Colloquium: Nonlinear collective interactions in quantum plasmas with degenerate electron fluids

The current understanding of some important nonlinear collective processes in quantum plasmas with degenerate electrons is presented. After reviewing the basic properties of quantum plasmas, we present model equations (e.g. the quantum hydrodynamic and effective nonlinear Schr\"odinger-Poisson equations) that describe collective nonlinear phenomena at nanoscales. The effects of the electron degeneracy arise due to Heisenberg's uncertainty principle and Pauli's exclusion principle for overlapping electron wavefunctions that result in tunneling of electrons and the electron degeneracy pressure. Since electrons are Fermions (spin-1/2), there also appears an electron spin current and a spin force acting on electrons due to the Bohr magnetization. The quantum effects produce new aspects of electrostatic (ES) and electromagnetic (EM) waves in a quantum plasma that are summarized in here. Furthermore, we discuss nonlinear features of ES ion waves and electron plasma oscillations (ESOs), as well as the trapping of intense EM waves in quantum electron density cavities. Specifically, simulation studies of the coupled nonlinear Schr\"odinger (NLS) and Poisson equations reveal the formation and dynamics of localized ES structures at nanoscales in a quantum plasma. We also discuss the effect of an external magnetic field on the plasma wave spectra and develop quantum magnetohydrodynamic (Q-MHD) equations. The results are useful for understanding numerous collective phenomena in quantum plasmas, such as those in compact astrophysical objects, in plasma-assisted nanotechnology, and in the next-generation of intense laser-solid density plasma interaction experiments.Comment: 25 pages, 14 figures. To be published in Reviews of Modern Physic

Dynamics of Patterns

This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects

The Krein Matrix: General Theory and Concrete Applications in Atomic Bose-Einstein Condensates

When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues (i.e., those with positive real part), but also those which are purely imaginary but have negative Krein signature. These latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues, i.e., they are a necessary building block in the mechanism leading to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a general theory for constructing a meromorphic matrix-valued function, the so-called Krein matrix, which has the property of not only locating the unstable eigenvalues, but also those with negative Krein signature. These eigenvalues are realized as zeros of the determinant. The resulting finite dimensional problem obtained by setting the determinant of the Krein matrix to zero presents a valuable simplification. In this paper the usefulness of the technique is illustrated through prototypical examples of spectral analysis of states that have arisen in recent experimental and theoretical studies of atomic Bose-Einstein condensates. In particular, we consider one-dimensional settings (the cigar trap) possessing real-valued multi-dark-soliton solutions, and two-dimensional settings (the pancake trap) admitting complex multi-vortex stationary waveforms.Comment: 26 pages, 16 figures (revised version on April 18 2013

Localized wave structures: Solitons and beyond

The review is concerned with solitary waves and other localized structures in the systems described by a variety of generalizations of the Korteweg–de Vries (KdV) equation. Among the topics we focus upon are “radiating solitons,” the generic structures made of soliton-like pulses, and oscillating tails. We also review the properties of solitary waves in the generalized KdV equations with the modular and “sublinear” nonlinearities. Such equations have an interesting class of solutions, called compactons, solitary waves defined on a finite spatial interval. Both the properties of single solitons and the interactions between them are discussed. We show that even minor non-elastic effects in the soliton–soliton collisions can accumulate and result in a qualitatively different asymptotic behavior. A statistical description of soliton ensembles (“soliton gas”), which emerges as a major theme, has been discussed for several models. We briefly outline the recent progress in studies of ring solitons and lumps within the framework of the cylindrical KdV equation and its two-dimensional extension. Ring solitons and lumps (2D solitons) are of particular interest since they have many features in common with classical solitons and yet are qualitatively different. Particular attention is paid to interactions between the objects of different geometries, such as the interaction of ring solitons and shear flows, ring solitons and lumps, and lumps and line solitons. We conclude our review with views of the future developments of the selected lines of studies of localized wave structures in the theory of weakly nonlinear, weakly dispersive waves

