative equilibria for equivariant evolution equations. We discuss several applications to parabolic systems with nonlinearities of FitzHugh-Nagumo and Ginzburg-Landau type.

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Stability of Travelling Waves

Cited by 352 publications

References 138 publications

Long-Range Interaction and Synchronization of Oscillating Dissipative Solitons

Long-Range Interaction and Synchronization of Oscillating Dissipative Solitons

Front propagation in a phase field model with phase-dependent heat absorption

Dynamics of Patterns in Nonlinear Equivariant PDEs

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