Symmetry Reductions and Conservation Laws of the Short Pulse Equation

short pulse equationLie Symmetry AnalysisConservation LawsIn this letter we study invariance properties of the nonlinear short pulse equation through Lie symmetry analysis. We show that solutions of the short pulse equation. Furthermore, we obtain two conservation of the equation through the direct method. We show that two resulting nonlocally related systems yield no nonlocal symmetries of the short pulse equation. Some remarks and appropriate conclusions are drawn at the end.Fakhar, K, Wang, G. & Kara, A.H. (2016). Symmetry reductions and conservation laws of the short pulse equation. Optik - International Journal for Light and Electron Optics, 127(21), 10201–10207.doi.org/10.1016/j.ijleo.2016.08.013 Peer reviewedMust link to publisher version with DOI Author's post-print must be released with a Creative Commons Attribution Non-Commercial No Derivatives LicensePeer reviewedThis article is embargoed until December 1, 2018

Existence, regularity, and symmetry of periodic traveling waves for Gardner-Ostrovsky type equations

We study the existence, regularity, and symmetry of periodic traveling solutions to a class of Gardner-Ostrovsky type equations, including the classical Gardner-Ostrovsky equation, the (modified) Ostrovsky, and the reduced (modified) Ostrovsky equation. The modified Ostrovsky equation is also known as the short pulse equation. The Gardner-Ostrovsky equation is a model for internal ocean waves of large amplitude. We prove the existence of nontrivial, periodic traveling wave solutions using local bifurcation theory, where the wave speed serves as the bifurcation parameter. Moreover, we give a regularity analysis for periodic traveling solutions in the presence as well as absence of Boussinesq dispersion. We see that the presence of Boussinesq dispersion implies smoothness of periodic traveling wave solutions, while its absence may lead to singularities in the form of peaks or cusps. Eventually, we study the symmetry of periodic traveling solutions by the method of moving planes. A novel feature of the symmetry results in the absence of Boussinesq dispersion is that we do not need to impose a traditional monotonicity condition or a recently developed reflection criterion on the wave profiles to prove the statement on the symmetry of periodic traveling waves

Phase Transition in a One-Dimensional Extended Peierls-Hubbard Model with a Pulse of Oscillating Electric Field: I. Threshold Behavior in Ionic-to-Neutral Transition

Photoinduced dynamics of charge density and lattice displacements is calculated by solving the time-dependent Schr\"odinger equation for a one-dimensional extended Peierls-Hubbard model with alternating potentials for the mixed-stack organic charge-transfer complex, TTF-CA. A pulse of oscillating electric field is incorporated into the Peierls phase of the transfer integral. The frequency, the amplitude, and the duration of the pulse are varied to study the nonlinear and cooperative character of the photoinduced transition. When the dimerized ionic phase is photoexcited, the threshold behavior is clearly observed by plotting the final ionicity as a function of the increment of the total energy. Above the threshold photoexcitation, the electronic state reaches the neutral one with equidistant molecules after the electric field is turned off. The transition is initiated by nucleation of a metastable neutral domain, for which an electric field with frequency below the linear absorption peak is more effective than that at the peak. When the pulse is strong and short, the charge transfer takes place on the same time scale with the disappearance of dimerization. As the pulse becomes weak and long, the dimerization-induced polarization is disordered to restore the inversion symmetry on average before the charge transfer takes place to bring the system neutral. Thus, a paraelectric ionic phase is transiently realized by a weak electric field. It is shown that infrared light also induces the ionic-to-neutral transition, which is characterized by the threshold behavior.Comment: 24 pages, 11 figure

Attosecond Coherent Control of Symmetry Breaking and Restoration in Atoms and Molecules

