Embedded Solitons in Second-Harmonic-Generating Systems

Yang, Jianke; Malomed, Boris A.; Kaup, D. J.

doi:10.1103/physrevlett.83.1958

Cited by 124 publications

(200 citation statements)

References 10 publications

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“…(26) yields k ES = 0.688, while the numerical value found in Ref. [7] was k ES = 0.696. Thus,the relative error of the criterion (22) for this case is 1.1%.…”

Section: Embedded Soliton In the Full Systemmentioning

confidence: 95%

“…It should be stressed that, as it was demonstrated in Ref. [7], the ansatz (13) yields exact soliton solutions to Eqs. (10) and (11) at some uniquely selected value of k [an expression for it is given below in Eq.…”

Section: The Variational Approximation For the Semi-lagrangian Systemmentioning

confidence: 99%

“…A condition (inequality) showing whether the exact soliton is embedded or ordinary has already been given in Ref. [7].…”

Section: Embedded Soliton In the Truncated Systemmentioning

confidence: 99%

“…In the case when the wavenumber k falls into the regions (7), ESs can exist in both the full and truncated models, i.e., (1), (2) and (3), (4) [7]. However, except for using the exact ES solutions in the truncated model to guess where ESs might exist in its full counterpart, the only known method for locating ESs in the full model was to search for them by means of direct numerical computations.…”

Section: An Analytical Criterion To Identify Embedded Solitons 41 Gementioning

confidence: 99%

“…[7], for which δ = 1, q = 1, and γ 1 = γ 2 = −0.05. Borrowing values of the amplitudes A and B directly from the numerical data, we find that A = 3.794 and B = 2.735.…”

Section: Embedded Soliton In the Full Systemmentioning

confidence: 99%

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Embedded solitons in Lagrangian and semi-Lagrangian systems

Kaup

Malomed

2003

Physica D: Nonlinear Phenomena

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We develop the technique of the variational approximation for solitons in two directions. First, one may have a physical model which does not admit the usual Lagrangian representation, as some terms can be discarded for various reasons. For instance, the second-harmonic-generation (SHG) model considered here, which includes the Kerr nonlinearity, lacks the usual Lagrangian representation if one ignores the Kerr nonlinearity of the second harmonic, as compared to that of the fundamental. However, we show that, with a natural modification, one may still apply the variational approximation (VA) to those seemingly flawed systems as efficiently as it applies to their fully Lagrangian counterparts. We call such models, that do not admit the usual Lagrangian representation, semi-Lagrangian systems. Second, we show that, upon adding an infinitesimal tail that does not vanish at infinity, to a usual soliton ansatz, one can obtain an analytical criterion which (within the framework of VA) gives a condition for finding embedded solitons, i.e., isolated truly localized solutions existing inside the continuous spectrum of the radiation modes. The criterion takes a form of orthogonality of the radiation mode in the infinite tail to the soliton core. To test the criterion, we have applied it to both the semi-Lagrangian truncated version of the SHG model and to the same model in its full form. In the former case, the criterion (combined with VA for the soliton proper) yields an exact solution for the embedded soliton. In the latter case, the criterion selects the embedded soliton with a relative error ≈ 1%.

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“…(26) yields k ES = 0.688, while the numerical value found in Ref. [7] was k ES = 0.696. Thus,the relative error of the criterion (22) for this case is 1.1%.…”

Section: Embedded Soliton In the Full Systemmentioning

confidence: 95%

Section: The Variational Approximation For the Semi-lagrangian Systemmentioning

confidence: 99%

“…A condition (inequality) showing whether the exact soliton is embedded or ordinary has already been given in Ref. [7].…”

Section: Embedded Soliton In the Truncated Systemmentioning

confidence: 99%

Section: An Analytical Criterion To Identify Embedded Solitons 41 Gementioning

confidence: 99%

“…[7], for which δ = 1, q = 1, and γ 1 = γ 2 = −0.05. Borrowing values of the amplitudes A and B directly from the numerical data, we find that A = 3.794 and B = 2.735.…”

Section: Embedded Soliton In the Full Systemmentioning

confidence: 99%

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Embedded solitons in Lagrangian and semi-Lagrangian systems

Kaup

Malomed

2003

Physica D: Nonlinear Phenomena

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Asymmetric Shape and Dynamic Stability of Exciton-Phonon Solitons Moving in a Medium

Roubtsov

Lépine

Önder

2002

phys. stat. sol. (b)

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Solitons are known to move ballistically through a medium without change of their shape. In practice, the shape of a soliton can change, and a long lasting tail appears behind the soliton moving in a periodic medium. Such a behavior can be described within the model of exciton-phonon coherent states as a dynamic effect in soliton transport. We argue that the coupling between the bosonic excitations of the medium, such as excitons, and the elastic modes of it, such as phonons, can be responsible for these effects. We derive nonlinear dynamic equations for the excited medium in the long wavelength approximation (a generalized Zakharov system) and apply a ballistic ansatz for the coherent state of bosons and displacement field (a kind of the condensate in this model). Like solitons, the quasistationary solution we obtained can move through the medium ballistically. Unlike solitons and kinks, there are inhomogeneous corrections to the ballistic velocity and coherent phase of the condensate that control the change of the packet shape. In the limit of T ! 0, laws of conservation prescribe formation of the cloud of collective excitations around the quasistationary Bose-core of the packet so that the density of the boson-phonon tail behind the moving coherent part can be estimated at any given t. The total packet can be associated with an exciton-phonon ''comet" with the quasistable coherent Bose-''nucleus" and incoherent tail moving in the medium.

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