Continuous Families of Embedded Solitons in the Third‐Order Nonlinear Schrödinger Equation

Yang, Jianke; Akylas, T. R.

doi:10.1111/1467-9590.t01-1-00238

Cited by 33 publications

(42 citation statements)

References 30 publications

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“…Further numerical results not presented here [4] compute representatives of both kinds of multi-humped ESs and follow their branches in the (k, q)-plane. Note that the latter kind also produces a family of curves that originate from the point Figure 2 [4, Fig 4.3] similar to the families computed by Yang and Akylas [17] for the third-order nonlinear Schrödinger equation. The aim of this paper has been to establish why in addition, an infinite family of single-humped solitons should emanate from the same critical point in the three-wave model.…”

Section: Discussionsupporting

confidence: 55%

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Accumulation of embedded solitons in systems with quadratic nonlinearity

Malomed

Wagenknecht

Champneys

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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Section: Discussionsupporting

confidence: 55%

“…[5,17,18]. In all other cases we are aware of the individual members of the family correspond to multi-humped solitary waves.…”

Section: Discussionmentioning

confidence: 98%

Accumulation of embedded solitons in systems with quadratic nonlinearity

Malomed

Wagenknecht

Champneys

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…has no single-humped localized solutions for any (Ω, V ) ∈ R 2 (see references in [YA03,PR05]). Therefore, the necessary condition for existence of single-humped traveling solutions is the constraint on parameters of the cubic polynomial function (1.4):…”

Section: Corollary 39mentioning

confidence: 99%

“…The normal form is represented by the third-order ODE related to the third-order derivative NLS equation [PR05]. Existence of embedded solitons in the third-order derivative NLS equation was considered in the past (see [YA03,PY05] In Section 2, the class of exceptional nonlinear functions in the general family of cubic polynomials (1.4)…”

Section: Introductionmentioning

confidence: 99%

Translationally invariant nonlinear Schrödinger lattices

Pelinovsky

2006

Nonlinearity

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“…But for femtosecond pulses another important physical effect namely, the third-order dispersion comes into play. In this case the appropriate evolution equation for the pulse propagation is given by [3] …”

Section: Introductionmentioning

confidence: 99%