Structure and dynamics of solitary waves in fluid media

This research deals with the study of nonlinear solitary waves in fluid media. The equations which model surface and internal waves in fluids have been studied and used in this research. The approach to study the structure and dynamics of internal solitary waves in near-critical situations is the traditional theoretical and numerical study of nonlinear wave processes based on the methods of dynamical systems. The synergetic approach has been exploited, which presumes a combination of theoretical and numerical methods. All numerical calculations were performed with the desktop personal computer. Traditional and novel methods of mathematical physics were actively used, including Fourier analysis technique, inverse scattering method, Hirota method, phase-plane analysis, analysis of integral invariants, finite-difference method, Petviashvili and Yang–Lakoba numerical iterative techniques for the numerical solution of Partial Differential Equation. A new model equation, dubbed the Gardner–Kawahara equation, has been suggested to describe wave phenomena in the near-critical situations, when the nonlinear and dispersive coefficients become anomalously small. Such near-critical situations were not studied so far, therefore this study is very topical and innovative. Results obtained will shed a light on the structure of solitary waves in near-critical situation, which can occur in two-layer fluid with strong surface tension between the layers. A family of solitary waves was constructed numerically for the derived Gardner–Kawahara equation; their structure has been investigated analytically and numerically. The problem of modulation stability of quasi-monochromatic wave-trains propagating in a media has also being studied. The Nonlinear Schrödinger equation (NLSE) has been derived from the unidirectional Gardner–Ostrovsky equation and a general Shrira equation which describes both surface and internal long waves in a rotating fluid. It was demonstrated that earlier obtained results (Grimshaw & Helfrich, 2008; 2012; Whitfield & Johnson, 2015a; 2015b) on modulational stability/instability are correct within the limited range of wavenumbers where the Ostrovsky equation is applicable. In the meantime, results obtained in this Thesis and published in the paper (Nikitenkova et al., 2015) are applicable in the wider range of wavenumbers up to k = 0. It was shown that surface and internal oceanic waves are stable with respect to selfmodulation at small wavenumbers when k → 0 in contrast to what was mistakenly obtained in (Shrira, 1981). In Chapter 4 new exact solutions of the Kadomtsev-Petviashvili equation with a positive dispersion are obtained in the form of obliquely propagating skew lumps. Specific features of such lumps were studied in details. In particular, the integral characteristics of single lumps (mass, momentum components and energy) have been calculated and presented in terms of lump velocity. It was shown that exact stationary multi-lump solutions can be constructed for this equation. As the example, the exact bilump solution is presented in the explicit form and illustrated graphically. The relevance of skew lumps to the real physical systems is discussed

Numerical Analysis and Theory of Oblique Alfvenic Solitons Observed in the Interplanetary Magnetic Field

Recently, there have been reports of small magnetic pulses or bumps in the interplanetary magnetic field observed by various spacecraft. Most of these reports claim that these localized pulses or bumps are solitons. Solitons are weakly nonlinear localized waves that tend to retain their form as they propagate and can be observed in various media which exhibit nonlinear steepening and dispersive effects. This thesis expands the claim that these pulses or bumps are nonlinear oblique Alfven waves with soliton components, through the application of analytical techniques used in the inverse scattering transform in a numerical context and numerical integration of nonlinear partial dierential equations. One event, which was observed by the Ulysses spacecraft on February 21st, 2001, is extensively scrutinized through comparison with soliton solutions that emerge from the Derivative Nonlinear Schrodinger (DNLS) equation. The direct scattering transform of a wave prole that has corresponding morphology to the selected magnetic bump leads to the implication of a soliton component. Numerical integration of the scaled prole matching the event in the context of the DNLS leads to generation of dispersive waves and a one parameter dark soliton

Die direkte und inverse nichtlineare Fourier-Transformation auf Grundlage der Korteweg-deVries-Gleichung (KdV-NLFT) - Eine Spektralanalyse nichtlinearer Wellen im Küstenbereich -

