scholarly journals Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

2021 ◽  
Author(s):  
Adam L. Binswanger ◽  
Mark A. Hoefer ◽  
Boaz Ilan ◽  
Patrick Sprenger
Keyword(s):  
Modulational Instability ◽  
General Criterion ◽  
Whitham Equations ◽  
Modulation Theory

scholarly journals Modulations of viscous fluid conduit periodic waves

2016 ◽  
Vol 472 (2196) ◽  
pp. 20160533 ◽  
Author(s):  
M. D. Maiden ◽  
M. A. Hoefer
Keyword(s):  
Viscous Fluid ◽  
Large Amplitude ◽  
Numerical Solutions ◽  
Modulational Instability ◽  
Periodic Wave ◽  
Periodic Waves ◽  
Wave Modulation ◽  
Envelope Solitons ◽  
Whitham Equations ◽  
Critical Wavenumber

Modulated periodic interfacial waves along a conduit of viscous liquid are explored using nonlinear wave modulation theory and numerical methods. Large-amplitude periodic-wave modulation (Whitham) theory does not require integrability of the underlying model equation, yet often either integrable equations are studied or the full extent of Whitham theory is not developed. Periodic wave solutions of the nonlinear, dispersive, non-integrable conduit equation are characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the nonlinear Schrödinger (NLS) equation are derived, depending on the carrier wavenumber. Dark and bright envelope solitons are found to persist in long-time numerical solutions of the conduit equation, providing numerical evidence for the existence of strongly nonlinear, large-amplitude envelope solitons. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted and observed. In the large-amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. The viscous fluid conduit system is a mathematically tractable, experimentally viable model system for wide-ranging nonlinear, dispersive wave dynamics.

scholarly journals Nonlinear Theory for Coalescing Characteristics in Multiphase Whitham Modulation Theory

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Thomas J. Bridges ◽  
Daniel J. Ratliff
Keyword(s):  
Nonlinear Theory ◽  
Boussinesq Equation ◽  
Travelling Wave ◽  
Elliptic Type ◽  
Travelling Wave Solutions ◽  
Two Phase ◽  
Whitham Equations ◽  
Complex Wave ◽  
Modulation Theory ◽  
Nonlinear Modulation

AbstractThe multiphase Whitham modulation equations with N phases have 2N characteristics which may be of hyperbolic or elliptic type. In this paper, a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly, a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation, that is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling wave solutions of coupled nonlinear Schrödinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.

Modulation instability of weakly nonlinear long internal wave packets

2020 ◽  
Author(s):  
Tatiana Talipova ◽  
Efim Pelinovsky
Keyword(s):  
Evolution Equations ◽  
Nonlinear Term ◽  
Modulation Instability ◽  
Modulational Instability ◽  
Wave Packets ◽  
Nonlinear Evolution Equations ◽  
Density Stratification ◽  
Gardner Equation ◽  
Weakly Nonlinear ◽  
Whitham Equations

<p>We exam the problem of the modulation instability of long internal waves. Such weakly nonlinear weakly dispersive wave packets in one-modal approximation are described by the Gardner equation (Korteweg-de Vries equation with both, quadratic and cubic nonlinearity and necessity condition for modulation instability of such quasi-harmonic waves is the positive coefficient of cubic nonlinear term, which is realized for certain density stratification. Nevertheless the linear dispersive relation used within the Gardner equation is valid for very long waves and does not describe waves of moderate length. It is why some other nonlinear evolution equations are applied in the theory of long surface waves like the Benjamin-Bona-Mahony (BBM) and Whitham equations. We use the extended versions of these equations including cubic nonlinear term and express all  coefficients through modal functions and density stratification. Then, the modulational instability of weakly modulated wave packets is investigated after deriving the nonlinear Schrodinger equation. Improved dispersion relation influences on the increment and size of modulational instability. Obtained results are compared with those, which known within the Gardner model.</p>

