Numerical investigations of dispersive shocks and spectral analysis for linearized quantum hydrodynamics

Lattanzio, Corrado; Marcati, Pierangelo; Zhelyazov, Delyan

doi:10.1016/j.amc.2020.125450

Cited by 9 publications

(11 citation statements)

References 15 publications

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“…In this paper we have proved the conjecture by Lattanzio et al [28,29] that subsonic viscous-dispersive shocks for the QHD system with linear viscosity (1.1) are spectrally stable in the small-amplitude regime. Small viscous-dispersive shocks comply with the compressivity of the shock, that is, they remain monotone, and exhibit exponential decay, sharing in this fashion important features with purely viscous shocks in fluid dynamics.…”

Section: Discussion and Open Problemssupporting

confidence: 62%

“…Thus, we believe that the study of the effects on stability of the nonlinear dispersive term of Bohmian type that appears in (1.1) is worth pursuing. It is to be observed that, according to the numerical calculations by Lattanzio et al [28,29], larger amplitude (and hence, oscillatory) dispersive profiles are also spectrally stable. Therefore, an important open problem is to analytically prove that spectral stability of dispersive shocks for system (1.1) holds beyond the smallamplitude regime.…”

Section: Discussion and Open Problemsmentioning

confidence: 96%

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Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

Folino¹,

Plaza²,

Zhelyazov³

2022

Preprint

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A compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The dispersive term is due to quantum effects described through the Bohm potential and the viscosity term is of linear type. It is shown that small-amplitude viscousdispersive shock profiles for the system under consideration are spectrally stable, proving in this fashion a previous numerical observation by Lattanzio et al. [28,29]. The proof is based on spectral energy estimates which profit from the monotonicty of the profiles in the small-amplitude regime.

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Section: Discussion and Open Problemssupporting

confidence: 62%

Section: Discussion and Open Problemsmentioning

confidence: 96%

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

Folino¹,

Plaza²,

Zhelyazov³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is shown that small-amplitude viscousdispersive shock profiles for the system under consideration are spectrally stable, proving in this fashion a previous numerical observation by Lattanzio et al. [28,29]. The proof is based on spectral energy estimates which profit from the monotonicty of the profiles in the small-amplitude regime.…”

supporting

confidence: 78%

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

Folino

Plaza

Zhelyazov

2022

CPAA

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<p style='text-indent:20px;'>A compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The dispersive term is due to quantum effects described through the Bohm potential and the viscosity term is of linear type. It is shown that small-amplitude viscous-dispersive shock profiles for the system under consideration are spectrally stable, proving in this fashion a previous numerical observation by Lattanzio <i>et al.</i> [<xref ref-type="bibr" rid="b28">28</xref>,<xref ref-type="bibr" rid="b29">29</xref>]. The proof is based on spectral energy estimates which profit from the monotonicty of the profiles in the small-amplitude regime.</p>

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“…In particular, for the case of this kind of hydrodynamic models involving dispersion terms, we recall here [19], where the spectral stability of traveling wave profiles for the p−system with real viscosity and linear capillarity has been discussed. Moreover, spectral analysis of the linearization around dispersive shocks for a variant of the QHD system (1.1) with linear viscosity can be found in [22], and the related Evans function computations in [23].…”

Section: Introductionmentioning

confidence: 99%

Dispersive shocks in quantum hydrodynamics with viscosity

Lattanzio

Marcati

Zhelyazov

2020

Physica D: Nonlinear Phenomena

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In this paper we study existence and stability of shock profiles for a 1-D compressible Euler system in the context of Quantum Hydrodynamic models. The dispersive term is originated by the quantum effects described through the Bohm potential; moreover we introduce a (linear) viscosity to analyze its interplay with the former while proving existence, monotonicity and stability of travelling waves connecting a Lax shock for the underlying Euler system. The existence of monotone profiles is proved for sufficiently small shocks; while the case of large shocks leads to the (global) existence for an oscillatory profile, where dispersion plays a significant role. The spectral analysis of the linearized problem about a profile is also provided. In particular, we derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of the eigenvalues in the unstable plane, using a careful analysis of the Evans function.

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Numerical investigations of dispersive shocks and spectral analysis for linearized quantum hydrodynamics

Cited by 9 publications

References 15 publications

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

Dispersive shocks in quantum hydrodynamics with viscosity

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