In the conventional analysis methods such as fast Fourier (FFT), wavelet (WT) and Hilbert-Huang transform (FFT) the spectral decomposition of the original data does not consider pos-sible effects of the relative water depth on the shape, stability or nonlinearity of the deter-mined spectral components. In the KdV-NFLT cnoidal waves are used as basis for the spec-tral decomposition of the original data. This allows the consideration of the water depth as a governing parameter for the analysis and the original surface data are decomposed adaptively into those nonlinear physical oscillatory waves and solitary waves that really occur in shallow water. Furthermore, the real nonlinear wave-wave interactions between the nonlinear cnoidal waves are considered and calculated explicitly in the KdV-NLFT. The main topics of the thesis are: (i) The numerical implementation and verification of the implemented inverse and direct scattering transform (IST and DST) of the Korteweg-deVries equation as a generalized Fourier transform (KdV-NLFT). (ii) The practical application of the implemented KdV-NLFT to selected shallow-water problems such as the reliable identifica-tion of solitons from signals measured over and behind submerged reefs (soliton fission) and the analysis of the nonlinear propagation of long-period waves in shallow water. (iii) A com-parative analysis using KdV-NLFT and conventional analysis methods such as the linear fast Fourier transform (FFT) in the frequency domain and the Hilbert-Huang transform (HHT) in the time-frequency domain. (iv) Then, based on the results of these comparative analyses rec-ommendations will be given for the practical application of the nonlinear KdV-NLFT and the conventional methods FFT and HHT for the spectral analysis of nonlinear shallow-water time and space series. Finally, the results of the thesis clearly show that the KdV-NLFT provides a decisive insight into the underlying nonlinear processes of the analysed shallow-water wave problems that cannot be obtained by application of the conventional analysis methods.In den herkömmlichen Analysemethoden schnelle Fourier- (FFT), Wavelet- (WT) und Hilbert-Huang-Transformation (HHT) erfolgt die spektrale Zerlegung von Originalsignalen ohne eine Berücksichtigung des möglichen Einflusses der relativen Wassertiefe auf die Form, die Stabilität oder die Nichtlinearität der gewählten spektralen Komponenten. In der KdV-NLFT werden cnoidalen Wellen als Basis für die spektrale Zerlegung der gegebenen freien Wasser-oberfläche verwendet. Hierdurch geht die Wassertiefe als maßgebender Parameter in die Ana-lyse ein, wodurch die Zerlegung adaptiv in die im Küstenbereich tatsächlich auftretenden physikalischen nichtlinearen oszillierenden und solitären Flachwasserwellen erfolgt. Zusätzlich werden die zwischen den cnoidalen Wellen auftretenden nichtlinearen Wellen-Wellen-Interaktionen in der KdV-NLFT explizit berücksichtigt und berechnet. Die Schwerpunkte der Arbeit sind: (i) Die numerische Implementierung und Verifizierung der implementierten Inversen und Direkten Streuungs-Transformation (IST und DST) für die Korteweg-deVries-Gleichung als eine generalisierte nichtlineare Fourier-Transformation (KdV-NLFT). (ii) Die beispielhafte praktische Anwendung der implementierten KdV-NLFT auf ausgewählte Problemstellungen aus dem Küsteningenieurwesen wie z.B. die zuverlässige Identifizierung von Solitonen in Signalen über und hinter getauchten Riffen (soliton fission) und die Analyse der nichtlinearen Ausbreitung langer Wellen im Flachwasser. (iii) Eine ver-gleichende Analyse zwischen der KdV-NLFT sowie den herkömmlichen Frequenz- und Zeit-Frequenz-Analyseverfahren FFT und HHT. (iv) Die Erarbeitung von Empfehlungen für die praktische Anwendung der KdV-NLFT sowie der herkömmlichen Verfahren FFT und HHT für die spektrale Analyse von nichtlinearen Daten im Flachwasser auf Grundlage der Ergeb-nisse der Vergleichsanalysen. Die in dieser Arbeit erzielten Ergebnisse zeigen eindeutig, dass die KdV-NLFT für die unter-suchten Fragestellungen weitreichende Einblicke in die zugrundeliegenden nichtlinearen Pro-zesse und Eigenschaften der Wellen im Flachwasser liefert, die mit den herkömmlichen Me-thoden nicht erzielt werden können