Short-wave transverse instabilities of line solitons of the two-dimensional hyperbolic nonlinear Schrödinger equation

Pelinovsky, Dmitry E.; Rouvinskaya, Ekaterina; Kurkina, Oxana; Deconinck, Bernard

doi:10.1007/s11232-014-0154-1

Cited by 13 publications

(19 citation statements)

References 13 publications

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“…It admits the family of Y -independent line solitons if α 1 + α 2 > 0, which includes both the case of the periodic (stripe) potentials with α 2 = 2α 1 > 0 and the case of the massive Thirring model with α 1 = 0 and α 2 > 0. From the previous literature, see, e.g., recent works [23,27] or pioneer work [37], it is known that the line solitons are unstable in the hyperbolic NLS equation with respect to the spatial translation, in agreement with the result of Lemma 3.5. Moreover, the instability region extends to all values of the transverse wave number p, in agreement with our numerical results on Figures 2 and 3.…”

Section: Discussionsupporting

confidence: 75%

“…Similar thresholds on the period of transverse instability exist in other models such as the elliptic nonlinear Schrödinger (NLS) equation [34] and the Zakharov-Kuznetsov (ZK) equation [22]. Nevertheless, this conclusion is not universal and the line solitons can be unstable for all periods of the transverse perturbations, as it happens for the hyperbolic NLS equation [23].…”

Section: Introductionmentioning

confidence: 67%

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Transverse Instability of Line Solitary Waves in Massive Dirac Equations

Pelinovsky

Shimabukuro

2015

J Nonlinear Sci

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Working in the context of localized modes in periodic potentials, we consider two systems of the massive Dirac equations in two spatial dimensions. The first system, a generalized massive Thirring model, is derived for the periodic stripe potentials. The second one, a generalized massive Gross-Neveu equation, is derived for the hexagonal potentials. In both cases, we prove analytically that the line solitons suffer from instability with respect to periodic transverse perturbations of large periods. The instability is induced by the spatial translation for the massive Thirring model and by the gauge rotation for the massive Gross-Neveu model. We also observe numerically that the instability holds for the transverse perturbations of any period in the massive Thirring model and exhibits a finite threshold on the period of the transverse perturbations in the massive Gross-Neveu model.

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Section: Discussionsupporting

confidence: 75%

Section: Introductionmentioning

confidence: 67%

Transverse Instability of Line Solitary Waves in Massive Dirac Equations

Pelinovsky

Shimabukuro

2015

J Nonlinear Sci

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“…(Specifically, it was derived in the stability analysis of one-dimensional solitons of the 'hyperbolic' nonlinear Schrödinger equation on the plane; see [62] and references therein.) Various regimes of eigenvalue trajectories have been analysed, numerically and theoretically, in [47,52,62,63].…”

mentioning

confidence: 99%

“…When h  -¥, there are two complex quadruplets of eigenvalues. The imaginary parts are given [47] by the same equation (G3) whereas the real parts undergo an exponential decay (at equal rates) [52]: Appendix H. Symplectic eigenvalues in the anti-continuum limit…”

mentioning

confidence: 99%

Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides

2017

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We consider a  -symmetric ladder-shaped optical array consisting of a chain of waveguides with gain coupled to a parallel chain of waveguides with loss. All waveguides have the focusing Kerr nonlinearity. The array supports two co-existing solitons, an in-phase and an antiphase one, and each of these can be centred either on a lattice site or midway between two neighbouring sites. We show that both bond-centred (i.e. intersite) solitons are unstable regardless of their amplitudes and parameters of the chain. The site-centred in-phase soliton is stable when its amplitude lies below a threshold that depends on the coupling and gain-loss coefficient. The threshold is lowest when the gain-to-gain and loss-to-loss coupling constant in each chain is close to the interchain gain-to-loss coupling coefficient. The antiphase site-centred soliton in the strongly-coupled chain or in a chain close to the  -symmetry breaking point, is stable when its amplitude lies above a critical value and unstable otherwise. The instability growth rate of solitons with small amplitude is exponentially small in this parameter regime; hence the small-amplitude solitons, though unstable, have exponentially long lifetimes. On the other hand, the antiphase soliton in the weakly or moderately coupled chain and away from the  -symmetry breaking point, is unstable when its amplitude falls in one or two finite bands. All amplitudes outside those bands are stable.

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“…Starting with the pioneer works [7,33], it is well known that the line solitons are spectrally unstable with respect to the long transverse perturbations in many nonlinear evolution equations such as the Kadometsev-Petviashvili (KP) and nonlinear Schrödinger (NLS) equations (see review in [10]). The spectral instability persists to the short transverse perturbations of any period in the hyperbolic version of the two-dimensional NLS equation [5,24], whereas it disappears for short transverse perturbations in the elliptic version of the two-dimensional NLS equation and in the KP-I equation [25,26]. Alternatively, for a fixed period of the transverse perturbation, the transverse instability occurs for the line solitons with larger-than-critical speeds of propagation and disappears for those with smaller-thancritical speeds.…”

Section: Introductionmentioning

confidence: 99%

Normal form for transverse instability of the line soliton with a nearly critical speed of propagation

Pelinovsky

2018

Math. Model. Nat. Phenom.

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In the context of the line solitons in the Zakharov-Kuznetsov (ZK) equation, there exists a critical speed of propagation such that small transversely periodic perturbations are unstable if the soliton speed is larger than the critical speed and orbitally stable if the soliton speed is smaller than the critical speed. The normal form for transverse instability of the line soliton with a nearly critical speed of propagation is derived by means of symplectic projections and near-identity transformations. Justification of this normal form is provided with the energy method. The normal form predicts a transformation of the unstable line solitons with larger-than-critical speeds to the orbitally stable transversely modulated solitary waves. Transverse instability of line solitons for the ZK equation (1.1)We are concerned here with the transverse instability of the line solitons under periodic transverse perturbations in the ZK equation (1.1). First, we review relevant properties of the line solitons of the KdV equation. Next, we obtain spectral transverse stability results for the line solitons with a nearly critical speed of propagation. Further, we study bifurcations of the transversely modulated solitary waves. Finally, we present the main result on the normal form for transverse instability of the line solitons with a nearly critical speed of propagation.

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Short-wave transverse instabilities of line solitons of the two-dimensional hyperbolic nonlinear Schrödinger equation

Cited by 13 publications

References 13 publications

Transverse Instability of Line Solitary Waves in Massive Dirac Equations

Transverse Instability of Line Solitary Waves in Massive Dirac Equations

Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides

Normal form for transverse instability of the line soliton with a nearly critical speed of propagation

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