When gap solitons become embedded solitons: a generic unfolding

Wagenknecht, Thomas; Champneys, Alan R

doi:10.1016/s0167-2789(02)00773-x

Cited by 21 publications

(27 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Rather than stating detailed bifurcation results, for which we refer to [419], we focus on the geometric arguments that lead to these results. Thus, consider a two-parameter family of reversible ODEs on R 4 that satisfies Hypothesis 5.26.…”

Section: Homoclinic Orbits To Nonhyperbolic Equilibriamentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

View full text Add to dashboard Cite

An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.

show abstract

Section: Homoclinic Orbits To Nonhyperbolic Equilibriamentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

View full text Add to dashboard Cite

show abstract

“…Note that geometric arguments have been successfully used in Ref. [10] to understand the transition from ESs to ordinary gap solitons (however, that work does not explain why this transition is absent in the system (5)). …”

Section: The Two-wave System As a Normal Formmentioning

confidence: 99%

Accumulation of embedded solitons in systems with quadratic nonlinearity

Malomed

Wagenknecht

Champneys

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

Previous numerical studies have revealed the existence of embedded solitons (ESs) in a class of multi-wave systems with quadratic nonlinearity, families of which seem to emerge from a critical point in the parameter space, where the zero solution has a fourfold zero eigenvalue. In this paper, the existence of such solutions is studied in a three-wave model.An appropriate rescaling casts the system in a normal form, which is universal for models supporting ESs through quadratic nonlinearities. The normal-form system contains a single irreducible parameter ε, and is tantamount to the basic model of type-I second-harmonic generation. An analytical approximation of WKB type yields an asymptotic formula for the distribution of discrete values of ε at which the ESs exist. Comparison with numerical results shows that the asymptotic formula yields an exact value of the scaling index, −6/5, and a fairly good approximation for the numerical factor in front of the scaling term. In this work, we resolve these issues, by developing an asymptotic analysis and verifying the predictions against numerical results. As a result, we derive a universal asymptotic approximation (normal form) for models with quadratic nonlinearity that support embedded solitons. The normal form amounts to a system of two second-order equations, which is well known as a basic model for ordinary solitons in second-harmonic-generating systems. There was a rather common belief that all soliton solutions had 3 Accumulation of embedded solitons been found in this much-studied system. Nevertheless, in this work we are able to predict the existence of an infinite series of previously unknown embedded solitons in the model. This is done in an analytical form, by realizing that the embedded solitons feature a broad inner zone, the solution in which must be matched to exponentially decaying ones in outer zones. Further, a method (Bohr-Sommerfeld quantization rule) borrowed from quantum mechanics is applied to the solution in the inner zone. The most essential part of the eventual analytical result is an asymptotic distribution law for values of the single control parameter of the normal-form system at which the embedded solitons exist. Comparison with numerical results clearly shows that the analysis has produced an exact value of the respective scaling index.

show abstract

“…As already mentioned, if the standing pulse converges to zero exponentially at onset, the resulting bifurcation can be analysed without using blow-up techniques [28,29].…”

Section: Hypothesis 3 (Standing Pulse)mentioning

confidence: 99%

Absolute instabilities of standing pulses

Sandstede

Scheel

2004

Nonlinearity

View full text Add to dashboard Cite

We analyse instabilities of standing pulses in reaction-diffusion systems that are caused by an absolute instability of the homogeneous background state. Specifically, we investigate the impact of pitchfork, Turing and oscillatory bifurcations of the rest state on the standing pulse. At a pitchfork bifurcation, the standing pulse continues through the bifurcation point where it selects precisely one of the two bifurcating equilibria. At a Turing instability, symmetric pulses emerge that are spatially asymptotic to the bifurcating spatially-periodic Turing patterns. These pulses exist for any wavenumber inside the Eckhaus stability band. Oscillatory instabilities of the background state lead to genuinely time-periodic pulses that emit small wave trains with a unique selected wavenumber. We analyse these three bifurcations by studying the standingwave and modulated-wave equations: In this setup, pulses correspond to homoclinic orbits to equilibria that undergo reversible bifurcations. We use blow-up techniques to show that the relevant stable and unstable manifolds can be continued across the bifurcation point and to investigate both existence and stability of the bifurcating waves.

show abstract

When gap solitons become embedded solitons: a generic unfolding

Cited by 21 publications

References 32 publications

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Accumulation of embedded solitons in systems with quadratic nonlinearity

Absolute instabilities of standing pulses

Contact Info

Product

Resources

About