Symmetry is a fundamental phenomenon in science, and symmetry breaking is often the origin of subsequent processes which are important in chemistry, physics and biology. As is well-known, a laser pulse can break the electronic symmetry in atoms and molecules by creating a superposition of electronic eigenstates with different irreducible representations, which typically initiates attosecond ultrafast charge migration. In the first part of this dissertation, an original theory of coherent laser control is proposed to induce the symmetry restoration of the electronic structure in atoms and molecules after symmetry breaking, with application to the oriented benzene molecule and to the ^{87}Rb atom. Four different strategies are proposed and corresponding sufficient conditions for symmetry restoration are derived analytically. The numerical and analytical results agree perfectly with each other. Meanwhile, the theoretical predictions for the ^{87}Rb atom have been confirmed by experimental partners in Japan, by means of high contrast Ramsey interferometry with a precision of about three attoseconds. The second part is devoted to the electronic flux during charge migration in oriented benzene molecule. Two different patterns of adiabatic attosecond charge migration are investigated by laser induced preparation of two different non-aromatic superposition states. From the knowledge of the time-dependent many-body wave functions as a linear combination of many-electron wave functions obtained from conventional quantum chemistry calculations, we derive expressions for the time-evolution of the one-electron density and the electronic flux. This allows to specify the number of electrons flowing during a given charge migration process, together with the mechanism of charge migration. In conclusion, this dissertation shows, for the first time, that the symmetry of electronic structure in atoms and molecules can not only be broken but also be restored by means of simple laser pulses. The coherent control strategies require strict control over the time-dependent phases of electronic wave functions. In practice, the precision required is few attoseconds - much shorter than the timescale of charge migration in such systems. The analysis of charge migration indicates that similar superposition states may lead to quantitative differences in the number of electrons flowing.Symmetrie ist ein fundamentales Phänomen in der Naturwissenschaft, und Symmetriebrechung kann bedeutende Folgeprozesse in der Chemie, Physik und Biologie auslösen. Insbesondere kann die elektronische Symmetrie von Atomen und Molekülen bekanntlich durch einen Laserpuls gebrochen werden, und zwar durch die Erzeugung einer Superposition von elektronischen Zuständen mit verschiedenen irreduziblen Darstellungen - dies bewirkt dann typischerweise ultraschnelle Ladungsmigration auf der Attosekundenzeit-skala. Der erste Teil dieser Dissertation entwickelt eine grundlegende Theorie der kohärenten Laserpuls-Kontrolle mit dem Ziel der Wiederherstellung der elektronischen Symmetrie in Atomen und Molekülen nach Symmetriebrechung, mit Anwendungen auf das orientierte Benzolmolekül sowie auf das ^{87}Rb Atom. Es werden insgesamt vier Strategien zur Wiederherstellung der Symmetrie vorgestellt, wobei hinreichend zielführende Bedingungen analytisch hergeleitet werden. Die analytischen Ergebnisse stimmen exzellent mit numerischen Quantendynamiksimulationen überein. Die theoretischen Vorhersagen für das ^{87}Rb Atom wurden inzwischen mit Hilfe der hoch-kontrastreichen Ramsey Interferometrie von experimentellen Partnern in Japan mit einer Genauigkeit von drei Attosekunden bestätigt. Der zweite Teil untersucht den Elektronenfluss während der Ladungsmigration in orientierten Benzolmolekülen. Dabei werden für zwei unterschiedliche nicht-aromatische elektronische Superpositionszustände verschiedene Typen der adiabatischen Ladungsmigration auf der Attosekundenzeitskala aufgezeigt. Aus der Kenntnis der zeitabhängigen Viel-Elektronen-Wellenfunktion als Linearkombination elektronischer Eigenfunktionen, die mit Hilfe konventioneller Verfahren der Quantenchemie berechnet werden, werden Ausdrücke für die Zeit-Evolution der Ein-Elektronen-Dichte und des Elektronenflusses hergeleitet. Daraus ergibt sich die jeweilige Zahl der Elektronen, die zur Ladungsmigration beitragen, sowie der Mechanismus der Ladungsmigration. Zusammenfassend zeigt diese Dissertation erstmals, dass die Symmetrie der Elektronenstruktur von Atomen und Molekülen mit Hilfe von einfachen Laserpulsen nicht nur gebrochen, sondern auch wiederhergestellt werden kann. Die Strategien der kohärenten Laserkontrolle verlangen dazu die strenge Kontrolle der zeitabhängigen Phasen der elektronischen Wellenfunktionen. Praktische Anwendungen erfordern dafür eine zeitliche Genauigkeit von wenigen Attosekunden - also noch viel genauer als die ohnehin schon ultrakurze Zeitskala der Ladungsmigration. Untersuchungen der Ladungsmigration zeigen, dass vergleichbar ähnliche Superpositionen elektronischer Wellenfunktionen zu quantitativ verschiedenen Elektronenflusszahlen führen können