Extreme events in discrete nonlinear lattices

We perform statistical analysis on discrete nonlinear waves generated though modulational instability in the context of the Salerno model that interpolates between the intergable Ablowitz-Ladik (AL) equation and the nonintegrable discrete nonlinear Schrodinger (DNLS) equation. We focus on extreme events in the form of discrete rogue or freak waves that may arise as a result of rapid coalescence of discrete breathers or other nonlinear interaction processes. We find power law dependence in the wave amplitude distribution accompanied by an enhanced probability for freak events close to the integrable limit of the equation. A characteristic peak in the extreme event probability appears that is attributed to the onset of interaction of the discrete solitons of the AL equation and the accompanied transition from the local to the global stochasticity monitored through the positive Lyapunov exponent of a nonlinear map.Comment: 5 pages, 4 figures; reference added, figure 2 correcte

Experiences of the Flipped Classroom method Does it make students more motivated?

The aim of this paper is to highlight use of the flipped classroom method, and how teachers perceive this teaching practice. More specific the research focus on whether the teachers’ experience that the model leads to increased motivation in the students learning process. The background for the research is generated from qualitative interviews with teachers, and the empirical data obtained is from semi-structured interviews with these informants. The results show that the flipped classroom method in fact did increase participation and cooperation, which in turn generated motivation and willing students. The teachers got more time for guidance of each student, which provided more solid knowledge on each student’s academic level

Solitary wave interaction in a compact equation for deep-water gravity waves

In this study we compute numerical traveling wave solutions to a compact version of the Zakharov equation for unidirectional deep-water waves recently derived by Dyachenko & Zakharov (2011) Furthermore, by means of an accurate Fourier-type spectral scheme we find that solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.Comment: 8 pages, 5 figures, 23 references. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/ . arXiv admin note: text overlap with arXiv:1204.288

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose’s method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin–Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength λ0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to δ spectra, where the standard Benjamin–Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes. Keywords: Rogue Waves; Wigner equation; Nonlinear Schrodinger equation; Penrose modes; Penrose conditio

Numerical instability of the Akhmediev breather and a finite-gap model of it

In this paper we study the numerical instabilities of the NLS Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be written in terms of different elementary functions in different time regions and, ultimately, it shows an exact recurrence of rogue waves described, at each appearance, by the Akhmediev breather. We discover a remarkable empirical formula connecting the recurrence time with the number of time steps used in the SSFM and, via our recent theoretical findings, we establish that the SSFM opens up a vertical unstable gap whose length can be computed with high accuracy, and is proportional to the inverse of the square of the number of time steps used in the SSFM. This neat picture essentially changes when the round-off error is sufficiently large. Indeed experiments in standard double precision show serious instabilities in both the periods and phases of the recurrence. In contrast with it, as predicted by the theory, replacing the exact Akhmediev Cauchy datum by its first harmonic approximation, we only slightly modify the numerical output. Let us also remark, that the first rogue wave appearance is completely stable in all experiments and is in perfect agreement with the Akhmediev formula and with the theoretical prediction in terms of the Cauchy data.Comment: 27 pages, 8 figures, Formula (30) at page 11 was corrected, arXiv admin note: text overlap with arXiv:1707.0565

Hamiltonian form and solitary waves of the spatial Dysthe equations

The spatial Dysthe equations describe the envelope evolution of the free-surface and potential of gravity waves in deep waters. Their Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. An accurate Fourier-type spectral scheme is used to solve for the wave dynamics and validate the new conservation laws, which are satisfied up to machine precision. Traveling waves are numerically constructed using the Petviashvili method. It is shown that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.Comment: 6 pages, 9 figures. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

From bore-soliton-splash to a new wave-to-wire wave-energy model

We explore extreme nonlinear water-wave amplification in a contraction or, analogously, wave amplification in crossing seas. The latter case can lead to extreme or rogue-wave formation at sea. First, amplification of a solitary-water-wave compound running into a contraction is disseminated experimentally in a wave tank. Maximum amplification in our bore–soliton–splash observed is circa tenfold. Subsequently, we summarise some nonlinear and numerical modelling approaches, validated for amplifying, contracting waves. These amplification phenomena observed have led us to develop a novel wave-energy device with wave amplification in a contraction used to enhance wave-activated buoy motion and magnetically induced energy generation. An experimental proof-of-principle shows that our wave-energy device works. Most importantly, we develop a novel wave-to-wire mathematical model of the combined wave hydrodynamics, wave-activated buoy motion and electric power generation by magnetic induction, from first principles, satisfying one grand variational principle in its conservative limit. Wave and buoy dynamics are coupled via a Lagrange multiplier, which boundary value at the waterline is in a subtle way solved explicitly by imposing incompressibility in a weak sense. Dissipative features, such as electrical wire resistance and nonlinear LED loads, are added a posteriori. New is also the intricate and compatible finite-element space–time discretisation of the linearised dynamics, guaranteeing numerical stability and the correct energy transfer between the three subsystems. Preliminary simulations of our simplified and linearised wave-energy model are encouraging and involve a first study of the resonant behaviour and parameter dependence of the device

Hidden expectations: Scaffolding subject specialists' genre knowledge of the assignments they set

Subject specialists’ knowledge of academic and disciplinary literacy is often tacit. We tackle the issue of how to elicit subject specialists’ tacit knowledge in order to develop their pedagogical practices and enable them to communicate this knowledge to students. Drawing on theories of genre and metacognition, a professional development activity was designed and delivered. Our aims were to (1) build participants’ genre knowledge and (2) scaffold metacognitive awareness of how genre knowledge can enhance their pedagogical practices. The findings reveal that participants built a genre-based understanding of academic literacy and that the tasks provided them with an accessible framework to articulate and reflect upon their knowledge of disciplinary literacy. Participants gained metacognitive awareness of misalignments between what they teach and what they expect from students, their assumptions about students’ prior learning and genre-based strategies to adapt their practice to students’ needs. Our approach provides a theoretically grounded professional development tool for the HE sector

Nonlinear wave interaction in coastal and open seas -- deterministic and stochastic theory

We review the theory of wave interaction in finite and infinite depth. Both of these strands of water-wave research begin with the deterministic governing equations for water waves, from which simplified equations can be derived to model situations of interest, such as the mild slope and modified mild slope equations, the Zakharov equation, or the nonlinear Schr\"odinger equation. These deterministic equations yield accompanying stochastic equations for averaged quantities of the sea-state, like the spectrum or bispectrum. We discuss several of these in depth, touching on recent results about the stability of open ocean spectra to inhomogeneous disturbances, as well as new stochastic equations for the nearshore