Materials Issues and Characterization of Low-k Dielectric Materials

MRS Bulletin ◽  
1997 ◽  
Vol 22 (10) ◽  
pp. 49-54 ◽  
Author(s):  
E. Todd Ryan ◽  
Andrew J. McKerrow ◽  
Jihperng Leu ◽  
Paul S. Ho
Keyword(s):  
Process Integration ◽  
Dielectric Materials ◽  
Propagation Delay ◽  
Thermomechanical Properties ◽  
General Criterion ◽  
Crosstalk Noise ◽  
Total Delay ◽  
Interlayer Dielectrics ◽  
Performance Improvements ◽  
And Performance

Continuing improvement in device density and performance has significantly affected the dimensions and complexity of the wiring structure for on-chip interconnects. These enhancements have led to a reduction in the wiring pitch and an increase in the number of wiring levels to fulfill demands for density and performance improvements. As device dimensions shrink to less than 0.25 μm, the propagation delay, crosstalk noise, and power dissipation due to resistance-capacitance (RC) coupling become significant. Accordingly the interconnect delay now constitutes a major fraction of the total delay limiting the overall chip performance. Equally important is the processing complexity due to an increase in the number of wiring levels. This inevitably drives cost up by lowering the manufacturing yield due to an increase in defects and processing complexity.To address these problems, new materials for use as metal lines and interlayer dielectrics (ILDs) and alternative architectures have surfaced to replace the current Al(Cu)/SiO2 interconnect technology. These alternative architectures will require the introduction of low-dielectric-constant k materials as the interlayer dielectrics and/or low-resistivity conductors such as copper. The electrical and thermomechanical properties of SiO2 are ideal for ILD applications, and a change to material with different properties has important process-integration implications. To facilitate the choice of an alternative ILD, it is necessary to establish general criterion for evaluating thin-film properties of candidate low-k materials, which can be later correlated with process-integration problems.

Private and Public Law

2020 ◽  
pp. 574-592
Author(s):  
Thomas W. Merrill
Keyword(s):  
Law And Economics ◽  
Public Law ◽  
Private Law ◽  
General Criterion ◽  
The Public ◽  
Private Ordering ◽  
Public Rights ◽  
Private And Public ◽  
The Relationship ◽  
Derivatives Of

This chapter explores the relationship between private and public law. In civil law countries, the public-private distinction serves as an organizing principle of the entire legal system. In common law jurisdictions, the distinction is at best an implicit design principle and is used primarily as an informal device for categorizing different fields of law. Even if not explicitly recognized as an organizing principle, however, it is plausible that private and public law perform distinct functions. Private law supplies the tools that make private ordering possible—the discretionary decisions that individuals make in structuring their lives. Public law is concerned with providing public goods—broadly defined—that cannot be adequately supplied by private ordering. In the twentieth and twenty-first centuries, various schools of thought derived from utilitarianism have assimilated both private and public rights to the same general criterion of aggregate welfare analysis. This has left judges with no clear conception of the distinction between private and public law. Another problematic feature of modern legal thought is a curious inversion in which scholars who focus on fields of private law have turned increasingly to law and economics, one of the derivatives of utilitarianism, whereas scholars who concern themselves with public law are increasingly drawn to new versions of natural rights thinking, in the form of universal human rights.

scholarly journals Modulational instability in a non-Kerr photonic Lieb lattice with metamaterials

2021 ◽  
Vol 103 (1) ◽  
Author(s):  
A. K. Shafeeque Ali ◽  
Andrei I. Maimistov ◽  
K. Porsezian ◽  
A. Govindarajan ◽  
M. Lakshmanan
Keyword(s):  
Modulational Instability ◽  
Lieb Lattice

scholarly journals Probing Band Topology Using Modulational Instability

2021 ◽  
Vol 126 (7) ◽  
Author(s):  
Daniel Leykam ◽  
Ekaterina Smolina ◽  
Aleksandra Maluckov ◽  
Sergej Flach ◽  
Daria A. Smirnova
Keyword(s):  
Modulational Instability
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