Dynamical behaviors of blowup solutions in trapped quantum gases: Concentration phenomenon

Xie, Yingying; Li, Lingfei; Zhu, S. J.

doi:10.1016/j.jmaa.2018.08.011

“…The wave collapse [27,28] leads to the formation of a singularity after a finite propagation distance (while an input with P < P crit decays). The wave collapse has been reported in plasmas [29], optics [30], Bose-Einstein condensates (BECs) [31,32], capillary-gravity waves in deep water [33] and astrophysics [34], etc. NLS equations that give rise to the critical collapse produce families of Townes solitons (TSs), which were first predicted in terms of the two-dimensional cubic NLS equation [21].…”

Section: Introductionmentioning

confidence: 99%

“…The wave collapse [27,28] leads to the formation of a singularity after a finite propagation distance (while an input with P<Pcrit decays). The wave collapse has been reported in plasmas [29], optics [30], Bose-Einstein condensates (BECs) [31,32], capillary-gravity waves in deep water [33] and astrophysics [34], etc.…”

Section: Introductionmentioning

confidence: 99%

Suppression of soliton collapses, modulational instability and rogue-wave excitation in two-Lévy-index fractional Kerr media

Zhong,

Chen,

Yan

et al. 2024

Proc. R. Soc. A.

7

0

View full text Add to dashboard Cite

We introduce a generalized fractional nonlinear Schrödinger (FNLS) equation for the propagation of optical pulses in laser systems with two fractional-dispersion/diffraction terms, quantified by their Lévy indices, α 1 α 2 ∈ ( 1 , 2 ] , and self-focusing or defocusing Kerr nonlinearity. Some fundamental solitons are obtained by means of the variational approximation, which are verified by comparison with numerical results. We find that the soliton collapse, exhibited by the one-dimensional cubic FNLS equation with only one Lévy index (LI) α = 1 , can be suppressed in the two-LI FNLS system. Stability of the solitons is also explored against collisions with Gaussian pulses and adiabatic variation of the system parameters. Modulation instability (MI) of continuous waves is investigated in the two-LI system too. In particular, the MI may occur in the case of the defocusing nonlinearity when two diffraction coefficients have opposite signs. Using results for the MI, we produce first- and second-order rogue waves on top of continuous waves, for both signs of the Kerr nonlinearity.

show abstract

“…Different versions of NLS equations have been also employed in other contexts, e.g., in the investigation of the dynamics of blow-up solutions in trapped quantum gases in Gross-Pitaevskii equation [24] which in a sense generalizes the three-dimensional NLS equation. Alternatively, plasma fluid model may be reduced to the Kortewegde Vries (KdV) equation, which has been recently investigated in a generalized form (i.e., with high power nonlinearities) with respect to solitary, compacton, and periodic types of solutions [25].…”

Section: Introductionmentioning

confidence: 99%

Coupled Nonlinear Schrodinger (CNLS) Equations for two interacting electrostatic wavepackets in a non-Maxwellian fluid plasma model

Lazarides

¹

,

Kourakis

²

2023

Preprint

View full text Add to dashboard Cite

The nonlinear dynamics of two co-propagating electrostatic wavepackets in a one-dimensional non-magnetized plasma fluid model is considered, from first principles. The coupled waves are characterized by different (carrier) wavenumbers and amplitudes. A plasma consisting of non-thermalized (κ−distributed) electrons evolving against a cold (stationary) ion background is considered. The original model is reduced, by means of a multiple-scale perturbation method, to a pair of coupled nonlinear Schr¨odinger (CNLS) equations for the dynamics of the wavepacket envelopes. For arbitrary wavenumbers, the resulting CNLS equations exhibit no known symmetry and thus intrinsically differ from the Manakov system, in general. Exact analytical expressions have been derived for the dispersion, self-modulation (nonlinearity) and cross-modulation (coupling) coefficients involved in the CNLS equations, as functions of the wavenumbers (k1, k2) and of the spectral index κ characterizing the electron profile. An analytical investigation has thus been carried out of the modulational instability (MI) properties of this pair of wavepackets, focusing on the role of the intrinsic (variable) parameters. Modulational instability is shown to occur in most parts of the parameter space. The instability window(s) and the corresponding growth rate are calculated numerically in a number of case studies. Two-wave interaction favors MI by extending its range of occurrence and by enhancing its growth rate. Growth rate patterns obtained for different κ index (values) suggest that deviation from thermal (Maxwellian) equilibrium, for low κ values, leads to enhances MI of the interacting wave pair. Although we have focused on electrostatic wavepacket propagation in nonthermal (non-Maxwellian) plasma, the results of this study are generic and may be used as basis to model energy localization in nonlinear optics, in hydrodynamics or in dispersive media with Kerr-type nonlinearities where modulational instability is relevant.

show abstract

“…In addition, the researcher in [25] utilized Lagrangian coordinates to extend the previous results through lowering the common condition of the initial value for the D-G-H equation to the class of continuous differential periodic initial value, i.e., for any u 0 ∈ C 1 (R), there is some T(u 0 ) > 0 and a unique solution u ∈ C([0, T); C 1 (R)) ∩ C 1 ([0, T); C(R)). Such method is also succeeded in proving the periodic the C-H equation that lives up to the least action principle [26,27] and giving the threshold for global existence and blowup solutions [28][29][30]. Regarding the conservative solution of the C-H equation [1,31,32], Bressan and Constantin transferred the equation to the ODE system at first, in terms of a set of variables.…”

Section: Introductionmentioning

confidence: 99%

Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

Wang

¹

,

Yan

²

,

Wu

³

2022

Journal of Mathematics

0

View full text Add to dashboard Cite

The current paper devotes to the continuation of the solutions for the Dullin–Gottwald–Holm equation with forcing. Due to forcing, this equation loses the conservation of energy and cannot be regarded as the Hamiltonian system. Thus, through employing the characteristic method and exploiting the balance law, we prove the existence of a semigroup of a global dissipative solution, which is defined for each initial value u 0 ∈ H 1 ℝ and depends continuity on it.

show abstract

Dynamical behaviors of blowup solutions in trapped quantum gases: Concentration phenomenon

Cited by 11 publications

References 20 publications

Suppression of soliton collapses, modulational instability and rogue-wave excitation in two-Lévy-index fractional Kerr media

Suppression of soliton collapses, modulational instability and rogue-wave excitation in two-Lévy-index fractional Kerr media

Coupled Nonlinear Schrodinger (CNLS) Equations for two interacting electrostatic wavepackets in a non-Maxwellian fluid plasma model

Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

Contact Info

Product

Resources

About